credit dynamics in a ﬁrst-passage time model with jumps

Credit dynamics in a

first-passage time model with jumps

and

Latin hypercube sampling

with dependence

Dissertation

zur Erlangung des akademischen Grades

Doktor der Wirtschaftswissenschaften

(Dr. rer. pol.)

der Frankfurt School of Finance & Management

vorgelegt von

Natalie Packham

Dezember 2008

Dekan:Prof. Dr. Thomas HeimerFrankfurt School of Finance and ManagementSonnemannstrasse 9 - 1160314 Frankfurt am Main

Gutachter:Prof. Dr. Wolfgang M. Schmidt (Frankfurt School of Finance & Management)Prof. Dr. Uwe Wystup (Frankfurt School of Finance & Management)Prof. Dr. Rudiger Kiesel (Universitat Ulm)

Tag der Disputation: 4.5.2009

i

Acknowledgments

I thank the many people who have helped and supported me while preparing thethesis. I would like to express my deep gratitude to my supervisor, Prof. Dr. Wolf-gang M. Schmidt. His guidance and support are invaluable. I am also grateful tomy co-supervisors Prof. Dr. Uwe Wystup and Prof. Dr. Rudiger Kiesel.

I would like to thank my colleagues and friends at Frankfurt School, in partic-ular Christoph Becker, Marie-Noelle Biemer, Dr. Susanne Griebsch, Prof. Dr. TomHeidorn, Matthias Hilgert, Dimitri Reiswich, Christian Schmaltz, Carlos Veiga andProf. Dr. Ursula Walther; the members of the Quant Centre at Frankfurt School;my friends Marcus Joachim and Dr. Steffi Kammer for interesting and inspiringdiscussions; Dr. Lutz Schlogl, whose idea it was to consider a model with jumps;Prof. Monique Jeanblanc and Giorgia Callegaro for helpful discussions; StefanieMuller and members of the Financial Mathematics group at Fraunhofer ITWM forvaluable feedback.

I am also grateful for travel support to: Frankfurt School of Finance & Man-agement; the sponsors of the European Summer School in Financial Mathematics(Paris); and to Prof. Dr. Rene Carmona and the Society of Industrial and AppliedMathematics (SIAM).

I thank my parents for proof-reading yet another thesis.I am most grateful to Dr. Michael Kalkbrener.

iii

iv Acknowledgments

Contents

Acknowledgments iii

List of Figures ix

List of Tables xi

List of Algorithms xiii

1. Introduction 1

2. A class of Ocone martingales 7

2.1 Time-change and DDS-Theorem . . . . . . . . . . . . . . . . . . . 72.2 Stochastic integrals with respect to Brownian motion as Ocone

martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Credit dynamics in a first-passage time model with jumps 13

3. Credit derivatives 15

3.1 Credit default swaps . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1.1 Mechanics and term structure . . . . . . . . . . . . . . . . 153.1.2 Shape and dynamics of the term structure . . . . . . . . . 17

3.2 Market model and risk-neutral valuation formula . . . . . . . . . . 183.3 Conditional default and survival probabilities . . . . . . . . . . . . 213.4 Risk-neutral valuation of CDS . . . . . . . . . . . . . . . . . . . . . 22

3.4.1 CDS valuation and fair CDS spread . . . . . . . . . . . . . 233.4.2 Mark-to-market of a CDS position . . . . . . . . . . . . . . 25

3.5 Credit derivatives with spread-sensitive payoff . . . . . . . . . . . . 253.5.1 Leveraged credit-linked note . . . . . . . . . . . . . . . . . 253.5.2 Gap option . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4. Credit spread models 29

v

vi Contents

4.1 Model requirements . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Hazard rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.3 Structural models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3.1 Merton model . . . . . . . . . . . . . . . . . . . . . . . . . 324.3.2 First-passage time models . . . . . . . . . . . . . . . . . . 33

4.4 Reduced-form models . . . . . . . . . . . . . . . . . . . . . . . . . 34

5. Overbeck-Schmidt model 35

5.1 Model specification . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2 Conditional default probabilities . . . . . . . . . . . . . . . . . . . 365.3 Short-term hazard rate and short-term credit spread . . . . . . . . 385.4 Dynamics of conditional default probabilities . . . . . . . . . . . . 395.5 Hazard rate dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 415.6 Distribution of conditional default probabilities . . . . . . . . . . . 425.7 Distribution of term hazard rate . . . . . . . . . . . . . . . . . . . 445.8 Extensions of the Overbeck-Schmidt model . . . . . . . . . . . . . 45

6. A first-passage time model with jumps 47

6.1 Credit quality process with stochastic volatility . . . . . . . . . . . 486.2 Conditional default probabilities . . . . . . . . . . . . . . . . . . . 486.3 Conditional default density . . . . . . . . . . . . . . . . . . . . . . 506.4 Variance as Levy-driven Ornstein-Uhlenbeck process . . . . . . . . 51

7. Dynamics 57

7.1 Jumps in default probabilities and credit spreads . . . . . . . . . . 577.2 Short-term hazard rate and short-term credit spread . . . . . . . . 607.3 Dynamics of default probabilities . . . . . . . . . . . . . . . . . . . 61

7.3.1 The distribution function of Lt,T . . . . . . . . . . . . . . . 667.4 Spread dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687.5 Distribution of conditional default probabilities . . . . . . . . . . . 717.6 Distribution of term hazard rate . . . . . . . . . . . . . . . . . . . 73

8. Implementation, calibration and examples 75

8.1 Computation of default probabilities and credit spreads . . . . . . 758.1.1 Jump size distribution of time-change Λ . . . . . . . . . . . 758.1.2 Panjer recursion . . . . . . . . . . . . . . . . . . . . . . . . 768.1.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

8.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808.2.1 Calibration to a term structure of default probabilities . . 818.2.2 Shape of credit-spread term structure . . . . . . . . . . . . 87

8.3 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Contents vii

8.3.2 Parameters and dynamics . . . . . . . . . . . . . . . . . . . 938.3.3 Evolution of the term structure shape . . . . . . . . . . . . 95

8.4 Uniqueness and stability of calibration parameters . . . . . . . . . 95

9. Valuation of credit derivatives 99

9.1 Information flow and the pricing filtration . . . . . . . . . . . . . . 999.2 Leveraged credit-linked note . . . . . . . . . . . . . . . . . . . . . . 1029.3 Default swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

10. Summary and conclusion 109

10.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10910.2 Model requirements revisited . . . . . . . . . . . . . . . . . . . . . 11110.3 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Latin hypercube sampling with dependence 115

11. Latin hypercube sampling with dependence 117

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11711.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

11.2.1 Stratified sampling . . . . . . . . . . . . . . . . . . . . . . 11811.2.2 Latin hypercube sampling . . . . . . . . . . . . . . . . . . 119

11.3 LHSD method and LHSD estimator . . . . . . . . . . . . . . . . . 12011.4 Consistency of the LHSD estimator . . . . . . . . . . . . . . . . . . 12311.5 Central Limit Theorem for LHSD and variance reduction . . . . . 12411.6 LHSD on random vectors with nonuniform marginals . . . . . . . . 133

12. Applications in finance 135

12.1 Valuing a first-to-default credit basket . . . . . . . . . . . . . . . . 13512.2 Valuing an Asian basket option . . . . . . . . . . . . . . . . . . . . 138

Appendix A Probability Theory and Stochastic Calculus 143

A.1 Regular conditional probabilities . . . . . . . . . . . . . . . . . . . 143A.2 Ito formula for semimartingales . . . . . . . . . . . . . . . . . . . . 144A.3 Classification of stopping times . . . . . . . . . . . . . . . . . . . . 144A.4 Martingale decomposition . . . . . . . . . . . . . . . . . . . . . . . 145A.5 Stochastic exponential . . . . . . . . . . . . . . . . . . . . . . . . . 145A.6 Other results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Appendix B Levy processes 147

B.1 Levy processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

viii Contents

B.2 Subordinators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148B.3 Compound Poisson processes . . . . . . . . . . . . . . . . . . . . . 149B.4 SDEs driven by Levy processes as Markov processes . . . . . . . . 150

Appendix C Integration by parts formula 151

Bibliography 153

List of symbols 159

Index 161

List of Figures

3.1 Term structure of CDS spreads . . . . . . . . . . . . . . . . . . . . . . . 163.2 BMW CDS spreads (daily) . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 BMW survival probabilities . . . . . . . . . . . . . . . . . . . . . . . . . 243.4 Leveraged credit-linked note (inception) . . . . . . . . . . . . . . . . . . 263.5 Leveraged credit-linked note (trigger time) . . . . . . . . . . . . . . . . . 27

5.1 Distributions of P(τ ∈ (t, T ]|Xt) and λ(t, T ) (OS-model) . . . . . . . . . 43

6.1 Example of variance process and credit quality process . . . . . . . . . . 55

7.1 Distributions of P(τ ∈ (t, T ]|Xt, σ2t ) and λ(t, T ) (LOU model) . . . . . . 73

8.1 Calibration example that leads to an inconsistent model. . . . . . . . . . 818.2 RMSE’s for different parameter sets . . . . . . . . . . . . . . . . . . . . 848.3 Several calibration results . . . . . . . . . . . . . . . . . . . . . . . . . . 858.4 Impact of individual parameters on default probability term structure . 868.5 RMSE’s for different parameter sets . . . . . . . . . . . . . . . . . . . . 878.6 Impact of individual parameters on credit spread term structure . . . . 898.7 Distributions of P(τ ∈ (t, T ]|Xt, σ

2t ) and λ(t, T ) (LOU model) II . . . . 90

8.8 First differences of term hazard rate distributions of Figure 8.7. . . . . . 928.9 Jump size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 948.10 Calibration parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

9.1 Prices and implied volatilities of swaptions . . . . . . . . . . . . . . . . 108

11.1 Example of Latin hypercube sampling . . . . . . . . . . . . . . . . . . . 12011.2 Example of LHSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

ix

x LIST OF FIGURES

List of Tables

8.1 Monte Carlo simulation vs. Panjer recursion . . . . . . . . . . . . . . . . 778.2 Parameters, RMSE’s and characteristics of dynamics examples . . . . . 918.3 RMSE’s and characteristics of bumped dynamics examples . . . . . . . 97

9.1 Valuation examples of leveraged credit-linked note . . . . . . . . . . . . 1039.2 Valuation examples of default swaptions . . . . . . . . . . . . . . . . . . 106

12.1 Parameters of FTD example . . . . . . . . . . . . . . . . . . . . . . . . . 13812.2 RMSE’s of FTD spread estimation . . . . . . . . . . . . . . . . . . . . . 13912.3 Parameters of Asian basket option . . . . . . . . . . . . . . . . . . . . . 14012.4 Simulated prices of Asian basket option . . . . . . . . . . . . . . . . . . 141

xi

xii LIST OF TABLES

List of Algorithms

1 Computation of conditional default probabilities . . . . . . . . . . . . . 792 FTD valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

xiii

xiv LIST OF ALGORITHMS

Chapter 1

Introduction

You are offered the following investment: deposit e 100 for five years and receivea very attractive coupon of 200 basis points above the LIBOR rate. However, theinvestment is not risk free; it is linked to the value in a position of a 5-year creditdefault swap (CDS) on company XYZ with notional e 500. The CDS position isa short position: it earns a regular premium, called the CDS spread or just thespread, in return for a lump payment should company XYZ default during thelifetime of the CDS. The amount paid at default is a fraction of the notional ofe 500 that compensates the counterparty for the loss incurred. Most likely, thedeposit of e 100 will not suffice to cover the payment at default. A trigger is agreedto terminate the investment while the cost of closing the CDS position is still likelyto be covered by the deposit amount. If the value of the CDS position hits orfalls below e−100 during the lifetime of the investment you stop receiving couponpayments and you are redeemed nothing at maturity. The investment offered iscalled a leveraged credit-linked note (LCLN)1.

The deposit of e 100 earns the LIBOR rate if placed in a money market account.You look up the fair 5-year CDS spread of company XYZ. It is currently at 60 basispoints, hence, the CDS position linked to your investment earns 300 basis pointsrelative to the deposit amount.

Why is the spread over LIBOR on the coupon smaller than the CDS spreadearned? Clearly, the issuer also faces some risk: if the value of the CDS positiondoes not hit e−100 exactly, but overshoots this value, then the issuer must cover themissing amount to unwind the position. This type of risk is called gap risk . Denoteby Vt the value of the CDS position at time t. The investment can be decomposed –from the investor’s point of view – into the deposit of e 100, the CDS short positionwith nominal e 500 and an option that pays max(−Vt − e 100, 0) at the first timebefore option expiry that Vt hits or falls below e−100. This type of option is called1The term leverage refers to the fact that a higher exposure in the underlying CDS is taken

than can be covered by the investment amount. Leveraging a CDS position is straightforward, ascredit default swaps are in general unfunded, i.e., there is no exchange of notional when enteringinto CDS position. Other products with a leverage component are constant proportion portfolioinsurance (CPPI) strategies and constant proportion debt obligations (CPDO). They all have incommon that gap risk is involved.

1

2 Introduction

a gap option. The premium for the gap option is embedded in the spread differenceof the CDS position and the investment coupon.

You may also want to know the probability of receiving the investment amountback. Moreover, the amount you earn on the coupons depends on the time whenthe investment is unwound. It is impossible to answer this question precisely. Youmay look into the history of company XYZ’s CDS spread to get an idea of how yourinvestment would have performed in the past. Apart from the question of whetherthe coupon you earn is “fair”, the investment decision finally depends on your viewand risk appetite.

The issuer on the other hand must be able to assess the value of the gap option.The foundation for modern option pricing was laid in the famous paper of [Black andScholes (1973)], where the price of a call or put option on a financial asset is derivedas the price of a self-financing trading strategy that replicates the option payoff bycontinuous re-balancing of a position in the underlying asset and a risk-free asset.Such a trading strategy is called a hedging strategy and the resulting price is calledthe no-arbitrage price. For many pricing problems, the asset price model adoptedby [Black and Scholes (1973)] – the asset price is assumed to follow a geometricBrownian motion – and its implications for asset returns are inappropriate. Thishas since lead to consider more complex models for asset pricing, involving e.g.stochastic volatility or jumps. Additionally, the market assumptions posed by [Blackand Scholes (1973)] – market completeness, endless liquidity, etc. – are often notfulfilled. In a complete market every financial claim can replicated by a hedgingstrategy in the market’s assets, and the no-arbitrage price of the claim is just theprice of this strategy. In an incomplete market, the issuer of a derivative mustdetermine a price in some other way, at the least consistent with the no-arbitrageprinciple.

Turning back to the example, the payoff of the gap option depends on theevolution of CDS spreads through time, and, provided there is no static hedgingstrategy, pricing of the option requires a model for the dynamics of CDS spreads. Infact, the value of the gap risk component is very sensitive to jumps in CDS spreads.For example, if it is assumed that CDS spreads evolve continuously, then it is easyto show that gap risk reduces to the risk of a sudden default of the underlyingcompany. But this contradicts empirical observation: CDS spreads exhibit jumps,often attributed to the arrival of bad news on the underlying company. It is thusimportant that a model for valuing gap options features jumps in the evolution ofCDS spreads.

The development of a model for CDS spreads with jumps is the topic of Part1. In general, a first-passage time model of default defines the default event asthe first time that a credit quality process hits a barrier, sometimes called defaultthreshold. In the model by [Overbeck and Schmidt (2005)], the barrier is constant,while the credit quality process is modelled as a time-changed Brownian motion,i.e., a Brownian motion that runs under a different clock. The time-change is deter-

3

ministic, continuous and strictly increasing. It is obtained by analytic calibrationto a set of given CDS spreads (or, equivalently, default probabilities). Essentially,we extend the Overbeck-Schmidt model to allow for jumps in default probabilitiesand credit spreads. We consider the credit quality process X to be a time-changedBrownian motion with a stochastic, continuous and strictly increasing time-change.Assuming that the time-changed Brownian motion has a representation as an Itointegral, X =

∫ ·0σu dWu, with W a Brownian motion, we model the volatility σ

as a cadlag process. Our standing example throughout will be a volatility processthat is the square root of an Ornstein-Uhlenbeck process driven by a compoundPoisson process. The representation of the credit quality process as a time-changedBrownian motion with a continuous time-change allows for deriving closed formulasfor conditional default probabilities (conditional on the flow of information) andCDS spreads, making use of well-known properties of Brownian motion. At thesame time, jumps in the volatility of the Ito integral translate into jumps in theevolution of CDS spreads. We derive the dynamics of conditional default probabili-ties and CDS spreads. In addition, we derive the distribution of conditional defaultprobabilities and a quantity known as the (term) hazard rate or (term) forwarddefault rate, many properties of which are easily derived from conditional defaultprobabilities. The hazard rate may be considered as a proxy of CDS spreads.

The model can be implemented efficiently by a combination of Monte Carlosimulation and numerical techniques. The state of the credit quality process is ob-tained by Monte Carlo simulation. Employing a technique called Panjer recursion,term structures of default probabilities (resp. credit spreads) conditional on thestate of the credit quality process and its volatility, are computed numerically. Anadvantage over many other models is that simulation is required only up to thosetime points where a term structure is needed, not for the actual maturities of CDSspreads and default probabilities themselves.

The model can be calibrated to a wide variety of market-given term structures,while at the same time providing different dynamics. However, contrary to theOverbeck-Schmidt model, analytic calibration is not possible. We calibrate by min-imising the error in a least-squares sense between market-given and model-deriveddefault probabilities. In fact, the calibration problem is ill-posed and we touch onthe implications regarding existence, uniqueness and stability of solutions.

The credit quality process is not in general an observable quantity, which re-quires care when specifying a filtration to be used for pricing. We show that undersome assumptions the credit quality process and its time-change (resp. variance,volatility) are progressively measurable with respect to the filtration generated bysome observable and tradeable assets. We then look into some valuation examplesof leveraged credit-linked notes and of default swaptions.

Finally, we assess the capabilities of the model based on a list of criteria, such ascalibration capabilities, consistency of the model, capabilities of the dynamics andstability of calibration parameters. The full list of criteria is given in Section 4.1.

4 Introduction

The credit quality process model specified is of course not the only model ca-pable of valuing spread-risky products such as gap options. Contrary to the usualstructural models, in particular firm-value models, where the input parameters arebased on economic observables, the model presented here is fully parameterised bymarket prices. Still, the credit quality process model does allow for some inter-pretation of the factors that trigger default. This may, for example, be useful forcalibrating the dynamics. (In the other class of credit models, reduced-form models,default is an exogenous event that happens totally unexpectedly.) Furthermore, themodel proposed is very efficient regarding numerical computation of spread-riskyclaims. For example, the jump-diffusion model of [Zhou (2001)] requires MonteCarlo simulation up to the maturity of the underlying credit spread. The structureof the time-changed Brownian motion may also prove powerful when valuing prod-ucts that depend on the joint dynamics of several correlated underlyings. This hasyet to be investigated.

A result that is needed for computing conditional default probabilities in Part 1,but that is not specific to the context of finance, is derived in Chapter 2. Here, weare concerned with a particular class of continuous local martingales, called Oconemartingales, whose representation as a time-changed Brownian motion (by the fa-mous Dambis, Dubins-Schwarz Theorem) features independent Brownian motionand time-change. We show that

∫ ·0σu dWu, where W is a Brownian motion and σ

a strictly positive cadlag process, is an Ocone martingale if and only if σ and W

are independent.

Part 2 is concerned with a variance reduction technique for Monte Carlo simu-lation. It is available as a working paper, [Packham and Schmidt (2008)]. Considerthe problem of reducing the variance of a Monte Carlo estimator targeted at a vec-tor of dependent random variables. Many existing variance reduction techniquesare powerful, but exploit particular properties of the problem at hand; see [Glasser-man (2004), Section 4.7] for a comparison of variance reduction techniques takinginto account their complexity and effectiveness. The method proposed here, Latinhypercube sampling with dependence (LHSD), is generally applicable, it is partic-ularly simple, and it achieves an effective variance reduction for many estimationproblems, including problems with rare events and high-dimensional problems. It isoften effective even for low sample sizes, and it may easily be combined with othervariance reduction techniques.

LHSD is a generalisation of a multivariate variance reduction technique knownas Latin hypercube sampling (LHS), introduced by [McKay et al. (1979)] and fur-ther studied by [Stein (1987)] and [Owen (1992)], amongst others. LHS relies onindependence of the components of the random vector involved. Essentially, LHSDextends LHS to random vectors with dependent components. The method is men-tioned by [Stein (1987)], but, to the best of our knowledge, it has not been analysedin detail and no results about its effectiveness have been derived yet.

On a probability space (Ω,F ,P), let (U1, . . . , Ud) be a random vector with uni-

5

form marginals and with copula2 C. Suppose the goal is to estimate Eg(U1, . . . , Ud)with g : [0, 1]d → R Borel-measurable and C-integrable.

The usual Monte Carlo estimator based on n independent samples (U1i , . . . , U

di ),

i = 1, . . . , n, is 1/n∑ni=1 g(U1

i , . . . , Udi ). It is a strongly consistent esti-

mator, i.e., 1/n∑ni=1 g(U1

i , . . . , Udi ) P–a.s.−→ Eg(U1, . . . Ud) as n → ∞. The

Central Limit Theorem for sums of independent random variables statesthat the scaled estimator converges in distribution to a Normal distribu-tion, i.e., 1/

√n∑ni=1

[g(U1

i , . . . , Udi )− Eg(U1, . . . , Ud)

] L→ N(0, σ2), with σ2 =Var(g(U1, . . . , Ud)). The Central Limit Theorem serves as an indicator of the speedof convergence via the approximation 1/n

∑ni=1

[g(U1

i , . . . , Udi )− Eg(U1, . . . Ud)

]≈

X, for some X ∼ N(0, σ2/n), from which we may derive confidence intervals andother statistics. In general, the variance of an estimator is a key figure for assessingthe quality of an estimation.

LHSD transforms n independent samples (U1i , . . . , U

di ), i = 1, . . . , n, in such a

way that for each dimension j, the marginals U ji , i = 1, . . . , n, are uniformly spreadover [0, 1]. At the same time, the transformation aims to preserve the copula. Weshow that the LHSD estimator of Eg(U1, . . . , Ud) is strongly consistent for boundedand C-a.e. continuous g. In the bivariate case, under some moderate conditions onthe copula C of the underlying random vector, we derive a Central Limit Theorem,which states that the LHSD estimator converges to a Normal distribution. The Cen-tral Limit Theorem is derived by applying a result from [Fermanian et al. (2004)].We show that, under some monotonicity conditions on g, the limit variance of theLHSD estimator is never greater than the respective Monte Carlo limit variance.

Monte Carlo simulation is widely used for the valuation of financial claims.The general approach to value a financial claim is to generate sample paths of theunderlying financial securities. The discounted expectation of the claim’s payoffunder a risk-neutral measure is then an estimator of the claim’s fair value. Fora comprehensive overview of Monte Carlo simulation in financial applications, werefer to [Glasserman (2004)].

We consider two examples of financial claims that depend on the joint distri-bution of several underlying assets. A first-to-default credit basket is valued basedon random numbers and Sobol sequences, both with and without LHSD. The vari-ance (resp. mean square error) of the LHSD estimators is between 2.25 and 4 timessmaller compared to the corresponding estimators without LHSD. An interestingfeature of the LHSD estimator is that, even though defaults are rare events, it guar-antees that a fixed number of default events are sampled. The second example isconcerned with the valuation of an Asian basket option, which may be formulatedas a high-dimensional estimation problem (dimension 2500 in the example). Thevariance reduction achieved depends on the strike of the option and lies betweenfactors of 6 and 200.

2A copula C is the distribution function of a random vector with uniform marginals, see e.g. [Joe(1997)] and [Nelsen (1999)]. We also associate with C the measure induced by the copula C.

6 Introduction

Chapter 2

A class of Ocone martingales

The famous Theorem of Dambis, Dubins-Schwarz (DDS-Theorem) states that a con-tinuous local martingale has a representation as a time-changed Brownian motion.Ocone martingales are those continuous local martingales whose DDS-Brownianmotion and associated time-change are independent. The basis for this class ofmartingales was laid by [Ocone (1993)] who studied certain properties of cadlag1

local martingales. Further work by [Dubins et al. (1993)] and [Vostrikova and Yor(2000)] has since lead to characterise Ocone martingales as the class of martingalesdescribed above. We study the relationship between stochastic integrals (with re-spect to Brownian motion) and Ocone martingales. The result will be applied laterwhen we define a model for the credit quality of a firm.

2.1 Time-change and DDS-Theorem

Throughout assume given a filtered probability space (Ω,F , (Ft)t≥0,P) that satis-fies the usual hypotheses.

Time-changing a process is like observing the process under a different “clock”.For a detailed account on time-changes see [Revuz and Yor (1999), Chapter V.1]and [Karatzas and Shreve (1998), Chapter 3.4.B].

Definition 2.1. A time-change C is a family (Cs)s≥0 of (Ft)t≥0-stopping timessuch that for P-almost all ω the map s 7→ Cs(ω) is increasing and right-continuous.

If A is an increasing, right-continuous, (Ft)t≥0-adapted process, then C = (Cs)s≥0

defined by

Cs = inft : At > s

is a time-change. Moreover, for every t ≥ 0, the random variable At is an (FCs)s≥0-

stopping time. If A is continuous and strictly increasing, and limt→∞At =∞, thenC is continuous and strictly increasing, and limt→∞ Ct =∞ and ACt = CAt = t.1A process X = (Xt)t≥0 is cadlag if it is right-continuous with left limits, i.e., lims↓t Xs = Xt

and lims↑t Xs exists, for all t ≥ 0. The word cadlag is an acronym of “continue a droite, limite agauche”.

7

8 A class of Ocone martingales

The Theorem of Dambis, Dubins-Schwarz characterises continuous local mar-tingales as time-changed Brownian motions. For a stochastic process X, we denoteits quadratic variation process by [X,X].

Theorem 2.2 (Dambis, Dubins-Schwarz). Let M = (Mt)t≥0 be an (Ft)t≥0-adapted continuous local martingale with limt→∞[M,M ]t = ∞ P–a.s.. Define, foreach 0 ≤ s <∞, the stopping time

Ts = inft ≥ 0 : [M,M ]t > s.

Then the time-changed process

Bs := MTs, Gs := FTs

, 0 ≤ s <∞,

is a Brownian motion. The filtration (Gt)t≥0 satisfies the usual conditions, and wehave, P–a.s.:

Mt = B[M,M ]t , 0 ≤ t <∞.

We call this special Brownian motion the DDS-Brownian motion of M . For a proofof the DDS-Theorem see [Revuz and Yor (1999), Chapter V.1] and [Karatzas andShreve (1998), Chapter 3.4.B].

2.2 Stochastic integrals with respect to Brownian motion as Oconemartingales

Definition 2.3. A continuous local martingale M = (Mt)t≥0, M0 = 0, is an Oconemartingale, if its DDS-Brownian motion is independent of [M,M ].

We are interested in identifying Ocone martingales that are defined as stochasticintegrals with respect to a Brownian motion. [Vostrikova and Yor (2000), Theorem3] provide a characterisation for adapted, continuous integrands. We shall allowadapted cadlag processes as integrands.2

Proposition 2.4. Let W = (Wt)t≥0 be an (Ft)t≥0-Brownian motion and let σ =(σt)t≥0 be an (Ft)t≥0-adapted, strictly positive cadlag process such that P(

∫ t0σ2s ds <

∞) = 1, t ≥ 0, and P(limt→∞∫ t0σ2s ds = ∞) = 1. Let M = (Mt)t≥0 be the

continuous local martingale given by

Mt =∫ t

0

σs dWs, t ≥ 0.

Then, W and σ are independent if and only if M is an Ocone martingale.2The proof of [Vostrikova and Yor (2000), Theorem 3], which draws on properties of Ocone

martingales, can be extended to cadlag integrands. The proof presented here is a direct proof.

2.2. Stochastic integrals with respect to Brownian motion as Ocone martingales 9

Observe that [M,M ] =∫ ·0σ2s ds. For the proof, we need the following Lemma.

Lemma 2.5. Let W,σ,M be as in Proposition 2.4. Then, σ([M,M ]t, 0 ≤ t <∞) =σ(σ2

t , 0 ≤ t <∞).

Proof. Clearly, σ([M,M ]t, 0 ≤ t < ∞) ⊆ σ(σ2t , 0 ≤ t < ∞). For the converse

statement, observe first that [M,M ] is differentiable a.e. (with respect to Lebesguemeasure), and for every such t, [M,M ]′t = σ2

t (this holds P–a.s.). Conversely, atany point t where σ2 is continuous, [M,M ] is differentiable, since suppose that σ2

is continuous at t, but

limh↓0

[M,M ]t+h − [M,M ]th

6= limh↑0


;

this is a contradiction to [M,M ]′ = σ2 a.e. and the continuity of σ2 at t. Now,let t be a point where [M,M ] is not differentiable; i.e., σ2 has a jump at t. Fromthe right-continuity of σ2, it follows that there exists h > 0 such that [M,M ] isdifferentiable on (t, t+ h). By the Mean Value Theorem, there exists sh ∈ (t, t+ h)such that


= σ2sh.

Moreover, by the right-continuity of σ2, taking the right limit,

limh↓0


= limh↓0

σ2sh

= σ2t ,

and we have shown that σ(σ2t , 0 ≤ t <∞) ⊆ σ([M,M ]t, 0 ≤ t <∞).

Proof of Proposition 2.4. Define the family of stopping times (τt)t≥0 by

τt = infs ≥ 0 :

∫ s0σ2u du > t

, t ≥ 0.

Then B = (Bt)t≥0, given by Bt = Mτt=∫ τt

0σs dWs, t ≥ 0, is the DDS-Brownian

motion of M . It is an (Fτt)t≥0-Brownian motion.The “only if” part: Let C be the space of real-valued continuous functions

on R+ and let D be the space of real-valued cadlag functions on R+. Denoteby (C,B(C)), resp. (D,B(D)), the measurable space of real-valued continuous,resp. cadlag, functions on R+ endowed with the σ-algebra generated by the finite-dimensional cylinder sets of C, resp. D.3 For every Γ ∈ B(C) and ∆ ∈ B(D) weshow that

P(B ∈ Γ, σ ∈ ∆) = P(B ∈ Γ) P(σ ∈ ∆). (2.2.1)

That B is independent of [M,M ] and hence an Ocone martingale then follows byLemma 2.5.3Under a suitable metric on C, resp. D, the σ-algebra B(C), resp. B(D), corresponds to the

σ-algebra generated by the open sets (with respect to the metric) of C, resp. D, see e.g. [Shiryaev(1996), Section II.§2] or [Karatzas and Shreve (1998), Sections 2.4 and 6.2].


It is straightforward that Equation (2.2.1) holds for sets ∆ with P(σ ∈ ∆) ∈0, 1. Choose ∆ such that P(σ ∈ ∆) ∈ (0, 1), and denote by D the σ-algebragenerated by σ ∈ ∆. By properties of conditional expectation,

P(B ∈ Γ, σ ∈ ∆) = E(1σ∈∆P(B ∈ Γ|D)). (2.2.2)

Writing D1 = σ ∈ ∆ and D2 = σ 6∈ ∆, it is easy to check that a version of theconditional probability of A ∈ F with respect to D is given by

P(A|D)(ω) =∑i=1,2

P(A ∩Di)/P(Di)1Di(ω), ω ∈ Ω.

Fix this version of the conditional probability. For every ω ∈ Ω, P(·|D)(ω) is aprobability measure (and thus it is a variant of the regular conditional probabilitywith respect to D ; cf. Appendix A.1). Moreover, P(·|D)(ω) P, i.e., P(·|D)(ω) isabsolutely continuous with respect to P. It follows, e.g. by Theorem 14 of [Protter(2005), Section II.5], that

∫ ·0σs dWs computed under the law P(·|D)(ω) and M are

P(·|D)(ω)-indistinguishable.By independence of W and D it follows that W is a Brownian motion under

P(·|D)(ω), and hence M is a continuous local martingale under P(·|D)(ω). Thequadratic variation, as a limit in probability, is invariant to absolutely continuouschanges in measure. Hence, by the Levy-characterisation of Brownian motion, B isan (Fτt

)-Brownian motion under P(·|D)(ω), in other words P(B ∈ ·|D) = P(B ∈ ·)is the Wiener measure on (C,B(C)). Finally, inserting into Equation (2.2.2) yields

E(1σ∈∆P(B ∈ Γ|D)) = E(1σ∈∆P(B ∈ Γ)) = P(B ∈ Γ) P(σ ∈ ∆). (2.2.3)

The “if” part: Now suppose that M is an Ocone martingale, i.e., B and [M,M ]are independent. We have

Wt =∫ t

0

1σs

dMs =∫ [M,M ]t

0

1στs

dBs, P–a.s.,

where the last part follows from [Karatzas and Shreve (1998), Proposition 3.4.8].Now it can be shown that, for Γ,∆ ∈ B(C)),

P(W ∈ Γ, [M,M ] ∈ ∆) = P(W ∈ Γ) P([M,M ] ∈ ∆)

using the same technique as in the “only if” part of the proof. That W and σ areindependent then follows from Lemma 2.5.

Remark 2.6. For simplicity, we have required that the integrand σ be strictlypositive. It is not hard to extend Proposition 2.4 to integrands that are P–a.s.nonzero, see e.g. [Vostrikova and Yor (2000), Theorem 3].

In Proposition 2.4 we started out with a continuous local martingale defined as astochastic integral with respect to a Brownian motion. We may also consider theconverse, where we define a continuous local martingale M as a Brownian motionand an independent time-change. Then, a sufficient condition for M to have a

2.2. Stochastic integrals with respect to Brownian motion as Ocone martingales 11

representation as a stochastic integral with respect to a Brownian motion is that[M,M ] be an absolutely continuous function of t for P-almost every ω ∈ Ω, cf.[Karatzas and Shreve (1998), Theorem 3.4.2].

We conclude this chapter by establishing some properties related to Proposition2.4 that will be used later. For a stochastic process X, we denote by (FXt )t≥0 thefiltration generated by X.

Corollary 2.7. Let M =∫ ·0σs dWs be as in Proposition 2.4, and let B the

DDS-Brownian motion of M . Define the family of stopping times (τt)t≥0 byτt = inf

s ≥ 0 :

∫ s0σ2u du > t

, t ≥ 0. Furthermore, let S be an P–a.s. finite (Fτt)-

stopping time and define B = (Bu)u≥0, with Bu := BS+u −BS. If M is an Oconemartingale, then

(i) B is an (F Bt )t≥0-Brownian motion independent of σ;(ii) (F Bt )t≥0 and FτS

are conditionally independent given Fσ∞, the σ-algebra gener-ated by σ;

(iii) B is independent of FτS∨ Fσ∞, the smallest σ-algebra containing FτS

and Fσ∞.

Proof.

(i) By the properties of Brownian motion, B is a Brownian motion independentof FτS

, cf. [Karatzas and Shreve (1998), Theorem 2.6.16]. In the notation ofthe previous proof, let ∆ ∈ B(D) and let D be the σ-algebra generated byσ ∈ ∆. Since B is a Brownian motion under P(·|D)(ω), for P-almost allω ∈ Ω, so is B, and the first claim follows in analogy to Equation (2.2.3).

(ii) The second claim is a generalisation of the first claim. Since B is independentof σ, we have P(B ∈ Γ|Fσ∞) = P(B ∈ Γ) P–a.s., Γ ∈ B(C). Moreover, thereexists a version Q of P(B ∈ ·|Fσ∞) that is a regular conditional distributionof B with respect to Fσ∞, cf. Theorem A.6. It follows that B is a Brownianmotion under Q and hence B is a Brownian motion independent of FτS

underQ, which is the required result.

(iii) Since F B∞ and FτSare conditionally independent given Fσ∞, P–a.s. (see

[Kallenberg (2001), Proposition 6.6] for the first statement),

P(B ∈ Γ|Fσ∞ ∨ FτS) = P(B ∈ Γ|Fσ∞) = P(B ∈ Γ), Γ ∈ B(C).

PART 1

Credit dynamics in a

first-passage time model with jumps

13

Chapter 3

Credit derivatives

Credit derivatives are financial instruments whose payoff is linked to a “credit event”such as the default of a firm or a sovereign entity. Initially set up to hedge the defaultrisk of corporate bonds or loans, credit derivatives are now considered an asset classof its own. Credit derivatives provide a means to isolate and transfer the credit riskcomponent of financial instruments in a standardised way.

The market for credit derivatives has grown considerably since its beginningin the mid-1990s. Statistics published by the Bank of International Settlements(BIS) reveal that the outstanding notional of credit derivatives has grown from $14trillion in December 2005 to over $42 trillion in June 2007 [BIS (2007)]. In December2007 and June 2008, the outstanding notional was approximately $57 trillion [BIS(2008)].

The plain vanilla credit derivative is the credit default swap (CDS), an instru-ment that provides insurance against the default of a borrower. Other single-namecredit derivatives include default swaptions and credit-linked notes. According tothe BIS statistics, the outstanding notional in single-name instruments was $24trillion in June 2007 and $33 trillion in June 2008. Multi-name credit derivativesprovide insurance against credit events in a portfolio of underlying entities. Exam-ples of multi-name credit derivatives are index trades, first-to-default credit basketsand collateralized debt obligations (CDOs) (outstanding notional according to BISstatistics: $18 trillion in June 2007, $24 trillion in June 2008).

3.1 Credit default swaps

3.1.1 Mechanics and term structure

The fundamental product of the credit derivatives market is the credit default swap(CDS). Given an underlying entity, such as a company, it is a contract betweentwo counterparties, the protection buyer and the protection seller, that insures theprotection buyer against the loss incurred by default of the underlying entity withina fixed time interval. The protection buyer regularly pays a constant premium, thecredit spread or CDS spread , that is fixed at inception, up until maturity of the CDS

15

16 Credit derivatives

0

5

10

15

20

1 2 3 4 5 6 7 8 9 10maturity [years]

BMW CDS spreads (15.1.2007) [bp]

Fig. 3.1 Term structure of CDS spreads. For example, on a 5-year CDS, the protection buyerpays 9.75bp per year (on a quarterly basis) in return for protection of the loss incurred by defaultof BMW over the next five years. (Source: Datastream)

or the default event, whichever occurs first. This stream of payments is termed thepremium leg of the CDS. In return, the protection seller agrees to compensate theprotection buyer for the loss incurred by default of the underlying entity at the timeof default in case this occurs before maturity. This constitutes the protection leg ofthe CDS.1

The fair CDS spread or fair credit spread is the CDS spread that makes thevalue of the premium leg and the protection leg equal. This is considered in detailin Section 3.4. The mapping of credit spreads with respect to their maturity iscalled the term structure of credit spreads.2 We shall assume market-given CDSspreads to be fair spreads and use the no-arbitrage principle to derive risk-neutraldefault probabilities; this relationship is made precise in Section 3.4. CDS for largefirms and sovereigns are liquidly traded, and typically CDS spreads for maturities1, 3, 5, 7, 10 years are quoted in the market. Credit spreads are quoted in basispoints (bp) with 1bp = 0.01%. An example of a term structure is given in Figure3.1.

1In practice, payment of the protection leg is triggered by a so-called credit event. Most CDScontracts are standardised under the terms of the International Swap Dealers Association (ISDA)contract, which classifies a credit event as either one of bankruptcy, obligation acceleration, obliga-tion default, failure to pay, repudiation/moratorium, restructuring; see e.g. [Duffie and Singleton(2003), Chapter 8] or [O’Kane (2008), Chapter 5] for more details. We follow the convention ofspeaking of default when the more subtle credit event is meant.2A term structure is also often called a curve. In the case of the term structure of credit spreads,

it is custom to use the term spread curve.

3.1. Credit default swaps 17

3.1.2 Shape and dynamics of the term structure

CDS spreads and the term structure of credit spreads change over time. We derivesome stylised facts about the shape and dynamics of the term structure.3

Before the advent of a liquid CDS market, the term credit spread referred tothe difference between the yield of a defaultable zero-coupon bond and default-freezero-coupon bond of the same maturity, see e.g. [Lando (2004), Chapter 2] for thisdefinition of credit spread, also called yield spread.

The corresponding term structure of credit spreads has been extensively studied.Since both term structures - CDS spreads and yield spreads - refer to credit risk, weshould expect them to share similar properties. It is important to note, however,that there are also differences between the two term structures: For example, theterm structure of yield spreads is affected by the liquidity of the underlying bonds,see e.g. [Duffie and Singleton (1999); Collin-Dufresne et al. (2001)]. On the otherhand, trading CDS is independent of the availability of an underlying financialsecurity. Also, it is not straightforward to take a short position in a bond, whichmay lead to significant transaction costs. Since there is no exchange of notional,CDS of arbitrary maturity can be incepted with minor transaction costs, and it ispossible to emulate a short bond position by entering into a position that involvesbuying CDS protection.

We assume that stylised facts of the yield spread term structure that can berelated to the credit risk component of the underlying entity apply to the CDSspread term structure as well. A wide variety of term structure shapes has beenobserved in the market, such as upward sloping, flat, hump-shaped and downwardsloping curve [Helwege and Turner (1999); Zhou (2001)]. [Fons (1994); Helwege andTurner (1999)] observe that investment-grade bond issuers usually face a low defaultrisk in the near future, with a higher default risk over a longer time horizon suchas ten years, hence the spread curve is typically upward sloping. Speculative-gradefirms, such as small but growing firms, are more prone to a default in the near-termwhile the risk of default over ten or more years is relatively low once the firm hassurvived the first few years, hence their spread curve is mostly downward sloping.

Another common observation is that short-term credit spreads do not tend tozero as maturity tends to zero, but are significantly greater than zero, see e.g.[Duffie and Lando (2001); Zhou (2001)], [Duffie and Singleton (2003), Ch. 3] and[Lando (2004), Ch. 2]. This indicates that, for any time to maturity, market partic-ipants presume a positive probability of unexpected and instantaneous default. Forexample, [Duffie and Lando (2001)] attribute this behaviour to the unavailability(publicly) of complete accounting information about a firm’s status. In other words,the observed firm value differs from the real firm value by some “accounting noise”.

3According to [Cont (2001)], stylised (empirical) facts comprise statistical properties shared byasset prices across a wide range of instruments, markets and time periods. We shall use the termstylised fact to denote common empirical properties of credit term structures, including statisticalproperties, shared by a wide range of underlying entities and time periods.


There is also a significant amount of research that indicates that credit spreadsare subject to jumps, i.e., in addition to diffusive (continuous) behaviour of yieldspreads through time, yield spreads may change by sudden and unexpected jumps,see [Johannes (2000); Das (2002); Dai and Singleton (2003); Tauchen and Zhou(2006); Zhang et al. (2008)]. According to [Zhou (2001)], a more volatile jumpcomponent indicates that a firm is more likely to default on its short-maturitybonds than a firm with a more volatile diffusion component.

In a recent empirical study on CDS spreads, [Schneider et al. (2007)] applied thebipower variation test of [Barndorff-Nielsen and Shephard (2006)] to a sample ofCDS spreads. The results indicate a minimum of 3 jumps per year for all maturities,with a high correlation of jumps across maturities. [Schneider et al. (2007)] inferthe following empirical stylised facts for CDS spreads:

• A jump affects broad ranges of the CDS maturity spectrum. This is econom-ically motivated by the fact that unfavourable events usually affect contractsof both short and long maturities, and similarly, when expectations about theoverall credit quality change, the entire term structure of CDS spreads reacts.

• Jumps in CDS spreads are mostly positive. The arrival of bad news suchas financial distress causes sudden upward moves in CDS spreads, becauseprotection sellers demand higher compensation for bearing higher risk. Goodnews, on the other hand, tend to propagate gradually. This observation isbacked by statistics of CDS spread time-series: CDS spreads of all maturitiesexhibit positive skewness and a significant excess kurtosis, and credit spreaddifferences exhibit a high excess kurtosis.

• The one-year CDS spread exhibits time-series variation different from CDSspreads of higher maturities.

As an example, consider Figure 3.2, which shows CDS spreads of BMW overa time period of approximately five years. We see the tendency of CDS spreadsof different maturities to evolve in a similar way. The bottom picture shows thelog returns of the 5-year CDS spread over the same time period. Obviously, themagnitude of CDS spread returns is not constant. This suggests that the volatilityof CDS spread returns changes over time. Furthermore, single large fluctuationssuggest the existence of jumps.

3.2 Market model and risk-neutral valuation formula

The valuation or pricing of financial securities requires a mathematical modelof the market. The standard approach for the valuation of derivatives is bythe principle of no-arbitrage, see e.g. [Bjork (2004); Dana and Jeanblanc (2003);Hunt and Kennedy (2004); Duffie (2001); Delbaen and Schachermeyer (2006);Bingham and Kiesel (1998)].

3.2. Market model and risk-neutral valuation formula 19

01020304050607080

0 1 2 3 4 5

spre

ad

[bp]

t [years]

BMW CDS spreads – 1 yr3 yrs5 yrs7 yrs

10 yrs

-0.4-0.3-0.2-0.1

00.10.20.30.4

0 1 2 3 4 5t [years]

BMW log-returns (5-yr-spread)

Fig. 3.2 BMW CDS spreads (daily), 2 Jan 2003 – 6 Aug 2007 (top) and log-returns of 5-yearCDS spread (bottom). (Source: Datastream)

Let (Ω,F , (Ft)t≥0,P) be a complete probability space endowed with a filtration(Ft)t≥0, i.e., a family of sub-σ-algebras of F such that Fs ⊆ Ft, s ≤ t. The filtrationrepresents the information available in the market, i.e., Ft contains all events thatare observable until time t. In particular, F0 is P-trivial, i.e., for all A ∈ F0, eitherP(A) = 0 or P(A) = 1. We shall assume throughout that (Ft)t≥0 satisfies the usualhypotheses, i.e., F0 contains all P-null sets of F and (Ft)t≥0 is right-continuous, seee.g. [Protter (2005), Chapter 1]. We shall also assume that the probability spaceis rich enough to support any objects that we define. If not otherwise stated, allprocesses are (Ft)t≥0-adapted.

In a market model, the price processes of financial assets are modelled as stochas-tic processes. Suppose there is a risk-free asset, the money market account β = (βt),which accrues at an instantaneous interest rate or short rate r = (rt)t≥0, i.e.,dβt = βt rt dt. The short rate satisfies rt > 0 and

∫ t0rs ds < ∞, for any t > 0,

and is assumed to be continuous and (Ft)t≥0-adapted. The accrual of the moneymarket account in the interval (t, T ] is

β(t, T ) = βT /βt.

The other price processes are risky assets. For the price process S = (St)t≥0 of afinancial asset, the discounted price process is S/β = (St/βt)t≥0; in other words, itis the price process of S expressed in units of the money market account.

Definition 3.1. A risk-neutral measure P is a probability measure that is equiv-


alent to the real-world probability measure and such that discounted prices are P-martingales.

Assumption 3.1. There exists a risk-neutral measure.

We assume that P is a risk-neutral measure and by E we denote the expectationoperator with respect to the measure P. One of the fundamental results of arbi-trage theory states that existence of a risk-neutral measure implies the absence ofarbitrage, see e.g. [Bjork (2004), Section 10.2].

We state a risk-neutral valuation formula for defaultable claims, where we follow[Bielecki and Rutkowski (2002), Chapters 2 and 8] and [Belanger et al. (2004)].

Denote by τ : Ω → R+ the random time of default of a firm.4 We assume thatτ is an (Ft)t≥0-stopping time, i.e., τ ≤ t ∈ Ft, for t ≥ 0. Accordingly, the defaultindicator process (1τ≤t)t≥0 is (Ft)t≥0-adapted. The cash flows associated with adefaultable claim with maturity T are as follows:5

• The promised cumulative cash flow process A is a cadlag, (Ft)t≥0-adaptedprocess of finite variation that denotes the cumulative stream of cash flowsreceived prior to default up until and including T . It does not include a cashflow at default.

• The recovery process Z represents the payoff at the time of default, if defaultoccurs prior to T . It is a nonnegative, (Ft)t≥0-adapted process.

We make the following assumptions:

Assumption 3.2.

(i) The promised cumulative cash flow process A satisfies E[∫ T

0β−1t dAt

]<∞.

(ii) The recovery process Z satisfies E[sup0<t≤T β

−1t Zt

]<∞.

Definition 3.2. The dividend process D of a defaultable claim with associated cashflows (A,Z) and maturity T is

Dt = At∧τ +∫ t

0

Zu d1τ≤u, 0 ≤ t ≤ T.

Note that∫ t0Zu d1τ≤u is a Lebesgue-Stieltjes integral.

The risk-neutral valuation formula for a defaultable claim is defined as follows:

Definition 3.3. The (ex-dividend) price process V = (Vt)0≤t≤T of a defaultable4Since we consider only one defaultable entity we omit explicit reference to the entity to avoid

linguistic clutter.5[Bielecki and Rutkowski (2002); Belanger et al. (2004)] require the processes A and Z defined

below to be (Ft)t≥0-predictable; we merely require that A and Z are (Ft)t≥0-adapted. Further-more, in the stated sources the process A is required to be nonnegative and nondecreasing; weallow for negative cash flows as well.

3.3. Conditional default and survival probabilities 21

claim (A,Z) with maturity T is

Vt = E

[∫ T

t

β(t, u)−1 dDu

∣∣∣Ft]

= E

[∫ T∧τ

t∧τβ(t, u)−1 dAu +

∫ T

t

β(t, u)−1Zu d1τ≤u∣∣∣Ft] , 0 ≤ t ≤ T. (3.2.1)

Definition 3.3 is justified as follows: If the payoff of the claim can be replicatedby a self-financing trading strategy in the market’s assets, then it can be shownthat the no-arbitrage price process is indeed given by Equation (3.2.1). Otherwise,Definition 3.3 is motivated by assuming existence of a risk-neutral measure for themarket extended by the claim’s price process, which implies that the extendedmarket is arbitrage-free. For a detailed discussion, see e.g. [Bjork (2004), Chapter10].

Example 3.4. The price process of a defaultable zero-coupon bond with maturityT with no recovery value at default is given by

Vt = E[β(t, T )−11τ>T

∣∣Ft] , 0 ≤ t ≤ T.

If we assume that the money market account and the default time, resp. defaultindicator process, are conditionally independent given Ft, for any 0 ≤ t ≤ T , then

Vt = E[e−

R Ttrs ds

∣∣Ft] P(τ > T |Ft)

= B(t, T ) P(τ > T |Ft), 0 ≤ t ≤ T, (3.2.2)

where B(t, T ) denotes the price of a default-free zero-coupon bond with maturityT at time t, and P(τ > T |Ft) is the risk-neutral probability of survival until Tconditional on Ft.

3.3 Conditional default and survival probabilities

The price process of the risky zero-coupon bond of Example 3.4 involves the risk-neutral conditional survival probabilities P(τ > T |Ft), 0 ≤ t ≤ T . To simplifynotation, we write P (t, T ) := P(τ ≤ T |Ft) and Q(t, T ) := P(τ > T |Ft), T ≥ t,t ≥ 0. Observe that, since τ is an (Ft)t≥0-stopping time,

P (t, T ) = 1τ≤t + 1τ>tP(τ ∈ (t, T ]|Ft). (3.3.1)

Before stating the valuation formula for CDS, we derive some straightforward resultson risk-neutral conditional survival probabilities, resp. default probabilities.

Lemma 3.5.

(i) For every t ≥ 0, there is a regular version of P (t, T ), T ≥ t, i.e., (assumingthat P (t, T ) is a regular version) for all ω ∈ Ω, P (t, T )(ω), T ≥ t, is adistribution function.


(ii) Q(t, T ) = 1τ>tP(τ > T |Ft), T ≥ t.(iii) For each T , the process (Q(t, T ))t≤T is a martingale with a cadlag modifica-

tion.6

(iv) For each T , the process (1τ>tP (t, T ))t≥0 is a semimartingale with a cadlagmodification.7

Proof.

(i) This follows directly from the existence of regular distribution functions forrandom variables, cf. Theorem A.5.

(ii) Fix T . For the first statement, observe that the events τ > T and τ >t, τ > T are equal, so that

P(τ > T |Ft) = E(1τ>t1τ>T|Ft

)= 1τ>tP(τ > T |Ft).

(iii) Clearly, Q(t, T ) is (Ft)t≥0-adapted and bounded. By the Tower Law for con-ditional expectations,

E(P(τ > T |Ft)|Fs) = P(τ > T |Fs), s ≤ t,

which establishes the martingale property. That Q(t, T ) has a cadlag modifi-cation follows from [Karatzas and Shreve (1998), Theorem 1.3.13].

(iv) That 1τ>tP (t, T ) = 1τ>t−Q(t, T ) is a cadlag semimartingale now followsdirectly.

We assume from now on that P (t, T ), T ≥ t, denotes a regular version of theconditional distribution of τ .

3.4 Risk-neutral valuation of CDS

In the following, let (C,B(C)) be the measurable space of real-valued, continuousfunctions on R+ endowed with the σ-algebra generated by the finite-dimensionalcylinder sets of C.

We shall make the following standard assumptions for CDS valuation:

Assumption 3.3.

(i) The recovery rate R, R ∈ (0, 1), of a CDS is deterministic and constant.(ii) There is no counterparty risk involved in a CDS transaction.6See [Karatzas and Shreve (1998), Chapter 1] for the following: Let X, Y be stochastic processes.

Then, Y is a modification of X if, for every t ≥ 0, we have P(Xt = Yt) = 1. Moreover, if Y is amodification of X, and if X and Y are both P–a.s. cadlag, then X and Y are indistinguishable,i.e., P(Xt = Yt, 0 ≤ t < ∞) = 1.7Recall that a process X is a semimartingale if it is of the form X = X0 + M + A, where X0 is

finite-valued and F0-measurable, M is a local martingale with M0 = 0 and A is a process of finitevariation with A0 = 0, cf. [Jacod and Shiryaev (2003), Section I §4.c].

3.4. Risk-neutral valuation of CDS 23

In addition, we shall make the following assumption on the relationship betweeninterest rates and the default indicator process:

Assumption 3.4. The short rate r and the default indicator process (1t≤τ)t≥0

are conditionally independent, i.e., for any sets A,B ∈ B(C) and any t ≥ 0, it holdsthat P(r ∈ A,1τ>· ∈ B|Ft) = P(r ∈ A|Ft) P(1τ>· ∈ B|Ft).

This assumption is fulfilled for example, if (Ft)t≥0 is generated by two independentstochastic processes, one of which drives the interest rate and the other drivingthe default indicator process. Assumption 3.4 is a common assumption, see e.g.[O’Kane (2008), Section 3.9] or [Schonbucher (2003), Section 3.1]; it is justified forCDS valuation, see e.g. [O’Kane (2008), Section 8.3.4] where the case is made thatthe CDS is an almost pure credit product. For defaultable claims that bear signif-icant interest rate sensitivity, this assumption may not be suitable. For example,[Longstaff and Schwartz (1995); Collin-Dufresne et al. (2001)] find that an increasein interest rates lowers the credit spread of defaultable bonds.

3.4.1 CDS valuation and fair CDS spread

We now turn to valuation of a CDS with maturity T assuming a notional of 1. Recallthat β(t, T )−1 = e−

R Ttru du, t ≤ T , and the price of a default-free zero coupon bond

with maturity T at time t is B(t, T ) = E[e−

R Ttru du

∣∣Ft].The protection buyer of a CDS continuously pays a spread s, i.e., the promised

cumulative cash flow on the premium leg is AT = s · T . The protection sellerin turn pays a fraction of the notional, (1 − R), at default if default occurs untilT , i.e., the recovery process is Zu = (1 − R), u ≤ T . The dividend process isDt = −s

∫ t01τ>u du + (1 − R) 1τ∈(0,t], t ≤ T . The ex-dividend price process

at time t from the point of view of the protection buyer is, according to Equation(3.2.1),

Vt = E

[−s∫ T

t

β(t, u)−11τ>u du+ (1−R)β(t, τ)−11τ∈(t,T ]

∣∣∣Ft]

= −s∫ T

t

B(t, u)Q(t, u) du+ (1−R) E[β(t, τ)−11τ∈(t,T ]|Ft

],

where we have applied Fubini’s Theorem A.18 and the conditional independence ofr and 1τ>·. Turning to the conditional expectation, write β(t, τ)−11τ∈(t,T ] =e−

R τtrw dw 1τ∈(t,T ]. Then,

E[β(t, τ)−11τ∈(t,T ]|Ft

]=∫

(0,∞]×Ce−

R utvw dw 1u∈(t,T ]P(τ ∈ du, r ∈ dv|Ft)

=∫ T

t

∫C

e−R u

tvw dw P(r ∈ dv|Ft) P(τ ∈ du|Ft)

=∫ T

t

B(t, u) P(τ ∈ du|Ft),


0.96

0.97

0.98

0.99

1

1 2 3 4 5 6 7 8 9 10maturity [years]

BMW survival probabilities (15.1.2007)

Fig. 3.3 BMW survival probabilities assuming R = 40%

using the fact that a regular version of the conditional probability exists (cf. Theo-rem A.2), using Fubini’s Theorem and using the conditional independence of r andτ . Putting things together, the valuation formula for a CDS of maturity T at timet is

Vt = −s∫ T

t

B(t, u)Q(t, u) du+ (1−R)∫ T

t

B(t, u)dP (t, u). (3.4.1)

At inception, a CDS involves no initial cash-flow, i.e., the market value is 0. ByEquation (3.4.1) and setting Vt = 0, we solve for the fair credit spread or fair CDSspread s(t, T ) at time t,

s(t, T ) =(1−R)

∫ TtB(t, u) dP (t, u)∫ T

tB(t, u)Q(t, u) du

on τ > t, t < T. (3.4.2)

On τ ≤ t or for t ≥ T , we set s(t, T ) = 0.Since the value of CDS quoted in the market is 0, market-quoted CDS spreads

are fair by definition. Here, for reasons of tractability, we consider only continuousspread payments, but in reality, spread payments are typically made quarterly, witha payment of the accrued spread in the period where default takes place. Given aterm structure of CDS spreads and a recovery rate, it is possible to back out impliedrisk-neutral survival probabilities by a bootstrapping algorithm, see e.g. [O’Kane(2008), Chapter 7]. As an example, the implied survival probabilities (Q(0, T ))T≥0

corresponding to the CDS term structure of Figure 3.1 are shown in Figure 3.3.The following Lemma is important for modelling the dynamics of CDS spreads.

Lemma 3.6. The fair spread (s(t, T ))t≥0) is a cadlag semimartingale on [0, τ ∧T [.

Proof. See [Schmidt (2007), Proposition 2] for a proof that (s(t, T ))t≥0 is a semi-martingale on [0, τ ∧ T [. That (s(t, T ))t≥0 has a cadlag modification follows fromthe fact that the numerator and the denominator of Equation (3.4.2) have cadlagmodifications.

3.5. Credit derivatives with spread-sensitive payoff 25

3.4.2 Mark-to-market of a CDS position

The value of an existing CDS position can be expressed as the cost of unwindingthe transaction by entering into an offsetting CDS position. We speak of the mark-to-market value of the existing CDS position.

Assume a CDS contract with maturity T entered at time v ≤ t from the pointof view of the protection buyer. Conditional on τ > t, the value of the positionat time t is, by Equation (3.4.1),

Vt = −s(v, T )∫ T

t

B(t, u)Q(t, u) du+ (1−R)∫ T

t

B(t, u) dP (t, u).

On the other hand, the value of selling CDS protection with maturity T at time t,is

0 = s(t, T )∫ T

t

B(t, u)Q(t, u) du− (1−R)∫ T

t

B(t, u) dP (t, u).

Combining these two equations, the mark-to-market value can be written as

Vt = (s(t, T )− s(v, T ))∫ T

t

B(t, u)Q(t, u) du.

Accordingly, for a protection seller the mark-to-market value is −Vt. For practicalreasons, if default occurs prior to T , we define the mark-to-market value at defaultto be Vτ∧T = −(1−R)1τ≤T. In practice, of course, when unwinding a transactionby buying the offsetting position, one has to take into account counterparty risk.

3.5 Credit derivatives with spread-sensitive payoff

Other than being subject exclusively to default risk, the payoff of some credit deriva-tives is determined explicitly by the level of CDS spreads. In this case the dynamicsof CDS spreads play a significant role in product valuation. Examples of such so-called spread products are default swaptions, which are options to enter into a CDSat some future date, and credit derivatives with a leverage component, such as lever-aged credit-linked notes (leveraged CLN). The payoff profile of a leveraged CLN isparticularly sensitive to the presence of jumps in CDS spreads.

3.5.1 Leveraged credit-linked note

A credit-linked note (CLN) is a note or bond paying an enhanced coupon to aninvestor for bearing the credit risk of a reference entity; see [Bielecki and Rutkowski(2002), Section 1.3.3] for a general description.

As an example of a spread product, consider the leveraged credit-linked note.This note is particularly sensitive to jumps in CDS spreads, even if a jump doesnot lead to default. The principal idea is that an investor sells protection on anamount of default risk that is a multiple k, the leverage factor, of his investment


Investorr + k s0

1

Issuerk s0

k(1−R)

k CDS

accountDefault-free

r 1

Fig. 3.4 Leveraged credit-linked note with leverage factor k and notional 1. Cash flows at incep-tion and while the note is alive.

amount. The motivation for taking leveraged exposure is to earn a certain multiplek of the credit spread. Most likely, his investment will not suffice to compensate theloss incurred by default. Therefore, a trigger is agreed to terminate the structurewhile the cost of closing the position is still likely to be sufficiently covered by theinvestment amount. The cost of closing the position depends on the level of creditspreads, hence the investor is exposed mainly to spread risk and to default risk onlyto a lesser extent.

In more detail, the issuer structures the note as follows (see Figure 3.4): Forsimplicity, assume a constant default-free interest rate r and an investment amountof 1, which is deposited in a default-free account earning a coupon r. In addition,protection is sold by entering a fair CDS with notional k earning a spread of k s0.The investor receives a fixed coupon until either maturity of the note or until atrigger event takes place. The size of the coupon is r + k s0 with k s0, k ≤ k, thepremium associated with the note. This premium is financed by the CDS position.The trigger event is determined as follows: denote by V kt the mark-to-market valueat time t of the underlying CDS position with notional k. The trigger event takesplace at time S = inft ∈ (0, T ] : −V kt ≥ K, with 0 < K ≤ 1 a pre-defined triggerlevel. At S, the note is unwound by withdrawing the investment amount 1 from thedeposit account and by closing the CDS position at a cost of −V kS (see Figure 3.5).Observe that possibly −V kS > 1, in which case the issuer must cover the missingamount required to unwind the CDS position. For this type of risk, called gap risk,the issuer is compensated with a premium of (k− k) s0. In the case where −V kS ≤ 1,the investor receives the remainder of the structure, 1 + V kS . Given K, valuation ofthe note essentially means determining the fair factor k.

Clearly, the trigger time S depends on the evolution of the underlying CDSspread. Furthermore, the amount of the redemption payment max(1 + V kS , 0) isundetermined until S. Assuming a model in which CDS spreads evolve continuously,the mark-to-market value V evolves continuously as well, and the trigger time is

3.5. Credit derivatives with spread-sensitive payoff 27

Default-freeaccountk CDS

k CDS Issuer Investor

1 + max(−VS − 1, 0)

max(1 + VS , 0)

k sS

k(1−R)

k(1−R)

k s0

Fig. 3.5 Leveraged credit-linked note with leverage factor k and notional 1. Cash flows at triggertime S.

S = inft ∈ (0, T ] : −V kt = K, unless a default takes place. Hence −V kS ≤ 1,and gap risk is limited to the default case. On the contrary, upward jumps in CDSspreads translate into upward jumps in the mark-to-market value of the CDS, andpossibly −V kS > 1, so the issuer faces gap risk even when no default takes place.

In some models, we can determine the fair factor k by no-arbitrage arguments.Assume first that CDS spreads evolve continuously through time, in which case,V k evolves continuously as well. Moreover, assume that there is no jump-to-defaultrisk.8 In this case, the trigger time is S = inft ∈ (0, T ] : −V kt = K and there isno gap risk. Consequently, the fair factor is k = k.

Now suppose that CDS spreads are constant, i.e., the note is exposed to defaultrisk only. The trigger time then coincides with the default time, in which case theinvestor loses his invested capital. The payoff of this position is equivalent to thepayoff of a short position in 1/(1−R) CDS, so k = 1/(1−R).

We can also infer upper and lower bounds for the factor k. The upper bound isk as the note’s coupon is funded by the underlying CDS position. To determine thelower bound, suppose first that k(1−R) < K. Since K ≤ 1, the investor has enoughcapital to enter into a CDS position with nominal k instead, earning the spreadk s(0, T ), which is the same spread that he earns by investing into the leveragedCLN. However, the CDS investment exposes him only to default risk, whereas if heholds the note he bears additionally spread risk. When the note triggers, he stopsearning the note’s spread, although possibly no default has happened; he wouldcontinue earning the spread if he held the CDS position instead. Additionally, theloss incurred by the CDS position is k(1 − R), whereas the minimum loss in thenote’s position is K. Hence, k (1−R) > K.

To determine the factor k, consider the discounted cash flows to the note issuerat time 0 (essentially, the issuer is exposed to gap risk), and apply the risk-neutralvaluation formula,

8We encounter models with this property later.


V gap0 = E

((k − k)s(0, T )

∫ T

0

e−ru1S>u du− e−rS max(−V kS − 1, 0)

)

= (k − k)s(0, T )∫ T

0

e−ruP(S > u) du

−∫

(0,T ]×(1,∞)

e−ru xP(S ∈ du,−V kS ∈ dx). (3.5.1)

The fair gap risk fee is obtained by setting V gap0 = 0, i.e.,

(k − k)s(0, T ) =

∫(0,T ]×(1,∞)

e−ru xP(S ∈ du,−V kS ∈ dx)∫ T0

e−ruP(S > u) du.

3.5.2 Gap option

The crucial component in valuation of the leveraged CLN is the gap risk. From thevaluation formula (3.5.1), it is clear that the gap risk component is in fact an optionthat the note issuer sells to the investor. The gap option has payoff max(−V kS −1, 0),and the option premium is the stream of payments

∫ T0

(k− k)s(0, T )1S>u du thatis earned while the note is alive, so in fact Equation (3.5.1) is the valuation formulafor a gap option. If (V kt )t≤T is a continuous process, then the payoff of the gapoption is 0, hence (k− k) = 0. Only if (V kt )t≤T has jumps does the gap option havea positive payoff and hence a non-zero premium.

Chapter 4

Credit spread models

One purpose of modelling the term structure of CDS spreads and its dynamics is toobtain prices from the model for derivatives whose price is not given by the market.Furthermore, a suitable model can be used to obtain hedging strategies for suchderivatives.

In this chapter, we introduce the hazard rate (or forward default rate) and theterm hazard rate as a useful quantities for modelling the term structure of creditspreads. We review some existing credit spread models. Such models are typicallyclassified as either structural or reduced-form models. Both approaches are thor-oughly described in the literature, so we give only a brief review; see e.g. [Lando(2004); Duffie and Singleton (2003); Bielecki and Rutkowski (2002)] for detailedexpositions.

Beforehand, however, we raise the question of how we can assess the quality ofa model.

4.1 Model requirements

We state some general criteria that allow us to assess the capabilities of a model.Most of these criteria can be found in one or several of [O’Kane (2008); Cairns(2004); Brigo and Mercurio (2006); Cont and Tankov (2004a)]. The list attemptsto highlight those aspects that should be considered in modelling. The individualcriteria are outlined in a descriptive manner; the precise statement of a modelrequirement depends on the intended application.

List of model requirements

(i) The model fits exactly the term structure of CDS spreads and prices of otherliquid credit derivatives in the market.

(ii) The model can be used to price products whose price is not given by themarket. In particular, the model is consistent, i.e., it produces arbitrage-freeprices. Note that even if market prices are assumed to be arbitrage-free, it isnot guaranteed that the prices produced by a model are arbitrage-free. (We

29

30 Credit spread models

give examples below.)(iii) The ranges of prices, spreads and other quantities produced by the model are

valid (e.g. rates and spreads are nonnegative).(iv) The model captures spread risk, that is the risk that market credit spreads

change even through default does not occur.(v) The model produces desired dynamics.(vi) The model captures any default timing. This is important as the timing of

payoffs may have a significant effect on a derivative’s value.(vii) The model captures effects of joint risks, such as co-movement in credit risk

and interest rates.(viii) The model takes into account an uncertain recovery payment at default.(ix) The model allows fast, stable and accurate pricing of CDS and other instru-

ments, either analytically or numerically.(x) The parameters obtained by calibration are stable in the sense that a recali-

bration based on a small change in market parameters leads only to a smallchange in calibration parameters.

(xi) Hedging strategies obtained from the model are stable in the same sense as inthe previous point.

(xii) The model is as simple as possible.

The list is not exhaustive of course; additional criteria may become importantfor particular modelling purposes. Some criteria may be considered as mandatory,e.g. (i)-(iii), whereas others may be relaxed, e.g. (viii). Again, this may dependheavily on the application. There are also criteria whose assessment is not clear, e.g.what exactly constitutes an accurate price, when exactly is pricing fast, and whatexactly are desirable dynamics? Again, these questions depend on the intendedapplication of the model and cannot be answered on a general level. In any case, itis desirable to understand whether or how a particular model fulfills a criterion.

As to criterion (ii), the no-arbitrage constraint for a term structure is thatsurvival probabilities are strictly decreasing with maturity, i.e.,

P (t, T1) < P (t, T2), T1 < T2, (4.1.1)cf. [O’Kane (2008), Section 7.7]. If P (t, T1) > P (t, T2), for any T1 < T2, then thedefault time τ is not well-defined. Moreover, the fact that in the real-world a riskyentity may default at any time applies also to the risk-neutral world; this followsfrom of the equivalence of the real-world and the risk-neutral probability measures.Hence, P (t, T1) 6= P (t, T2) whenever T1 < T2.

Similarly, recall the valuation example of a gap option from Section 3.5.1. In amodel where credit spreads evolve continuously and jump-to-default risk is excluded,there is no gap risk. On the other hand, if the real-world probability of a gap event isstrictly positive, then this must also be captured by the model, again by equivalenceof real-world and risk-neutral probability measures.

[Buraschi and Corielli (2005)] discuss time-inconsistency of dynamic models.This arises for example, if a model cannot fit observed prices over time, forcing

4.2. Hazard rate 31

recalibration, which leads to a change in model-implied probability distributions;see also [Backus et al. (1998); Dai and Singleton (2003)] on this issue.

Finally, recall that in Section 3.1.2, we have already discussed the dynamics ofthe CDS term structure (criterion (v)), and gathered some stylised facts. Again,equivalence of the real-world and the risk-neutral probability measures implies thatrisk-neutral dynamics of the credit spread term structure must be such that termstructures of different shapes can be obtained, that credit spreads do not vanishas time-to-maturity tends to zero and that credit spreads exhibit jumps. Theseproperties should be produced by a model if this is relevant for a specific productto be priced.

4.2 Hazard rate

The hazard rate or forward default rate is useful for modelling the term structureof credit spreads.1 Furthermore, as outlined below, it is a useful approximation forcredit spreads.

Assume that the default time τ of a firm admits a (conditional) density, i.e.,for t > 0, on τ > t, there exists a nonnegative function p(t, T ), T ≥ t suchthat P (t, T ) =

∫ Ttp(t, u) du, T ≥ t. The hazard rate at time t is the mapping

T 7→ λ(t, T ), defined by

λ(t, T ) =p(t, T )

1− P (t, T )= − d

dTln(1− P (t, T )), T ≥ t.

It follows that the survival probability at time t is

Q(t, T ) = e−R T

tλ(t,u) du.

Clearly, the hazard rate at time t is (Ft)t≥0-adapted.The hazard rate has an important interpretation: formally, λ(t, T ) dT is the

instantaneous probability of default at T conditional on no default until T (but nofurther information), for, letting Q be a regular version of P(·|Ft),

lim∆↓0

1∆

Q(τ ≤ T + ∆|τ > T ) = lim∆↓0

1∆

Q(τ ∈ T + ∆)Q(τ > T )

= λ(t, T ).

The hazard rate bears a close analogy to the instantaneous forward rate ininterest rate modelling. Indeed, recall the valuation formula for a risky zero bondwith maturity T , Example 3.4. Denote by f(t, T ), T ≥ t, the instantaneous forwardrate curve at time t. Then, assuming independence of interest rates and the survivalprocess (1τ>T)T≥t, on τ > t,

Vt = B(t, T )Q(t, T ) = e−R T

t[f(t,u)+λ(t,u)] du.

1The term hazard rate is ambiguous in the literature: some authors consider the hazard rate tobe a synonym for default intensity, which we encounter later.


From Equation (3.4.2) we obtain the following relationship between λ(t, ·) andthe CDS spread s(t, T ):

s(t, T )1−R

∫ T

t

B(t, u)Q(t, u) du =∫ T

t

λ(t, u)B(t, u)Q(t, u) du. (4.2.1)

A simple and useful approximation is the credit triangle, which describes therelationship between the credit spread, a constant hazard rate and the recoveryrate.2 Assuming a constant hazard rate λ, the fair spread is constant regardless ofthe maturity so that, setting s(t, T ) = s, T ≥ t, Equation (4.2.1) becomes

s

1−R= λ. (4.2.2)

Observe that in this case Q(t, T ) = e−λ(T−t), T ≥ t.We shall often consider the term hazard rate at time t, λ(t, T ), T > t, defined

by

λ(t, T ) =

∫ Ttλ(t, u) duT − t

. (4.2.3)

The term hazard rate fulfills

e−R T

tλ(t,u) du = e−λ(t,T ) (T−t) (4.2.4)

λ(t, T ) = − ln(1− P (t, T ))T − t

, T > t. (4.2.5)

The relationship (4.2.5) makes computation of dynamics, statistics and other quanti-ties of λ(t, T ) particularly easy if the corresponding quantities of P (t, T ) are known.

4.3 Structural models

The remainder of this chapter contains a brief account of the two established modeltypes for credit risk, structural and reduced-form models.

4.3.1 Merton model

The classical paradigm of structural models is the assumption that observable eco-nomic variables, such as the value of the firm under consideration, trigger default.The approach was introduced by [Merton (1974)], where the principal idea is thatthe value of a firm’s assets and its debt are traded assets. Let (Ω,F , (Ft)t≥0,P)be a filtered probability space that satisfies the usual hypotheses and that is richenough to accommodate a Brownian motion W . As before, P is assumed to be arisk-neutral measure. The firm’s asset value V is modelled as a Geometric Brownianmotion

Vt = V0 exp((r − 1/2σ2

)t+ σWt

), t ≥ 0,

2The term credit triangle is taken from [O’Kane (2008), Section 3.10].

4.3. Structural models 33

where r denotes the constant default-free interest rate and σ is the volatility of theasset value. The firm’s debt is a zero-coupon bond with face value D and maturityT . The difference between V and the value of the zero-coupon bond is the firm’sequity. At T , the payoffs to the bond holders and to the equity holders are

BT = min(D,VT ) = D −max(D − VT , 0)

ST = max(VT −D, 0).

For t ≤ T , the values of Bt and St are computed according to the Black-Scholesformula. The yield spread of the risky bond can be computed using the bond price.

The event VT < D corresponds the default event, as the bond holders donot receive the face value of their debt. The conditional probability of default atmaturity T can be computed explicitly, and is given by

P(τ ≤ T |Ft) = P(VT ≤ D|Ft) = N(− ln(Vt/D)− r(T − t) + 1/2σ2(T − t)

σ√T − t

),

for t ≥ T , where N denotes the standard normal distribution function.

4.3.2 First-passage time models

In the Merton model default can only occur at the debt’s maturity T . In a first-passage time model , default occurs at the first point in time that the firm value Vhits a default threshold (not necessarily the face value of the debt). Computation ofdefault probabilities is based on the hitting time distribution of Brownian motion.This class of models was introduced by [Black and Cox (1976)] whose principalidea is as follows: Let dVt/Vt = (r − κ) dt + σ dWt, where κ represents the firm’spayout ratio. For γ, K and a fixed time horizon Υ, define the barrier function tobe v(t) = K e−γ(Υ−t), t ≥ 0. Define the stopping time τ = inft ≥ 0 : Vt ≤ v(t).Then, for every t < T , on the set τ > t, setting ν = r − κ− γ − 1/2σ2,

P(τ ≤ T |Ft) = N(− ln(Vt/v(t))− ν(T − t)

σ√T − t

)+ e−2ν σ−2 ln(Vt/v(t)) N

(− ln(Vt/v(t)) + ν(T − t)

σ√T − t

).

cf. [Bielecki and Rutkowski (2002), Section 3.1.1]. There are many extensions,for example the approach by [Longstaff and Schwartz (1995)], which incorporatesstochastic interest rates that are correlated with the firm’s asset value.

Often the firm value and the barrier function are continuous, in which case creditspreads vanish as time-to-maturity tends to zero, see e.g. [Lando (2004), Section2.2.2], contradicting empirical observation, see Section 3.1.2. This can be overcomeby introducing jumps in the asset value, as in [Merton (1976); Zhou (2001); Hilberinkand Rogers (2002)]. Often, tractability of the model declines. For example, in [Zhou(2001)], where the asset value follows a jump-diffusion, valuation of a risky zero bondrequires Monte Carlo simulation. Recent approaches involving Levy-processes are


[Kiesel and Scherer (2007); Cariboni and Schoutens (2007); Baxter (2007); Madanand Schoutens (2007); Garcia et al. (2007)]. An empirical analysis of the capabilitiesof several structural models is found in [Eom et al. (2004)].

In a more general setting, we may consider a model to be of structural type whendefault is the first hitting time of a certain threshold by an abstract observable creditquality process, where we understand “observable” as the property of being adaptedto the underlying flow of information.

4.4 Reduced-form models

In the reduced-form approach, the default event is not directly linked to economicobservables, but it is an unpredictable Poisson event. This approach is followed by[Jarrow and Turnbull (1995); Madan and Unal (1998); Lando (1998); Duffie andSingleton (1999); Duffie and Lando (2001)] and many others. Its main advantageslie in the capability of reproducing a given credit spread term structure well and inits close analogy to interest-rate term structure modelling.

The overall idea is to model the (default) intensity process of a doubly stochasticprocess or Cox process (which is a pure jump process) and consider the default timeto be the first jump of this process.

More specifically, as before let (Ω,F ,P) be a probability space. Now considera filtration (Gt)t≥0 that represents the information available to investors. Thisfiltration is sometimes called the background filtration. Let λ = (λt)t≥0 be a (Gt)t≥0-adapted nonnegative process, and let N be a jump process with λ the Ft-intensity ofN , where Ft = Gt ∨Ht and (Ht)t≥0 = σ(Ns, 0 ≤ s ≤ t). Default occurs at the timeof the first jump of N . It turns out that 1τ≤t−

∫ τ∧t0

λs ds is an (Ft)t≥0-martingale;λ is called the stochastic intensity for the jump time τ .

In this framework the price of a risky zero-coupon bond with maturity T is

V0 = E

(exp

(−∫ T

0

[rs + λs] ds

)),

which highlights the analogy of the intensity process to the short rate. Moreover,conditional probabilities of survival at t are given by

Q(t, T ) = P(τ > T |Ft) = 1τ>tE

(exp

(−∫ T

t

λs ds

)∣∣∣Gt) , T ≥ t.

A predominant feature of reduced-form models is that for τ to have an inten-sity, it must be a totally inaccessible stopping time, which implies that it is notpredictable, see Appendix A for the definitions.

Many extensions exist; for a systematic development of the mathematical tools,see e.g. [Elliott et al. (2000); Bielecki and Rutkowski (2002)]. [Duffie and Lando(2001)] link the structural and the reduced-form approach by assuming that in-vestor’s do not have access to full information about the firm’s assets. This is anexample of a structural model for which a default intensity process exists.

Chapter 5

Overbeck-Schmidt model

The Overbeck-Schmidt model [Overbeck and Schmidt (2005)] (OS-model) allowsfor straightforward analytic calibration to a given term structure of default prob-abilities. The model is intended for pricing credit derivatives whose payoff doesnot depend on spread risk. However, the model does exhibit dynamics, which weexplore to some extent. A thorough analysis of the dynamics is found in [Kam-mer (2007), Chapter 5]. [Kammer (2007)] also studies extensions of the Overbeck-Schmidt model for modelling the dynamics of credit spreads in a purely continuoussetting.

5.1 Model specification

We specify the Overbeck-Schmidt model and give an overview of some of the resultsfound in [Overbeck and Schmidt (2005)]. In the Overbeck-Schmidt model, thedefault time τ is determined as the first time that a credit quality process X =(Xt)t≥0 hits a barrier b < X0, i.e.,

τ = inft ≥ 0 : Xt ≤ b. (5.1.1)

The principal idea is to model X as a time-changed Brownian motion. Given aBrownian motion B and a deterministic, strictly increasing and continuous timetransformation Λ = (Λt)t≥0, with Λ0 = 0, set

Xt := BΛt , t ≥ 0. (5.1.2)

Assume given the distribution of the default time, F (t), t ≥ 0. If the time-changeΛ is given by

Λt =

b

N(−1)(F (t)

2

)2

, t ≥ 0, (5.1.3)

where N(−1) denotes the inverse of the Normal distribution function, then τ , definedby Equation (5.1.1), admits the distribution F (t), t ≥ 0. (This follows easily by

35

36 Overbeck-Schmidt model

Equation (5.2.1) below.) Furthermore, if the distribution of τ , defined by Equation(5.1.1), admits a density, then the time change Λ is absolutely continuous, and

Λt =∫ t

0

σ2s ds, (5.1.4)

with σ : [0,∞) → [0,∞) a square-integrable function. The quadratic variation[X,X] of X is just [X,X] = Λ, so that there exists a representation of X as astochastic integral

Xt =∫ t

0

σs dWs, (5.1.5)

for some Brownian motion W . The volatility σ can be interpreted as the defaultspeed in the sense that the higher the default speed the higher the likelihood ofcrossing the default barrier.

In [Overbeck and Schmidt (2005)], the model is used to value products whosepayoff does not depend on the level of CDS spreads; for example, by modellingseveral correlated credit quality processes, one can price multiname products suchas first-to-default credit baskets. Although the OS-model is not intended to valuespread products, it exhibits dynamics by specification of the process X. Thesedynamics are fully determined by calibration to market-given default probabilities,and it is not possible to assign different dynamics to the same term structure ofdefault probabilities. We shall extend the OS-model to allow for different dynamics.It is interesting in its own right to study the properties and dynamics of the OS-model first.

5.2 Conditional default probabilities

Recall from Section 3.3 that P (t, T ) = P(τ ≤ T |Ft), T ≥ t, is a regular distributionfunction, and that (1τ>tP (t, T ))t≥0 is a cadlag semimartingale. In this section,we derive some properties of conditional default probabilities in the OS-model. Itis easy to see that the default time is (Ft)t≥0-predictable. We state a formula forP (t, T ), and we show that the default probability process (P (t, T ))t≥0 is continuous.A consequence of the last statement is that CDS spreads evolve in a continuousmanner, as is easily checked with the CDS valuation formula (3.4.2).

Proposition 5.1. Let X = (Xt)t≥0, with Xt = BΛt and (Λt)t≥0 given by Equation(5.1.3), and let τ = inft ≥ 0 : Xt ≤ b, b < X0. Then:

(i) τ is an (Ft)t≥0-predictable stopping time (cf. Definition A.9).(ii) On τ > t, P (t, T ) = P(τ ∈ (t, T ]|Xt); in particular, P (t, T ) is σ(Xt)-

measurable on τ > t.(iii) For t ≤ T , on τ > t, the probability of default until T conditional on Ft is

given by

P (t, T ) = 2N(

b−Xt√ΛT − Λt

). (5.2.1)

5.2. Conditional default probabilities 37

(iv) For any T ≥ 0, the conditional default probability process (P (t, T ))t≤T iscontinuous in t.

Proof of Proposition 5.1.

(i) Define τn = inft > 0 : Xt ≤ b + 1/n, n ≥ 1. It is straightforward to checkthat (τn)n≥1 announces τ .

(ii) Recall Equation (3.3.1),

P (t, T ) = 1τ≤t + 1τ>tP(τ ∈ (t, T ]|Ft).

The events τ ∈ (t, T ] and mint<u≤T Xu ≤ b are equivalent, and the claimfollows from the Markov property of X.

(iii) The hitting-time distribution of a Brownian motion starting at 0 is, cf.[Karatzas and Shreve (1998), Section 2.6.A],

P(

mins≤t

Bs < b

)= 2 N

(b√t

), b < 0. (5.2.2)

On τ > t, making use of the fact that X is a Markov process and takinginto account that the time-change Λ is continuous,

P (t, T ) = P(τ ∈ (t, T ]|Ft) = P(

mint<u≤T

Xu ≤ b∣∣Ft)

= P(

mint<u≤T

Xu ≤ b∣∣Xt

)= P

(mint<u≤T

BΛu −BΛt ≤ b−BΛt

∣∣BΛt

)= P

(min

0<u≤ΛT−Λt

BΛt+u −BΛt≤ b−BΛt

∣∣BΛt

)= 2N

(b−BΛt√ΛT − Λt

),

where the last step is an application of the Independence Lemma A.17, since(BΛt+u −BΛt

)u≥Λtis a Brownian motion independent of Ft and (b−BΛt

) isFt-measurable.

(iv) Taking into account that N(·), X,√· and Λ are continuous, for any sequence

tn → t, as n→∞, on τ > t,

limtn→t

P (tn, T ) = limtn→t

2N

(b−Xtn√ΛT − Λtn

)= 2N

(b−Xt√ΛT − Λt

)= P (t, T ).

On τ < T, to analyse the behaviour around default, observe that by thecontinuity of X and since Xτ = b, it follows easily that limtn↑τ P (tn, T ) = 1,as n→∞ by Equation (5.2.1).

Inspection of Equation (5.2.1) reveals that P (t, T ) depends on the time-changeincrement and on the distance of Xt to the barrier b. The time-change, beingdeterministic, has limited impact on the dynamics of (P (t, T ))t≥0.

The last part of the Proposition tells us that it is impossible to generate jumps ina default probability process when the time-change is deterministic and continuous.It is thus clear that the model is unsuitable for valuing gap risk – the probability


of a gap event in the OS-model is zero. It is not hard to see that for deterministictime-changes with discontinuities any jumps in a default probability process aredeterministic (gap events are predictable in this case).

Furthermore, an important consequence of Proposition 5.1 is that in the set-ting of the OS-model a necessary condition for jumps at random times in defaultprobability processes is that the time-change be stochastic.

5.3 Short-term hazard rate and short-term credit spread

Recall from Section 4.2 that the hazard rate corresponds to the instantaneous prob-ability of default conditional on no prior default. The behaviour of the hazard ratein the OS-model at vanishing time-to-maturity gives us insight into the short-timebehaviour of the credit spread via the relationship (4.2.1).

Fix a time t, assume that τ > t, and let P (t, T ), T > t, be a term structureof default probabilities satisfying Equation (5.2.1). Suppose further that the time-change Λ is absolutely continuous with Λ =

∫ ·0σ2s ds. This implies that P (t, T ),

T > t, is differentiable almost everywhere, cf. Equation (5.2.1). (For simplicity,assume that P (t, T ) is differentiable for every T > t.) The hazard rate λ(t, T ) is

λ(t, T ) =∂∂T P (t, T )

1− P (t, T ), T > t. (5.3.1)

Proposition 5.2. Let λ(t, T ) be the hazard rate as given by Equation (5.3.1) andlet s(t, T ) be the credit spread with maturity T at time t. Then, on τ > t, asT ↓ t, λ(t, T )→ 0 and s(t, T )→ 0.

Proof. For the numerator of Equation (5.3.1), denoting by n(·) the Normal densityfunction,

∂

∂TP (t, T ) =

∂

∂T2N(


)= − b−Xt

(ΛT − Λt)3/2n(


)σ2T ,

and for the latter (discarding any constants),

1(ΛT − Λt)3/2

e−1/(2(ΛT−Λt))σ2T = e−1/(2(ΛT−Λt))−ln((ΛT−Λt)

3/2)σ2T .

For the exponent,

− 12(ΛT − Λt)

− 3/2 ln(ΛT − Λt) = −1− 3 ln(ΛT − Λt)(ΛT − Λt)2(ΛT − Λt)

→ −∞ as T ↓ t,

since ln(ΛT − Λt)(ΛT − Λt)→ 0, as T → t. Since σ2T <∞, it follows that

limT↓t

∂

∂TP (t, T ) = 0.

5.4. Dynamics of conditional default probabilities 39

For the denominator of Equation (5.3.1), limT↓t(1−P (t, T )) = 1, so that λ(t, T )→0, as T ↓ t, follows. For the spread, observe that by Equation (4.2.1),

s(t, T )1−R

≤ supt<u≤T

λ(t, u),

and the claim follows, since limT↓t supt<u≤T λ(t, u) = 0. (The last statement iseasily seen since λ(t, T ) → 0 as T ↓ t implies that for any ε > 0 there exists δ > 0such that for all T ≥ t with |T − t| < δ it holds that λ(t, T ) < ε.)

It also follows that the term hazard rate from Equation (4.2.3) vanishes:

Corollary 5.3. Let λ(t, T ), T > 0, be the term hazard rate at time t. Then, onτ > t, limT↓t λ(t, T ) = 0.

Proof. The claim follows since λ(t, T ) ≤ supt<u≤T λ(u, T ) andlimT↓t supt<u≤T λ(t, u) = 0.

The limit of the spread was obtained by fixing a term structure and analysing thebehaviour as time-to-maturity of the spread tends to zero. Now fixing a maturityT , consider the credit spread s(t, T ) as t ↑ T . Assuming that the short rate evolvescontinuously, it is not hard to establish that the paths of s(t, T ) are continuous(the paths of P (t, T ) are continuous for all T > t, cf. Equation (5.2.1); now applyEquation (3.4.2)). The next statement then follows directly.

Corollary 5.4. Let s(t, T ) be the credit spread with maturity T at time t. Then,on τ > T, limt↑T s(t, T ) = 0.

5.4 Dynamics of conditional default probabilities

Proposition 5.5. Assume that the time transformation Λ is absolutely continuous,i.e., Λ has a representation as in Equation (5.1.4). On τ > 0, for fixed T > 0,

P (t, T ) = P (0, T )−∫ t

0

1τ>u2√

ΛT − Λun(

b−Xu√ΛT − Λu

)σu dWu, t ≥ 0.

(5.4.1)

Proof. Consider first 1τ>tP(τ ∈ (t, T ]|Xt). Define

g(t, x, z) := z 2N(

b− x√ΛT − Λt

).


The corresponding partial derivatives are

gx(t, x, z) = −z 2√ΛT − Λt

n(


)gxx(t, x, z) = −z 2(b− x)

(ΛT − Λt)3/2n(


)gz(t, x, z) = 2N

(b− x√ΛT − Λt

)gt(t, x, z) = −z b− x

(ΛT − Λt)3/2n(


)· Λ′t = −1

2gxx σ

2t .

Observe that gx and gz are continuous.1 By application of the Ito formula, cf.Theorem A.8,

g(t,Xt,1τ>t) = g(0, 0, 1)−∫ t

0

12gxx σ

2u du+

∫ t

0

gx dXu +∫ t

0

gz d1τ>u

+12

∫ t

0

gxx d[X,X]u +∑

0<u≤t

(∆g(u,Xu,1τ>u)− gz∆1τ>u

).

Since∑

0<u≤t ∆g <∞, we may separate the terms in the sum and simplify accord-ingly. Moreover, for any u < T ,

∆g(u,Xu,1τ>u) = 1τ>u2N(


)− lim

s↑u1τ>s2N

(b−Xs√ΛT − Λs

)=

−1, if τ = u,

0, else.

Finally,

1τ>tP(τ ∈ (t, T ]|Xt)

= P(τ ≤ T )−∫ t

0

1τ>u2√

ΛT − Λun(


)σu dWu − 1τ≤t,

and Equation (5.4.1) now follows by part (ii) of Proposition 5.1 and P (0, T ) =P(τ ≤ T ).

We can make some interesting observations from Equation (5.4.1). First, themartingale property of (P (t, T ))T>t is easily established. We also know thatP (t, T ) ∈ [0, 1] for any t ≥ 0, and that P (T, T ) = 1τ≤T ∈ 0, 1. These propertiesare guaranteed by the level of the integrand of the Brownian integral, which de-pends on the distance-to-default b−Xu and the remaining time-change incrementuntil maturity ΛT − Λu.1Observe that Λ′t exists only for almost all t ∈ (0, T ]. For any t where Λ′t does not exist, we just

define gt(x, y, z) = −1/2gxxσ2t . Moreover, gt may fail to be continuous, which is required to apply

the Ito formula. However, we shall see that the terms involving gxx cancel, so they do not affectthe dynamics anyway.

5.5. Hazard rate dynamics 41

5.5 Hazard rate dynamics

Using the dynamics of (P (t, T ))t≥0, it is straightforward to compute the dynamicsof the term hazard rate λ(t, T ) as defined by Equation (4.2.3). Recall Equation(4.2.5),

λ(t, T ) = − ln(1− P (t, T ))T − t

, T > t, on τ > t.

The reason for computing the dynamics of λ(t, T ) is that it allows for an interestingobservation (see Remark 5.7 below), which we encounter again later, when comput-ing the dynamics of credit spreads in the more general model. For the dynamics ofcredit spreads in the OS-model see [Kammer (2007), Section 5.4].

Proposition 5.6. Fix T > 0, and let λ(t, T ) be as in Equation (4.2.5), t < T .Assume that the time transformation Λ is absolutely continuous and given by Λt =∫ t0σ2u du, t ≥ 0. Then, on τ > t,

λ(t, T ) = λ(0, T ) +∫ t

0

λ(u, T )T − u

+n(

b−Xu√ΛT−Λu

)2

(ΛT − Λu)(1− P (t, u))2(T − u)σ2u

du

−∫ t

0

2n(

b−Xu√ΛT−Λu

)√

ΛT − Λu(1− P (t, u))(T − u)σu dWu. (5.5.1)

Proof. Define

f(p, t) = − ln(1− p)T − t

.

The corresponding partial derivatives are

ft(p, t) = − ln(1− p)(T − t)2

=f(p, t)T − t

fp(p, t) =1

(1− p)(T − t)

fpp(p, t) =1

(1− p)2(T − t).

The claim follows by applying the Ito formula

λ(t, T ) = f(P (t, T ), t) = f(0, t) +∫ t

0

ft(P (u, T ), u) du

+∫ t

0

fp(P (u, T ), u) dP (u, T ) +12

∫ t

0

fpp(P (u, T ), u) d[P (·, T ), P (·, T )]u,

together with the dynamics of P (t, T ) from Proposition 5.5.


Remark 5.7. Observe that the drift term in Equation (5.5.1) is nonnegative. Onthe other hand, we have established that limt→T λ(t, T ) = 0 P–a.s. on τ > T,which further implies limt→T E(λ(t, T ) = 0) on τ > T; The co-existence of thesestatements is not obvious. To understand this behaviour heuristically recall thatτ > t = min0<s≤tXs > b, and the process λ(·, T ) is defined only on the subsetof Ω where the paths of X =

∫ ·0σu dWu are above the default barrier. Furthermore,

the integrands of both integrals entering Equation (5.5.1) must ensure that λ(t, T )vanishes as t tends to T .

5.6 Distribution of conditional default probabilities

We compute the distribution of P (t, T ). Recall from part (ii) of Proposition 5.1that P (t, T ) = 1τ≤t + 1τ>tP(τ ∈ (t, T ]|Xt). The following identity is useful forthe results of this section: for 0 ≤ x < 1,

P (t, T ) ≤ x = P(τ ∈ (t, T ]|Xt) ≤ x, τ > t . (5.6.1)

Proposition 5.8. For T ≥ 0 and t ≤ T , the distribution of P (t, T ) is

P(P (t, T ) ≤ x) = N (h1(x))−N (h2(x)) , 0 ≤ x < 1,

with

h1(x) =N(−1)

(x2

)√ΛT − Λt − b√Λt

h2(x) =N(−1)

(x2

)√ΛT − Λt + b√

Λt.

Proof. Using the fact that Λ is continuous, by Equation (5.6.1),

P (P (t, T ) ≤ x) = P(

2N(


)≤ x, min

0<s≤tXs > b

)= P

(2N(

b−BΛt√ΛT − Λt

)≤ x, min

0<s≤Λt

Bs > b

)= P

(BΛt ≥ −

(N(−1)

(x2

)√ΛT − Λt − b

),mΛt > b

),

where mt = min0<s≤tBs denotes the running minimum of Bt. For a Brownianmotion B and its running maximum Mt = max0<s≤tBs we have [Karatzas andShreve (1998), Section 2.8.A],

P(Bt ≤ a,Mt ≥ b) =1√2πt

∫ ∞

2b−ae−x

2/(2t) dt, a ≤ b, b ≥ 0.

5.6. Distribution of conditional default probabilities 43

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2

P

P(τ ∈ (t, T ]|Xt)

1%2%3%

0

0.2

0.4

0.6

0.8

1

0 0.01 0.02 0.03 0.04 0.05 0.06

P

λ(t, T )

1%2%3%

Fig. 5.1 Distributions of P(τ ∈ (t, T ]|Xt) (left) and λ(t, T ) (right) conditional on τ > t,cf. Corollary 7.15 and Proposition 5.11. The value of λ(t, T ) is an approximation of the creditspread (in percent; s(t, T ) ≈ λ(t, T ) · 104). Parameters are t = 1, T = 5 and initial hazard rates1%, 2%, 3%. The diamonds mark the initial 5-year default probability, resp. hazard rate. Thechoice of the barrier is arbitrary, see Remark 5.10.

By symmetry, for a Brownian motion B started at 0, the events Bt ≥ x,mt ≤ band Bt ≤ −x,Mt ≥ −b have the same probability. Continuing above, yields

P(BΛt≥ −

(N(−1)

(x2

)√ΛT − Λt − b

),mΛt > b

)= P

(BΛt ≥ −N(−1)

(x2

)√ΛT − Λt + b

)−P

(BΛt ≥ −N(−1)

(x2

)√ΛT − Λt + b,mΛt ≤ b

)= N

(N(−1)

(x2


)−

(1−N

(−2b−N(−1)

(x2

)√ΛT − Λt + b

√Λt

))

= N

(N(−1)

(x2


)−N

(N(−1)

(x2

)√ΛT − Λt + b√

Λt

).

The density can be obtained by differentiation.From Equation (5.6.1) we get the following result:

Corollary 5.9. For T ≥ 0 and t ≤ T , the distribution of P(τ ∈ (t, T ]|Xt) condi-tional on τ > t is, for 0 ≤ x < 1,

P(P(τ ∈ (t, T ]|Xt) ≤ x

∣∣∣τ > t)

=P(P (t, T ) ≤ x)

P(τ > t).

An example of conditional default distributions is given in the left picture of Fig-ure 5.1. Here, the probability distribution of P(τ ∈ (t, T )|Xt) with t = 1 andT = 5 is shown, for initial default distributions F (T ) = 1 − e−hT , T ≥ 0,h ∈ 0.01, 0.02, 0.03. Consider for example the distribution of the default proba-bility with h = 0.03. The initial 5-year default probability is approx. 0.14. With


a probability of roughly 1/2 the default probability in one year is below 0.05, andwith a probability of roughly 1/5 it is above 0.2. Loosely speaking, with a highprobability, the underlying entity will have either a very high or a very low creditquality in one year. This suggests a very volatile movement of the default probabil-ity process, which is explained as follows: In order to match the initial 5-year defaultprobability, the credit quality process must adopt a high default speed. The highvolatility of the credit quality process then leads to a volatile default probabilityprocess.

Apart from the fact that these dynamics are fully specified by calibration to thegiven default probability distribution, we may ask how we can obtain distributionsthat are more realistic. The first issue will be solved by incorporating a stochas-tic volatility (i.e., stochastic time-change); the second issue will be addressed byincorporating jumps into the stochastic volatility.

Remark 5.10. An interesting point to note is that the dynamics are not influencedby the choice of the barrier b, although at first sight of Proposition 5.8, one maybe lead to think so. This follows from the relationship between the barrier and thetime transformation. Indeed, using Equation (5.1.3), we can re-write h1 and h2 asfollows: For any T ≥ t,

h1,2(x) =N(−1)(x/2)

(b

N(−1)(P(τ≤T )/2)− b

N(−1)(P(τ≤t)/2)

)∓ b

bN(−1)(P(τ≤t)/2)

= N(−1)(x/2)(

N(−1)(P(τ ≤ t)/2)N(−1)(P(τ ≤ T )/2)

− 1)∓N(−1)(P(τ ≤ t)/2). (5.6.2)

5.7 Distribution of term hazard rate

The distribution of the term hazard rate λ(t, T ), which exists only conditional onτ > t, is computed easily using Corollary 7.15.

Proposition 5.11. For T ≥ 0 and t ≤ T , let g(x) = 1 − e−x(T−t), x ≥ 0. ByEquation (4.2.4), λ(t, T ) is such that P(τ ∈ (t, T ]|Xt) = g(λ(t, T )) (clearly, λ(t, T )is σ(Xt)-measurable). Then, for x ≥ 0,

P(λ(t, T ) ≤ x|τ > t) =(

N(h1(g(x))

)−N

(h2(g(x))

)) (1− 2N

(b√Λt

))−1

with h1, h2 as in Proposition 5.8.

Proof. Since g is monotone increasing,

P(λ ≤ x

∣∣τ > t)

= P(g(λ) ≤ g(x)

∣∣τ > t)

= P(P(τ ∈ (t, T ]|Xt) ≤ g(x)

∣∣∣τ > t).

Application of Corollary 7.15 establishes the claim.

5.8. Extensions of the Overbeck-Schmidt model 45

The density may be obtained by differentiation. For an example of the term hazardrate distribution see Figure 5.1.

Using Proposition 5.11, it is easy to show that λ(t, T )→ 0 as t→ T . For x > 0,and using Equation (5.6.2) for h1,2,

limt↑T

h1,2(x) = ∓N(−1)

(P(τ ≤ T )

2

).

We obtain

limt↑T

P(λ ≤ x|τ > t) =N(−N(−1)(P(τ ≤ T )/2))−N(N(−1)(P(τ ≤ T )/2))

1−P(τ ≤ T )= 1,

and the claim follows since x > 0 was arbitrary.

5.8 Extensions of the Overbeck-Schmidt model

[Kammer (2007)] studies extensions of the OS-model with a stochastic time-change.She treats several examples of time-changes, such as a CIR-type time-change drivenby a Brownian motion. The dynamics of the credit spread process are computed,and multivariate models incorporating several correlated credit quality processes areconsidered. The time-changes considered have in common that the resulting defaultprobability processes and spread processes are continuous, so that the issue of thevery volatile movement of the processes remains (cf. Figure 5.1 and the commentsin the text).

Chapter 6

A first-passage time model with jumps

We have seen in the last chapter that the dynamics of the Overbeck-Schmidt modelare unsuitable for valuing spread products. The principal reason for this is thatthe dynamics are completely determined by calibrating the model to a given term-structure of credit spreads. We have also seen that the model is unable to producerandom jumps in credit spreads (Proposition 5.1, part (iv)), and that default prob-abilities and the hazard rate are quite volatile (Figure 5.1). On the other hand, theOS-model is easy to calibrate (Equation (5.1.3)) and conditional default probabili-ties are easily computed (Proposition 5.1, part (iii)).

The natural question is then: Can we extend the OS-model to allow for mean-ingful dynamics, while at the same time maintaining its tractability?

More specifically, in the OS-model, if X is a credit quality process (Equations5.1.2 or 5.1.5), then P (t, T ) = F (t, T,Xt) for some function F by the Markovproperty of X (Proposition 5.1), and (P (t, T ))t≤T is driven by the credit qualityprocess X (Proposition 5.5). The dynamics of X are completely determined bycalibration to the spot curve, however. We rephrase our question as follows: Can weenrich the dynamics in the sense that P (t, T ) = F (t, T,Xt, Yt), so that (P (t, T ))t≤Tis driven by a Markov process (X,Y ) with X the credit quality process and Y someother process that produces desired dynamics?

We proceed as follows: The credit quality process X will be defined as a stochas-tic integral with respect to a Brownian motion, X =

∫ ·0σs dWs, where the integrand

σ is a cadlag stochastic process. If, for example, σ is the solution to an SDE drivenby a Levy process, then, under some mild conditions, (X,σ) is a Markov processand σ plays the role of the process Y mentioned above. We derive a formula for con-ditional default probabilities. In a more concrete setting, we specify the volatilityprocess σ as the square root of a Levy-driven Ornstein-Uhlenbeck process. Typ-ically, the driving Levy process will be a compound Poisson process; this reflectsthe occurrence of jumps as rare events. This setup will be our working examplethroughout, which we analyse in detail in the following chapters.

47

48 A first-passage time model with jumps

6.1 Credit quality process with stochastic volatility

Throughout, let (Ω,F , (Ft)t≥0,P) be a complete filtered probability space thatsatisfies the usual hypotheses and that is rich enough to accommodate any objectsthat we define.

Definition 6.1. The credit quality process X = (Xt)t≥0 of a risky entity is definedto be

Xt =∫ t

0

σs dWs, t ≥ 0,

where W is an (Ft)t≥0-Brownian motion and σ is a strictly positive (Ft)t≥0-adapted cadlag process independent of W with P(

∫ t0σ2s ds < ∞) = 1, t ≥ 0, and

limt→∞∫ t0σ2s ds =∞ P–a.s..1

To emphasise the association with σ, we speak of X as a credit quality process withvolatility σ. To ease notation we may also speak of (X,σ) as a credit quality process.Denote the quadratic variation process of X by Λ = (Λt)t≥0, with Λt =

∫ t0σ2s ds.

Observe that Λ is continuous, strictly increasing and (Ft)t≥0-adapted.Define the family of (Ft)t≥0-stopping times τt = infs ≥ 0 : Λs > t, t ≥ 0.

By application of the Theorem of Dambis, Dubins-Schwarz (see Theorem 2.2) theprocess B, with Bt = Xτt , t ≥ 0, is an (Fτt)-Brownian motion. Conversely, given B,the credit quality process X can be expressed as a time-changed Brownian motion,Xt = BΛt , t ≥ 0. We refer to B as the DDS-Brownian motion of X. By Section 2.1,for every t > 0, Λt is an (Fτt

)t≥0-stopping time. Moreover, since Λ is continuous,strictly increasing and limt→∞ Λt =∞ P–a.s., it follows that τΛt

= Λτt= t.

The default time τ of the risky entity associated with the credit quality processX is the first time that X hits a barrier b < 0:

τ = inft ≥ 0 : Xt ≤ b.

Clearly, the credit quality process of Definition 6.1 is a generalisation of theOS-model with an absolutely continuous time-change.

6.2 Conditional default probabilities

We analyse the properties of conditional default probabilities as we did for the OS-model in Section 5.2. In particular, we are interested in deriving results analogousto Proposition 5.1. As in the OS-model, τ is an (Ft)t≥0-predictable stopping time;the proof is the same as in Proposition 5.1. For an analogue of the last part ofProposition 5.1, which states that the default probability process (P (t, T ))t≥0 can-not jump, we have to be more specific about the volatility process. We shall showin Section 7.1 that, although the credit quality process is continuous, it is possible1The requirement limt→∞

R t0 σ2

s ds = ∞ P–a.s. will ensure that τ < ∞ P–a.s..

6.2. Conditional default probabilities 49

to specify a model in which default probability processes and credit spreads exhibitjumps. In the following, we state a formula for conditional default probabilities,analogous to Equation (5.2.1). Moreover, if (X,σ) is a Markov process, then onτ > t, P (t, T ) is σ(Xt) ∨ σ(σt)-measurable.

By Proposition 2.4, a credit quality process as defined above is an Ocone mar-tingale, i.e., its DDS-Brownian motion B and associated time-change Λ are inde-pendent. This property together with Corollary 2.7 are the key for establishing aformula for conditional default probabilities P (t, T ) = P(τ ≤ T |Ft), t ≥ 0, T > t.

Proposition 6.2. Let X be a credit quality process with volatility process σ. Letτ = inft ≥ 0 : Xt ≤ b be the associated default time. On τ > t, the probabilityof default until time T > t, conditional on Ft, is given by

P(τ ≤ T |Ft) = E(

2N(


) ∣∣∣Ft) P–a.s.. (6.2.1)

Proof. Let B be the DDS-Brownian motion of X, and recall that BΛt = Xt, t ≥ 0.By continuity of Λ and by properties of conditional expectation, P–a.s., on τ > t,

P(τ ≤ T |Ft) = P(

mint<u≤T

Xu ≤ b∣∣∣Ft) = P

(mint<u≤T

BΛu ≤ b∣∣∣Ft)

= P(

minΛt<u≤ΛT

Bu ≤ b∣∣∣Ft) = P

(min

0<u≤ΛT−Λt

BΛt+u ≤ b∣∣∣Ft)

= P(

min0<u≤ΛT−Λt


∣∣∣Ft)= E

(P(

min0<u≤ΛT−Λt


∣∣∣Ft ∨ σ(ΛT )) ∣∣∣Ft) .

(6.2.2)

The random time Λt is an (Fτt)t≥0-stopping time, and with FτΛt

= Ft it followsfrom Corollary 2.7 that (BΛt+u − BΛt

)u≥0, is a Brownian motion independent ofFt ∨ σ(ΛT ) ⊆ Ft ∨ Fσ∞. On the other hand, the random variables ΛT − Λt andb − BΛt are Ft ∨ σ(ΛT )-measurable. By Lemma A.17 and the first-passage timedistribution of Brownian motion, cf. Equation (5.2.2), P–a.s.,

P(

min0<u≤ΛT−Λt

BΛt+u −BΛt ≤ b−BΛt

∣∣∣Ft ∨ σ(ΛT ))

= 2N(

b−BΛt√ΛT − Λt

).

Inserting into Equation (6.2.2) yields Equation (6.2.1).

Remark 6.3. It is easily verified that the conditional default distribution is well-defined, i.e., that the mapping T 7→ P (t, T ) fulfills the properties of a distributionfunction, for every t ≥ 0. This follows from the condition that σ be strictly positive,which implies that the time-change Λ is strictly increasing. Moreover, it is easilyseen that the no-arbitrage constraint P (t, T1) < P (t, T2), for T1 < T2, (cf. Equation(4.1.1)) is satisfied.


Corollary 6.4. Let X be a credit quality process with volatility process σ, andassume further that (X,σ) has the Markov property. Let τ be the associated defaulttime. Then, for T > t, on τ > t, the conditional default distribution is

P(τ ≤ T |Ft) = P(τ ≤ T |Xt, σt) = E(

2N(


) ∣∣∣Xt, σt

)P–a.s..

(6.2.3)

Proof. For the first step, observe that on τ > t the events t < τ ≤ T andmint<u≤T Xu ≤ b are equal, and the latter is conditionally independent of Ftgiven (Xt, σt) by the Markov property. For the second step, taking into accountthat ΛT − Λt =

∫ Ttσ2u du, it follows that 2N((b − Xt)/

√ΛT − Λt) is a bounded

random variable that is measurable with respect to σ(Xu, u ≥ t) ∨ σ(σu, u ≥ t).The claim now follows from Proposition 6.2 and the Markov property, cf. [Protter(2005), Definition I.5.2].

Finally, by setting t = 0 we obtain a formula for unconditional default probabilities:

Corollary 6.5. Let X be a credit quality process with volatility process σ, and letτ be the associated default time. Assume further that σ0 is non-random. Then, thedefault distribution is given by

P(τ ≤ T ) = 2E(

N(

b√ΛT

)), T ≥ 0. (6.2.4)

6.3 Conditional default density

Proposition 6.6. Let X be a credit quality process with volatility σ as in Definition6.1, and let τ = inft ≥ 0 : Xt ≤ b be the associated default time. Suppose furtherthat

∫ T0

E(σ2u) du < ∞, T ≥ 0. Then, the distribution of τ conditional on Ft is

absolutely continuous and admits a density, and on τ > t, P–a.s.,

P(τ ≤ T |Ft) =∫ T

t

E(

n(

b−Xt√Λu − Λt

)−(b−Xt)

(Λu − Λt)3/2σ2u

∣∣∣Ft) du, T > t. (6.3.1)

Proof. This is an application of the Conditional Fubini Theorem A.18. For everyT > t and ω ∈ Ω such that ΛT (ω) is differentiable,

∂

∂T2 N(


)= n

(b−Xt√ΛT − Λt

)−(b−Xt)

(ΛT − Λt)3/2σ2T

= −b−Xt√2π

exp(− (b−Xt)2

2(ΛT − Λt)− ln((ΛT − Λt)3/2)

)σ2T .

The exponential is continuous in T and tends to 0 as either T → t or T →∞, henceit is bounded, say by K ∈ R. Hence, for any T ≥ 0,∫ T

0

E(

n(


)−(b−Xt)


)du ≤

∫ T

0

E(Kσ2

u

)du <∞.

6.4. Variance as Levy-driven Ornstein-Uhlenbeck process 51

By the Conditional Fubini Theorem, P–a.s.,∫ T

t

E(

n(


)−(b−Xt)


∣∣∣Ft) du

= E

(∫ T

t

n(


)−(b−Xt)

(Λu − Λt)3/2σ2u du

∣∣∣Ft)

= E(

N(


) ∣∣∣Ft) ,where the last step follows since Λ is differentiable almost everywhere.

6.4 Variance as Levy-driven Ornstein-Uhlenbeck process

We put the model to work by specifying the variance process σ2 to be a mean-reverting process with jumps. Candidates as drivers for the variance are Levyprocesses: they incorporate jumps, and we can build Markov processes by specify-ing the dynamics of the variance with respect to Levy-driven SDEs (see TheoremB.23). A review of the definition and some properties of Levy processes, especiallysubordinators and compound Poisson processes, are given in Appendix B.

For our modelling purpose, it is sufficient to consider variance processes driven bycompound Poisson processes, where jumps are rare events – the economic rationale isthat jumps in CDS spreads are triggered by the arrival of “bad news” in the market.Nonetheless, the statements in this section are general enough to incorporate infiniteactivity processes.

As an explicit example we model the variance process as a Levy-driven Ornstein-Uhlenbeck process, which we now introduce.

Definition 6.7. Let Z = (Zt)t≥0 be a subordinator. The process Y = (Yt)t≥0,defined by

Yt = Y0 e−at +∫ t

0

e−a(t−s) dZs, t ≥ 0, a ∈ R,

is a Levy-driven Ornstein-Uhlenbeck process (LOU process).

The LOU process Y is the solution of the SDE

dYt = −aYt− dt+ dZt, t ≥ 0.

Provided that Z is a pure-jump subordinator, it is easily seen that Y moves up byjumps and decays exponentially in-between the jumps.

Models where an asset price’s variance is driven by an LOU process were firstconsidered by [Barndorff-Nielsen and Shephard (2001)]. For details on LOU pro-cesses, see also [Norberg (2004)], [Schoutens (2003), Chapter 5] and [Cont andTankov (2004a), Chapter 15.3.3].


We specify the variance process as an LOU process that incorporates in additiona deterministic function of time – denoted by θ below –, which will turn out to beuseful for calibration. In the following, when we speak of an LOU process, we meanthe more general process given by Equation (6.4.2) below.

Proposition 6.8. Let Z be an (Ft)t≥0-subordinator, let a ∈ R+ and let θ be abounded and cadlag function. Then the solution σ2 to the SDE

dσ2t = a(θ(t)− σ2

t−) dt+ dZt (6.4.1)

exists and is given by

σ2t = e−at σ2

0 +∫ t

0

e−a(t−u) a θ(u) du+∫ t

0

e−a(t−u) dZu, t ≥ 0. (6.4.2)

Moreover, for any T > t, the increment of the integrated process is∫ T

t

σ2u du =

(1− e−a(T−t)

) σ2t

a

+∫ T

t

θ(u)(

1− e−a(T−u))

du+1a

∫ T

t

(1− e−a(T−u)

)dZu. (6.4.3)

Proof. The solution is verified by applying the Ito formula to eat σ2t , which estab-

lishes a fortiori that a solution exists (alternatively, existence and uniqueness ofthe solution to (6.4.1) follow from [Protter (2005), Theorem V.7]). By integratingeach term of Equation (6.4.2), the integrated process

∫ t0σ2u du is obtained. Equation

(6.4.3) then follows by computing the difference∫ t0σ2u du−

∫ s0σ2u du.

The credit quality process (X,σ) with σ2 an LOU-process is then as follows:

Proposition 6.9. Let W be an (Ft)t≥0-Brownian motion, and let σ = (σt)t≥0,σ2

0 > 0, be as in Proposition 6.8, with Z independent of W and with θ such thatσ2t > 0, t ≥ 0. Then the stochastic process X = (Xt)t≥0,

Xt =∫ t

0

σs dWs, t ≥ 0, (6.4.4)

is a credit quality process in the sense of Definition 6.1. Moreover, (X,σ) is aMarkov process with respect to (Ft)t≥0.

Proof. That P(∫ t0σ2s ds < ∞) = 1, t ≥ 0, is a consequence from Equation (6.4.3)

and the fact that Zt < ∞ P–a.s. for any t ≥ 0, since Z is a subordinator, cf.Section B.2. Similarly, limt→∞

∫ t0σ2s ds = ∞ follows from Z being a subordinator,

cf. Section B.2. That (X,σ) is a Markov process follows from Theorem B.23.

A sample path of σ2 and Λ =∫ t0σ2s ds is given in Figure 6.1.

In the following, we shall often assume that Z is a compound Poisson processand make use of the following statements.


Remark 6.10. If Z is a compound Poisson process, then Z is a convergent sum ofjumps, so that Equations (6.4.2) and (6.4.3) can be re-written as

σ2t = e−at σ2

0 +∫ t

0

e−a(t−u) a θ(u) du+∑

0<u≤t

e−a(t−u) ∆Zu (6.4.2’)

∫ T

t

σ2u du =

(1− e−a(T−t)

) σ2t

a

+∫ T

t

θ(u)(

1− e−a(T−u))

du+1a

∑t<u≤T

(1− e−a(T−u)

)∆Zu.

(6.4.3’)Suppose that Z is a compound Poisson process with jump intensity λ and jumpsize Y . Application of Lemma B.21 yields that, for every t ≤ T , the random partof Equation (6.4.3’) follows a compound Poisson distribution (cf. Section B.3),∑

t<u≤T

(1− e−a(T−u)) ∆Zu ∼ CPO(λ(T − t), (1− e−a(T−S))Y ), (6.4.5)

with S a uniformly distributed random variable on (t, T ] and independent of Y .

Corollary 6.11. Let X be a credit quality process with LOU variance process σ2

as in Proposition 6.9, and suppose further that σ2 is driven by a compound Poissonprocess Z with jump intensity λ and jump size Y > 0, EY < ∞. Let τ = inft ≥0 : Xt ≤ b be the associated default time. Then, on τ > t, the conditionaldistribution of τ admits a density, given by Equation (6.3.1).

Proof. Let σ2T be given by Equation (6.4.2’). By application of Lemma B.21, the

random part of σ2T is distributed as∑0<u≤T

e−a(T−u) ∆Zu ∼ CPO(λT, e−a(T−S)Y ),

where S ∼ U(0, T ) independent of Y . By properties of the compound Poissondistribution,

E

∑0<u≤T

e−a(T−u) ∆Zu

= λT Ee−a(T−S) EY.

For every T ≥ t, there is a constant MT such that∫ T

0

Eσ2u du ≤MT · T + λEY

T 2

2<∞,

and the claim follows from Proposition 6.6.

Remark 6.12. We introduce some notation that is used frequently in the followingchapters. Assume that X is a credit quality process with LOU variance process σ2.For T > t, on τ > t, define

gt,T (x, y) = 2E

N

b− x√(1− e−a(T−t)

)ya +

∫ Ttθ(u)

(1− e−a(T−u)

)du+ Lt,T

a

,


with

Lt,T :=∑

t<u≤T

(1− e−a(T−u)

)∆Zu

(the denominator of gt,T (x, y) is just ΛT − Λt, see Equation (6.4.3)). Since Xt andσ2t are Ft-measurable, and Lt,T is independent of Ft, it follows by the Independence

Lemma A.17 that

gt,T (Xt, σ2t ) = P(τ ≤ T |Ft) P–a.s.,

i.e., gt,T (Xt, σ2t ) is a version of the conditional default probability P(τ ≤ T |Ft).


1

2

3

4

5

6

0 1 2 3 4 50

2

4

6

8

10

t

σ2

Λ

-1

0

1

2

3

4

0 1 2 3 4 50

2

4

t

Xσ

0

0.1

0.2

0.3

0.4

0 1 2 3 4 5t

(5− t)-yr. default pr.5-yr. default pr.

0

250

500

750

1000

0 1 2 3 4 5t

(5− t)-yr. hazard rate5-yr. hazard rate

Fig. 6.1 Example of variance process and credit quality process. Top: variance process σ2,Equation (6.4.2) (continuous line, left axis); time-change Λ, Equation (6.4.3) (dashed line, rightaxis).Second from top: credit quality process X, Equation (6.4.4) (continuous line, left axis); volatilityσ (dashed line, right axis). Second from bottom: 5-year default probability process, with decayingtime-to-maturity (continuous line) and with fixed time-to-maturity (dashed line).Bottom: Term hazard rate computed from default probabilities, i.e., − ln(1 − P (t, T ))/(T − t)(continuous line), − ln(1− P (t, T + t))/T (dashed line).Parameters: a = 3, θ ≡ 1, σ2

0 = 1; σ2 is driven by a compound Poisson process with jump intensityλ = 3 and exponential jump size distribution with parameter η = 2 (mean 1/η, variance 1/η2).The barrier is b = −3.

Chapter 7

Dynamics

7.1 Jumps in default probabilities and credit spreads

The continuity of the credit quality process X and the associated time-change Λwere essential to derive the formula for conditional default probabilities, Equation(6.2.1), from which credit spreads can be computed. But recall that we wished tobuild a model that incorporates jumps in credit spreads. We show that for a creditquality process as in Proposition 6.9, with the variance an LOU process driven bya compound Poisson process, jumps in the variance process propagate to defaultprobabilities and credit spreads. Intuitively, this is explained as follows: Recall thatthe conditional probability of default until T at time t, on τ > t is

P (t, T ) = 2E(

N(


) ∣∣∣Ft) ,with ΛT − Λt =

∫ Ttσ2u du. In a suitable model, a jump in σ2 at time t implies a

greater level in the variance σ2u, u > t, hence a greater variance in the credit quality

process, hence a higher probability of hitting the barrier, than if no jump had takenplace. Jumps in conditional default probablities then propagate into jumps in creditspreads.

Proposition 7.1. Let X be a credit quality process with LOU variance processσ2 as in Proposition 6.9, and suppose that σ2 is driven by a compound Poissonprocess. Let τ = inft > 0 : Xt ≤ b be the associated default time. Fix T > 0 andlet (P (t, T ))t≤T be the associated conditional default probability process. Then, forP-almost all ω ∈ τ > t, (P (t, T ))t≤T is a process whose jumps are positive and

∆σ2t (ω) = 0 ⇐⇒ ∆P (t, T )(ω) = 0, t < T.

Moreover, for any t < T , P–a.s.,

∆P (t, T ) = 2E(

N(


) ∣∣Xt, σ2t

)− 2E

(N(


) ∣∣Xt, σ2t−

). (7.1.1)

57

58 Dynamics

Proof. Abbreviate

h(t, T ) =∫ T

t

θ(u)(

1− e−a(t−u))

du

Lt,T =∑

t<u≤T

(1− e−a(T−u)

)∆Zu,

so that, by Equation (6.4.3’),

ΛT − Λt =(

1− e−a(T−t)) σ2

t

a+ h(t, T ) +

Lt,Ta

.

By the Markov property of (X,σ) the conditional default probability at t until T ,on τ > t is given by Equation (6.2.3), and by Lemma A.17, a version of thisconditional probability is given by gt,T (Xt, σt) with

gt,T (x, y) := E

2N

b− x√(1− e−a(T−s)

)y2/a+ h(t, T ) + Lt,T /a

. (7.1.2)

To derive the claim of the Proposition we require the following:

(i) For any t ≤ T , Lt−,T = Lt,T P–a.s.,(ii) for any sequence (tn, xn, yn)→ (t, x, y),

gtn,T (xn, yn)→ gt,T (x, y), P–a.s., (7.1.3)

(iii) for (b− x) < 0, gt,T (x, y) is strictly increasing in y.

Property (i) was established in Equation (B.1.1). For (ii) observe that

b− xn√(1− e−a(T−tn)

)y2n/a+ h(tn, T ) + Ltn,T /a

→ b− x√(1− e−a(T−t)

)y2/a+ h(t, T ) + Lt,T /a

, P–a.s., as n→∞,

by (i) and since all the terms in the sum of the denominator converge and thelimit of the denominator is greater 0. Equation (7.1.3) is obtained by continuityof the Normal distribution and Dominated Convergence. For (iii) observe that thedenominator in Equation (7.1.2) is strictly increasing in y and that for (b− x) < 0,t 7→ N((b− x)/

√t) is strictly increasing.

Fix gt,T (Xt, σt) as the version of the conditional default probability P (t, T ) onτ > t. Then, taking into account that X is continuous P–a.s., and that on τ > twe have (b−Xt) < 0, we obtain P–a.s. for every sequence tn ↑ t,

P (t−, T ) = limtn↑T

gtn,T (Xtn , σtn)

= gt,T (Xt, σt−)

= gt,T (Xt, σt), if ∆σt = 0< gt,T (Xt, σt), if ∆σt > 0

= P (t, T ).

7.1. Jumps in default probabilities and credit spreads 59

A straightforward consequence of this Proposition is that a jump at σ2t triggers a

jump in all conditional default probability processes (P (t, T ))t≤T , T ≥ 0.

Remark 7.2. We postulate that this result can be generalised to LOU processesdriven by subordinators other than compound Poisson processes. However, thisrequires care with the representation of the process as the sum of jumps may notconverge. Furthermore, in the proof we have not used properties specific to themean-reversion feature of the LOU process. In order to establish the statement forother Markov processes (X,σ2) where σ2 is driven by a Levy process, essentially,one must show that a corresponding version of Equation (7.1.3) holds.

To compute CDS spreads from default probabilities, assume for simplicity thatthe short rate is constant, rt = r, t ≥ 0, so that the formula for CDS spreads,Equation (3.4.2), becomes on τ > t, P–a.s.,

s(t, T )1−R

=

∫ Tt

e−r(u−t) dP (t, u)∫ Tt

e−r(u−t)Q(t, u) du

=e−r(T−t) P (t, T ) +

∫ Ttr e−r(u−t) P (t, u) du∫ T

te−r(u−t)Q(t, u) du

, (7.1.4)

where the second line is obtained by integration by parts. Recall that by Assumption3.3, R ∈ (0, 1).

Proposition 7.3. Let (P (t, u))0≤t≤u), u > 0, be a family of cadlag conditionaldefault probability processes, and let (s(t, T ))0≤t≤T be the CDS spread process formaturity T . Then (s(t, T ))0≤t≤T is cadlag, and for t ≤ T and for P–a.a. ω ∈ τ >t,

(i) ∆s(t, T )(ω) = 0, if ∆P (t, u)(ω) = 0, for all u ∈ (t, T ],(ii) ∆s(t, T )(ω) > 0, if ∆P (t, u)(ω) > 0, for all u ∈ (t, T ].

Proof. Consider first the integral of the numerator of Equation (7.1.4). For anysequence tn → t, as n→∞,

limn→∞

∫ T

tn

r e−r(u−tn) P (tn, u) du

= limn→∞

∫ t

tn

r e−r(u−tn) P (tn, u) du︸︷︷︸=0

+ limn→∞

∫ T

t

r e−r(u−tn) P (tn, u) du.

Then, for the numerator of Equation (7.1.4), by Dominated Convergence, for anysequence tn ↑ t as n→∞,

limn→∞

[e−r(T−tn)P (tn, T ) +

∫ T

t

r er(u−tn) P (tn, u) du

]

= e−r(T−t)P (t−, T ) +∫ T

t

r e−r(u−t) P (t−, u) du.

60 Dynamics

Similarly, we obtain for the denominator,

limn→∞

∫ T

tn

e−r(u−tn)Q(tn, u) du =∫ T

t

e−r(u−t) (1− P (t−, u)) du.

It follows that∆s(t, T )

1−R=

e−r(T−t)P (t, T ) +∫ Ttr e−r(u−t) P (t, u) du∫ T

te−r(u−t) (1− P (t, u)) du

−e−r(T−t)P (t−, T ) +

∫ Ttr e−r(u−t) P (t−, u) du∫ T

te−r(u−t) (1− P (t−, u)) du

= 0, if ∆P (t, u) = 0, t < u ≤ T> 0, if ∆P (t, u) > 0, t < u ≤ T.

In a similar way, noting that Λ is continuous, the corresponding result for CDSspread processes with a fixed time-to-maturity is obtained.

Corollary 7.4. Let (s(t, t+T ))t≥0 be the CDS spread process for time-to-maturityT . Then, under the assumptions of Proposition 7.3, (s(t, t + T ))t≥0 is cadlag andfor t ≤ T and P–a.a. ω ∈ τ > t,

(i) ∆s(t, t+ T )(ω) = 0, if ∆P (t, u)(ω) = 0, for all u ∈ (t, T + t],(ii) ∆s(t, t+ T )(ω) > 0, if ∆P (t, u)(ω) > 0, for all u ∈ (t, T + t].

Obviously, the credit quality process model does not include events where creditspreads jump for selected maturities only. However, this is compatible with theobservation that credit spreads tend to jump together, cf. Section 3.1.2.

Remark 7.5. Finally, it is easily seen that a jump in the variance process cannotlead to default P–a.s.. It suffices to recall that τ = inft > 0 : Xt ≤ b is apredictable stopping time, whereas the jumps of the driving compound Poissonprocess are totally inaccessible (cf. Section B.1).

This observation is closely connected to the result of the following section, wherewe establish that the short-term credit spread vanishes.

7.2 Short-term hazard rate and short-term credit spread

In Section 5.3, we established that the short-term hazard rate and the short-termcredit spread in the OS-model vanish as time-to-maturity tends to 0. We establishthe corresponding result for the model with stochastic volatility with jumps.

Assume given at time t, on τ > t, a term structure of default probabilitiesP (t, T ), T > t, that admits a density. As before, the hazard rate is1

λ(t, T ) =∂∂T P (t, T )

1− P (t, T ), T > t. (7.2.1)

1Strictly speaking, P (t, T ) may be differentiable only almost everywhere.

7.3. Dynamics of default probabilities 61

Proposition 7.6. Let λ(t, T ) be the hazard rate as given by Equation (7.2.1) andlet s(t, T ) be the credit spread with maturity T at time t (assuming no default untilt). Then, as T ↓ t, λ(t, T )→ 0 and s(t, T )→ 0.

Proof. For the numerator of Equation (7.2.1), P–a.s.,

∂

∂TP (t, T ) =

∂

∂T2E(

N(


) ∣∣∣Xt, σ2t

)= −E

(b−Xt

(ΛT − Λt)3/2n(


)σ2T

∣∣∣Xt, σ2t

).

For P-almost all ω ∈ τ > t, just as in the proof of Proposition 5.2,

limT↓t

b−Xt(ω)(ΛT (ω)− Λt(ω))3/2

n

(b−Xt(ω)√

ΛT (ω)− Λt(ω)

)σ2T (ω) = 0,

That limT↓t∂∂T P (t, T ) = 0 follows by the Dominated Convergence Theorem. For

the denominator of Equation (7.2.1), limT↓t(1 − P (t, T )) = 1, so that λ(t, T ) → 0as T ↓ t follows. For the spread, observe that by Equation (4.2.1),

s(t, T )1−R

≤ supt<u≤T

λ(t, u),

and the claim follows, since limT↓t supt<u≤T λ(t, u) = 0 (cf. proof of Proposition5.2).

By the same argument as in Corollary 5.4, we obtain the short-term behaviour asthe spread moves through time.

Corollary 7.7. Let s(t, T ) be the credit spread with maturity T at time t. Then,on τ > T, limt↑T s(t, T ) exists. Moreover, if the credit quality process is drivenby an LOU variance process, then limt↑T s(t, T ) = 0 P–a.s..

7.3 Dynamics of default probabilities

As for the OS-model in Section 5.4, we compute the dynamics of default probabili-ties, now for the credit quality process X with LOU variance process σ2 (cf. Propo-sition 6.9), where we assume that the subordinator Z driving σ2 is a compoundPoisson process with integrable jump size.2 We fix a maturity T > 0 throughout(primarily to unclutter notation), and we introduce the following notation:

F (t, σ2t , Lt,T ) := ΛT − Λt =

1− e−a(T−t)

aσ2t +

∫ T

t

θ(u)(

1− e−a(T−u))

du+Lt,Ta

,

(7.3.1)2The results may be generalised to accommodate other types of subordinators, in which case care

has to be taken with the representation of the process as the sum of jumps may not converge.

62 Dynamics

with

Lt,T =∑

t<u≤T

(1− e−a(T−u)

)∆Zu. (7.3.2)

We introduce the following notation for the distribution function of Lt,T ,

G(t, u) = P (Lt,T ≤ u) , u ≥ 0,

and we write Gt(t, u) = ∂∂tG(t, u) provided the partial derivative exists. In Propo-

sition 7.8, we derive the dynamics under some assumptions on the distribution ofLt,T . The general case is treated in Corollary 7.9.

Proposition 7.8. Let X be a credit quality process with LOU variance process σ2

as in Proposition 6.9 and suppose that σ2 is driven by a compound Poisson process.Assume further that G(t, u), u ≥ 0, is continuously differentiable with boundedderivative Gt(t, u) for all u ≥ 0. Then,

P (t, T ) = P (0, T )

−∫ t

0

1τ>uE

(2√

F (u, σ2u, Lu,T )

n

(b−Xu√

F (u, σ2u, Lu,T )

)∣∣∣Xu, σ2u

)σu dWu

+∑

0<u≤t

1τ>u∆P (u, T )

−∫ t

0

∫ ∞

0

−1τ>uGt(u, v)b−Xu

aF (u, σ2u, v)3/2

n

(b−Xu√F (u, σ2

u, v)

)dv du.

(7.3.3)

Recall that (P (t, T ))t≥0 is a martingale, so that there exists a decomposition

P (t, T ) = P (0, T ) +M ct +Md

t , t ≥ 0, (7.3.4)

where M c is a continuous martingale and Md is a purely discontinuous martingale,cf. Theorem A.12. It is straightforward by Equation (7.3.3) that M c is the Brownianintegral, so that Md is the remainder, consisting of the sum of jumps and the du-integral. It follows in particular that the du-integral is the compensator of the sumof jumps. Intuitively, the compensator reflects that the expected remaining totalnumber of jumps of the random part of ΛT − Λt decays with time. It is also easilyseen that the compensator is nonnegative, by observing that Gt(t, u) ≥ 0, u ≥ 0,and b−Xt < 0 on τ > t.

Proof. This is an application of the Ito formula, cf. Theorem A.8. As in thederivation of the dynamics of the OS-model default probabilities, Proposition 5.5,we derive first the dynamics of 1τ>tP(τ ∈ (t, T ]|Xt, σ

2t ).

Define

h(t, x, y, u) := N

(b− x√F (t, y, u)

)g(t, x, y, z) := z 2 Eh(t, x, y, Lt,T ).


By the Independence Lemma A.17, P–a.s.,

1τ>tP(τ ∈ (t, T ]|Xt, σ2t ) = g(t,Xt, σ

2t ,1τ>t).

Denote the respective partial derivatives of g by gx, gxx, gy, gt and the partial deriva-tives of h by hx, hxx, hy, ht. A sufficient condition for the existence of the partialderivatives gx, gxx is that h(t, x, y, Lt,T ) and hx(t, x, y, Lt,T ) are Lipschitz in x. Thepartial derivatives are then obtained by Dominated Convergence. For the Lipschitzcondition, it is sufficient that h(t, x, y, Lt,T ) and hx(t, x, y, Lt,T ) are differentiable inx and that hx and hxx are bounded and cadlag , see e.g. [Klebaner (2005), Chapter1] or [Protter (2005), Section V.3]. It is not hard to verify that hx, hxx are con-tinuous and bounded for any t < T (using the same technique as in the proof ofProposition 5.2), so that

gx(t, x, y, z) = 2z E

(− 1√

F (t, σ2t , Lt,T )

n

(b− x√

F (t, σ2t , Lt,T )

))

gxx(t, x, y, z) = 2z E

(− b− x

(F (t, σ2t , Lt,T ))3/2

n

(b− x√

F (t, σ2t , Lt,T )

)).

Similarly, the partial derivative with respect to y is given by

gy(t, x, y, z) = zE

(− (b− x)(1− e−a(T−t))

a(F (t, σ2t , Lt,T ))3/2

n

(b− x√

F (t, σ2t , Lt,T )

))

=1− e−a(T−t)

2agxx(t, x, y).

Next, we derive gt. First, by integration by parts,

g(t, x, y, z) = 2z∫ ∞

0

h(t, x, y, u)G(t,du)

= 2z limu→∞

h(t, x, y, u)− 2z∫ ∞

0

G(t, u)hu(t, x, y, u) du,

where

hu(t, x, y, u) =∂

∂uh(t, x, y, u) = − b− x

2aF (t, y, u)3/2n

(b− x√F (t, y, u)

).

It is straightforward that limu→∞ h(t, x, y, u) = 1/2. In order to derive gt, observefirst that G(t, u) is continuously differentiable in t and bounded by assumption.Second,

∂

∂thu(t, x, y, u) =

b− x4aF (t, y, u)5/2

n

(b− x√F (t, y, u)

)∂

∂tF (t, y, u)

(1− (b− x)2

F (t, y, u)

),

and it is not hard to show that ∂∂thu(t, x, y, u) is bounded in u, for t ∈ (0, T ] .

Additionally, observe that limu→∞ ht(t, x, y, u) = 0, where

ht(t, x, y, u)

=12

b− xF (t, y, u)3/2

n

(b− x√F (t, y, u)

) (y e−a(T−t) − θ(t) (1− e−a(T−t))

).

64 Dynamics

Putting things together, using Dominated Convergence in the first step, we obtain,

gt(t, x, y, z) = −2z∂

∂t

∫ ∞

0

G(t, u)hu(t, x, y, u) du

= −2z∫ ∞

0

Gt(t, u)hu(t, x, y, u) du− 2z∫ ∞

0

G(t, u)ht(t, x, y,du)

= −2z∫ ∞

0

Gt(t, u)hu(t, x, y, u) du+ 2z∫ ∞

0

ht(t, x, y, u)G(t,du) (?)

=: g2,t(t, x, y, z) + g1,t(t, x, y, z), (7.3.5)

where (?) is obtained by applying integration by parts to the second term. For g1,twe obtain,

g1,t(t, x, y, z) = −(y e−a(T−t) + θ(t)(1− e−a(T−t))

)2

gxx(t, x, y).

By the Ito formula, using that σ2 is of finite variation, which implies that thecontinuous part of [σ2, σ2] is 0,

1τ>tP(τ ≤ T |Xt, σ2t ) = P (0, T )

+∫ t

0

[g1,t(u,Xu, σ

2u,1τ>u) + g2,t(u,Xu, σ

2u,1τ>u)

]du

+∫ t

0

gx(u,Xu, σ2u,1τ>u) dXu +

∫ t

0

gy(u,Xu, σ2u−,1τ>u−) dσ2

u

+12

∫ t

0

gxx(u,Xu, σ2u,1τ>u) d[X,X]u

+∑

0<u≤t

[∆g(u,Xu, σ

2u,1τ>u)− gy(u,Xu, σ

2u−,1τ>u−) ∆σ2

u

].

Putting together all du-integrals except∫ t0g2,t du, yields,

∫ t

0

[g1,t + gy a(θ(u)− σ2

u) +12gxx σ

2u

]du

=∫ t

0

−σ2u e−a(T−u) + θ(u)(1− e−a(T−u))

2gxx(t, x, y, z) du

+∫ t

0

[1− e−a(T−u)

2aa(θ(u)− σ2

u) +12σ2u

]gxx(t, x, y) du

= 0.


Since Z is a compound Poisson process,∑

0<u≤t ∆g(u,Xu, σ2u−,1τ>u−) < ∞

P–a.s., so that we obtain

1τ>tP(τ ≤ T |Xt, σ2t ) = P (0, T )

− 2∫ t

0

1τ>uE

(1√

F (u, σ2u, Lu,T )

n

(b−Xu√

F (u, σ2u, Lu,T )

∣∣∣Xu, σ2u

))σu dWu

+∫ t

0

g2,t(u,Xu, σ2u,1τ>u) du

+∑

0<u≤t

∆

[1τ>u 2E

(N

(b−Xu√

F (u, σ2u, Lu,T )

)∣∣∣Xu, σ2u

)].

(7.3.6)

Now, observe that since τ is a predictable stopping time and since the jump timesof σ2 are totally inaccessible (cf. Section B.1), it follows that 1τ>· and σ2 do notjump together P–a.s. (cf. Definition A.10). If for some u < T , ∆1τ>u 6= 0, thenP–a.s.,

∆

[1τ>u 2E

(N

(b−Xu√

F (u, σ2u, Lu,T )

)∣∣∣Xu, σ2u

)]

= − lims↑u

1τ>s2E

(N

(b−Xs√

F (s, σ2s , Ls,T )

)∣∣∣Xs, σ2s

)= −1,

taking into account that F (s, σ2s , Ls,T ) > 0 for any s < T . On the other hand, when

∆σ2u > 0, P–a.s.,

∆

[1τ>u 2E

(N

(b−Xu√

F (u, σ2u, Lu,T )

)∣∣∣Xu, σ2u

)]

= 1τ>u∆2E

(N

(b−Xu√

F (u, σ2u, Lu,T )

)∣∣∣Xu, σ2u

)= 1τ>u∆P (u, T ).

Hence, for the sum of jumps in Equation (7.3.6), we obtain

∑0<u≤t

∆

[1τ>u 2E

(N

(b−Xu√

F (u, σ2u, Lu,T )

)∣∣∣Xu, σ2u

)]= −1τ≤t +

∑0<u≤t

1τ>u∆P (u, T ).

Equation (7.3.3) now follows from P (t, T ) = 1τ≤t + 1τ>tP(τ ∈ (t, T ]|Xt, σ2t ).

The only reason to require that the distribution of Lt,T be continuously differen-tiable with bounded derivative in the previous proposition was to derive the partialderivative gt (see Equation (7.3.5)) to apply the Ito formula. However, by the mar-tingale decomposition, we can derive the continuous martingale part by the Ito

66 Dynamics

formula as before and then deduce that the purely discontinuous martingale partmust be the sum of compensated jumps. Moreover, by Lemma A.15, the com-pensator is continuous, as the jump times of the driving Levy process are totallyinaccessible. We obtain:

Corollary 7.9. Let X be a credit quality process with LOU variance process σ2 asin Proposition 6.9 and suppose that σ2 is driven by a compound Poisson process.Then,

P (t, T ) = P (0, T )

−∫ t

0

1τ>uE

(2√

F (u, σ2u, Lu,T )

n

(b−Xu√

F (u, σ2u, Lu,T )

)∣∣∣Xu, σ2u

)σu dWu

+∑

0<u≤t

1τ>u∆P (u, T )−A(t, T ),

where A(t, T ) is the compensator of the jumps of P (t, T ) taking place on τ > t.

Again, we postulate that the results in this Section can be generalised to varianceprocesses other than the LOU process.

7.3.1 The distribution function of Lt,T

In Proposition 7.8 we required that the distribution function of Lt,T be continuouslydifferentiable in t with bounded derivative. A simple condition for this to hold isthat the jump size distribution of the underlying compound Poisson process admitsa density.

Proposition 7.10. Let Lt,T , t ≤ T , be as in Equation (7.3.2), with Z a compoundPoisson process with jump intensity λ and jump size distribution F , F (0) = 0.Suppose further that F is absolutely continuous with density f . Then, for t < T , thedistribution function G(t, x), x ≥ 0, of Lt,T is continuously differentiable in t withbounded derivative Gt(t, x) = ∂

∂tG(t, x). Moreover, for every x ≥ 0, limt↑T Gt(t, x)exists.

Proof. By Equation (6.4.5), Lt,T ∼ CPO(λ(T − t), (1 − e−a(T−S))Y ) with S uni-formly distributed on (t, T ] and Y ∼ F . In order words, setting

H(t, x) = P(

(1− e−a(T−S))Y ≤ x)

= P(

(1− e−a(T−t)(1−U)Y ≤ x), x ≥ 0,

where U is uniformly distributed on (0, 1], and writing Hn∗(t, x) for n-fold convo-lution of H(t, x) with respect to x, we have

G(t, x) = P(Lt,T ≤ x) = e−λ(T−t) +∞∑n=1

Hn∗(t, x)e−λ(T−t) (λ(T − t))n

n!, x ≥ 0.

(7.3.7)


Let us first make some preliminary computations regarding the jump size dis-tribution H. By application of the Independence Lemma A.17, H may be writtenas

H(t, x) =1

T − t

∫ T

t

P((

1− e−a(T−u))Y ≤ x

)du

=1

T − t

∫ T

t

P(Y ≤ x

1− e−a(T−u)

)du

=1

T − t

∫ T

t

F

(x

1− e−a(T−u)

)du.

It follows that H(t, x) is differentiable in t with derivative

Ht(t, x) =1

T − t

[1

T − t

∫ T

t

F

(x

1− e−a(T−u)

)du− F

(x

1− e−a(T−t)

)]

=1

T − t

[H(t, x)− F

(x

1− e−a(T−t)

)].

The derivative is continuous in t since F is continuous. Furthermore, the densityh(t, x) of H(t, x) is given by

h(t, x) =1

T − t

∫ T

t

f

(x

1− e−a(T−u)

)1

1− e−a(T−u)du,

which is differentiable in t with derivative

ht(t, x) =1

T − t

[h(t, x)− f

(x

1− e−a(T−t)

)1

1− e−a(T−t)

].

Let us now examine the derivative of the convolution product Hn?(t, x), whichwe shall need for computing the derivative of Equation (7.3.7). Using the equality

Hn∗(t, x) =∫ x

0

H(n−1)∗(t, x− y)H(t,dy), n ∈ N,

we obtain by induction, using the Leibniz rule for parameter integrals, that

∂

∂tHn∗(t, x) =

∂

∂t

∫ x

0

H(n−1)∗(t, x− y)H(t,dy)

=∫ x

0

H(n−1)∗(t, x− y)Ht(t,dy) +∫ x

0

∂

∂tH(n−1)∗(t, x− y)H(t,dy)

(?)

=[Ht ∗H(n−1)∗

](t, x) +

[H ∗ ∂

∂tH(n−1)∗

](t, x)

= n[Ht ∗H(n−1)∗

](t, x), n ∈ N.

To obtain Equation (?) we have made use of the fact that H(t, x) has a densityh(t, x) that is differentiable in t, and that Ht(t, x) is of bounded variation.

68 Dynamics

Let us now compute Gt(t, x) explicitly. For t < T and x ≥ 0,

Gt(t, x) = λe−λ(T−t) + λ

∞∑n=1

Hn∗(t, x)e−λ(T−t) (λ(T − t))n

n!

−∞∑n=1

Hn∗(t, x)e−λ(T−t) λn n (T − t)n−1

n!

+∞∑n=1

n[Ht ∗H(n−1)∗

](t, x)

e−λ(T−t) (λ(T − t))n

n!

= λG(t, x)− λ∞∑n=1

Hn∗(t, x)e−λ(T−t)(λ(T − t)n−1)

(n− 1)!

+ λ

∞∑n=1

[(H − F

(·

1− e−a(T−t)

))∗H(n−1)∗

](t, x)

e−λ(T−t) (λ(T − t))n−1

(n− 1)!

= λG(t, x)− λ∞∑n=0

P(

(1− e−a(T−t))X +n∑k=1

Yk ≤ x) e−λ(T−t) (λ(T − t))n

n!

(7.3.8)where Yk ∼ H, k ∈ N, X ∼ F , and all random variables are independent. (For thelast step recall that convolution is a distributive operation.) It follows that G(t, x)is continuously differentiable in t and that Gt(t, x) is bounded, as the second term ofEquation (7.3.8) is bounded by a compound Poisson distribution function. Finally,it is straightforward by Equation (7.3.8) that the limit limt↑T Gt(t, x) exists forevery x ≥ 0.

The computation of Gt(t, x) in the previous proof may be used to compute thecompensator in Equation (7.3.3) explicitly, which in turn, may provide some insightsabout expected jump sizes, etc.

7.4 Spread dynamics

We compute the dynamics of credit spreads using the Ito formula, see TheoremA.8. For simplicity, assume that interest rates are 0, i.e., r ≡ 0. Fixing a maturityT , denote by s(t,Xt, σ

2t ) the credit spread with maturity T at time t. By Equation

(3.4.2), the spread satisfies

s(t,Xt, σ2t ) = (1−R)

P (t, T )∫ Tt

[1− P (t, u)] du, on τ > t,

with R the recovery rate. In the following, all statements are on τ > t. We alsoassume that the compensator of the jumps of P (t, T ) allows a representation as inEquation (7.3.3).


as in Proposition 6.9 and suppose that σ2 is driven by a compound Poisson process.

7.4. Spread dynamics 69

Assume further that G(t, u), u ≥ 0, is continuously differentiable with boundedderivative Gt(t, u) for all u ≥ 0. Then, on τ > t,

s(t,Xt, σ2t ) = s(0, X0, σ

20) +

∫ t

0

s(u,Xu, σ2u) [Gu −Hu] σu dWu

+∫ t

0

s(u,Xu, σ2u)[H2u −GuHu

]σ2u du

+∫ t

0

s(u,Xu, σ2u)

[−Au +

1∫ Tu

[1− P (u, v)] dv

]du

+∑

0<u≤t

∆s(u,Xu, σ2u) (7.4.1)

with

Gt := G(t,Xt, σ2t ,1τ>t) =

∂

∂xlnP (t, T )

Ht := H(t,Xt, σ2t ,1τ>t) =

∂

∂xln

(∫ T

t

(1− P (t, v)) dv

)

At := A(t,Xt, σ2t ,1τ>t) =

aT (t,Xt, σ2t ,1τ>t)

P (t, T )−∫ Tt−av(t,Xt, σ

2t ,1τ>t) dv∫ T

t(1− P (t, v)) dv

,

with∫ t0aT (u,Xu, σ

2u,1τ>u) du the compensator of

∑0<u≤t ∆P (u, T ).

Proof. Define

gT (t, x, y, z) := z 2EN

(b− x√

F (t, y, Lt,T )

),

with F (t, y, Lt,T ) given by Equation (7.3.1) and observe that gT (t,Xt, σ2t ,1τ>t) =

P (t, T ) P–a.s.. Furthermore, the partial derivatives of gT are denoted by gT,t, gT,x,gT,xx and gT,y, respectively. The partial derivatives are given explictly in the proofof Proposition 7.8 where the index T is omitted.

Denote by st, sx, sxx and sy the partial derivatives of s, given by

st(t, x, y) = s(t, x, y)

(gT,t(t, x, y, z)gT (t, x, y, z)

−∫ Tt−gu,t(t, x, y, z) du− 1∫ T

t(1− gu(t, x, y, z)) du

)

sx(t, x, y) = s(t, x, y)

(gT,x(t, x, y, z)gT (t, x, y, z)

−∫ Tt−gu,x(t, x, y, z) du∫ T

t(1− gu(t, x, y, z)) du

)

sxx(t, x, y) =sx(t, x, y)2

s(t, x, y)+ s(t, x, y)

(gT,xx(t, x, y, z)gT (t, x, y, z)

− gT,x(t, x, y, z)2

gT (t, x, y, z)2

)

+ s(t, x, y)

− ∫ Tt −gu,xx(t, x, y, z) du∫ Tt

(1− gu(t, x, y, z)) du+

(∫ Tt−gu,x(t, x, y, z) du

)2

(∫ Tt

(1− gu(t, x, y, z)) du)2

sy(t, x, y) = s(t, x, y)

(gT,y(t, x, y, z)gT (t, x, y, z)

−∫ Tt−gu,y(t, x, y, z) du∫ T

t(1− gu(t, x, y, z)) du

)

70 Dynamics

By the Ito formula, using that∑

0<u≤t ∆s(u,Xu, σ2u) <∞ P–a.s.,

s(t,Xt, σ2t ) = s(0, X0, σ

20) +

∫ t

0

[st(u,Xu, σ

2u) +

12sxx(u,Xu, σ

2u)σ2

u

]du

+∫ t

0

sx(u,Xu, σ2u) dXu +

∫ t

0

sy(u,Xu, σ2u) dσ2

u +∑

0<u≤t

∆s(u,Xu, σ2u),

(7.4.2)

where σ2u denotes the continuous part of σ2

u. As in the previous Proposition, theterms involving g·,y, g·,xx and parts of g·,t cancel. Set

Gt =gT,x(t,Xt, σ

2t ,1τ>t)

gT (t,Xt, σ2t ,1τ>t)

Ht =

∫ Tt−gv,x(t,Xt, σ

2t ,1τ>t) dv∫ T

t(1− gv(t,Xt, σ2

t ,1τ>t)) dv,

and observe thatsx(t,Xt, σ

2t )

s(t,Xt, σ2t )

= (Gt −Ht).

After some computations,

s(t,Xt, σ2t ) = s(0, X0, σ

20) +

∫ t

0

s(u,Xu, σ2u) [Gu −Hu] σu dWu

+12

∫ t

0

s(u,Xu, σ2u)[(Gu −Hu)2 −G2

u +H2u

]σ2u du

+∫ t

0

s(u,Xu, σ2u)

[−Au +

1∫ Tu

[1− P (v, T )] dv

]du

+∑

0<u≤t

∆s(u,Xu, σ2u)

Remark 7.12. We see a similar behaviour to the behaviour of the OS-model (seeRemark 5.7): It is straightforward that Ht ≥ 0 and that Gt ≤ 0, for any t ≥ 0.Consequently, the drift part of Equation (7.4.1), excluding the compensator part,is nonnegative, whereas the integrand of the Brownian integral is nonpositive. Onthe other hand, we have already established that limt↑T s(t, T ) = 0 P–a.s., which isnot obvious from the dynamics of the spread.

We can also make a heuristic observation regarding the behaviour as default isapproached. For T > τ ,

limt↑τ

gT,x(t,Xt, σ2t ,1τ>t) = lim

t↑τ1τ>t2E

(− 1√

ΛT − Λtn(


))= 2E

(− 1√

ΛT − Λτn(0)

)= c < 0.

Consequently, limt↑τ−Gt = c and limt↑τ−Ht = −∞. Taking into account that thecompensator At, t ≥ 0 is positive (cf. Remark after Proposition 7.8), it is easily seen

7.5. Distribution of conditional default probabilities 71

that the drift of Equation (7.4.1) tends to ∞. As Xt → b, the Brownian integral isalso positive, hence the credit spread tends to ∞ as default is approached.

This is of course what we expect: As default becomes more likely, the instanta-neous spread payments, which are presumably paid only for a short time, have tocompensate the now very likely payment on the default leg of the CDS.

The variance process, as an LOU process, has positive jumps and decays con-tinuously. Does this property translate into credit spreads? We already know thatjumps in the variance process translate into positive jumps in credit spreads. How-ever, decreasing movement in the variance is deterministic and as such alreadyreflected in the conditional expectations that determine conditional default proba-bilities, and hence credit spreads. On the other hand, loosely speaking, a decreasein the variance at any time implies that no jump has taken place although thepossibility of a jump entered conditional default probabilities and credit spreadsbeforehand. This is reflected in Equation (7.4.1) in the drift term involving theprocess A, which is derived from the compensator of the jumps of conditional de-fault probabilities. It remains open whether the overall drift in Equation (7.4.1) ispositive or negative when σ decays.

We state the stochastic exponential of the credit spread, cf. Section A.5.

Proposition 7.13. In the setup and notation of the previous proposition, thestochastic exponential of s(t,Xt, σ

2t ) is, on τ > t,

s(t,Xt, σ2t ) = s(0, X0, σ

20) exp

(∫ t

0

[Gu −Hu] σu dWu +12

∫ t

0

[H2u −G2

u

]σ2u du

)· exp

(∫ t

0

[−Au +

1∫ Tu

(1− P (v, T )) dv

]du

)·∏

0<u≤t

(1 +

∆s(u,Xu, σ2u)

s(u,Xu, σ2u−)

).

Proof. The stochastic exponential is an application of Theorem A.16 using Equa-tion (7.4.1).

In a continuous model [Gt−Ht]σt is called the instantaneous volatility or the spreadvolatility.

7.5 Distribution of conditional default probabilities

We compute the distribution of conditional default probabilities as we did for theOS-model in Section 5.6. Fixing t ≥ 0 and T ≥ t, the goal is to compute thedistribution of P (t, T ). Define

fσt(Xt) := P(τ ∈ (t, T ]|Xt, σ2t ) = 2E

(N(


) ∣∣∣Xt, σ2t

).

Observe that fσt is continuous and strictly decreasing so that its inverse f (−1)σt exists.

Similarly to Equation (5.6.1), we have for 0 ≤ x < 1,P (t, T ) ≤ x =

P(τ ∈ (t, T ]|Xt, σ

2t ) ≤ x, τ > t

. (7.5.1)

72 Dynamics

Proposition 7.14. For T ≥ 0 and t ≤ T , the distribution of P (t, T ) isP(P (t, T ) ≤ x) = E (N(h1,σt(x,Λt))−N(h2,σt(x,Λt))) , 0 ≤ x < 1,

with

h1,σt(x,Λt) =

−f (−1)σt (x)√

Λt

h2,σt(x,Λt) =2b− f (−1)

σt (x)√Λt

.

Proof. Using Equation (7.5.1), and by the Independence Lemma A.17 togetherwith the independence of the DDS Brownian motion B and the pair σt,Λt,

P (P (t, T ) ≤ x|Λt, σt) = P(fσt

(Xt) ≤ x, min0<s≤t

Xs > b∣∣∣Λt, σt)

= P(fσt

(BΛt) ≤ x, min

0<s≤Λt

Bs > b∣∣∣Λt, σt)

= P(BΛt≥ f (−1)

σt(x), min

0<s≤Λt

Bs > b∣∣∣Λt, σt)

= P(BΛt≥ f (−1)

σt(x)∣∣∣Λt, σt)−P

(BΛt ≥ f (−1)

σt(x), min

0<s≤Λt

Bs ≤ b∣∣∣Λt, σt)

= N

(−f (−1)

σt (x)√Λt

)−N

(2b− f (−1)

σt (x)√Λt

),

using the joint distribution of a Brownian motion and its running minimum as inSection 5.6. Finally,

P(P (t, T ) ≤ x) = E

(N

(−f (−1)

σt (x)√Λt

)−N

(2b− f (−1)

σt (x)√Λt

)).

From Equation (7.5.1) we get the following result, which is similar to Corollary 7.15:

Corollary 7.15. For T ≥ 0 and t ≤ T , the distribution of P(τ ∈ (t, T ]|Xt, σ2t )

conditional on τ > t is, for 0 ≤ x < 1,

P(P(τ ∈ (t, T ]|Xt) ≤ x

∣∣∣τ > t)

=P(P (t, T ) ≤ x)

P(τ > t)

= E (N(h1,σt(x,Λt))−N(h2,σt(x,Λt)))(

1− 2EN(

b√Λt

))−1

. (7.5.2)

Figure 7.1 shows an example for the distribution of P(τ ∈ (t, T ]|Xt, σ2t ) con-

ditional on τ > t, where the variance process is an LOU process. Recall thatthe distributions in the OS-model suggested a very volatile movement of the creditquality process. By the introduction of jumps in the variance, the default speed inthis example has been significantly reduced, while maintaining the default proba-bilities implied by the given term structures. Other distributions, and hence otherdynamics, are obtained by changing the parameters of the LOU process. This willbe discussed in detail in Section 8.3.

7.6. Distribution of term hazard rate 73

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2

P

P(τ ∈ (t, T ]|Xt, σ2t )

1%2%3%

0

0.2

0.4

0.6

0.8

1

0 0.01 0.02 0.03 0.04 0.05 0.06

P

λ(t, T )

1%2%3%

Fig. 7.1 Distributions of P(τ ∈ (t, T ]|Xt, σ2t ) (left) and λ(t, T ) (right) conditional on τ > t, cf.

Equation (7.5.2) and Proposition 7.16. As in Figure 5.1, we choose t = 1, T = 5 and initial hazardrates 1%, 2%, 3%. The diamonds mark the initial 5-year default probability, resp. hazard rate.Recall that λ(t, T ) is an approximation of the credit spread (in percent; s(t, T ) ≈ λ(t, T ) · 104).The parameters are as follows: for h = 0.01: a = 3, jump intensity 1.5 and jump size 0.01, 20with probabilities 0.975, 0.025, barrier b = −3, σ2

0 = 3.33; for h = 0.02: jump intensity 1.5,jump size 0.01, 20 with probabilities 0.95, 0.05, barrier b = −3, σ2

0 = 3.89; for h = 0.03: jumpintensity 2, jump size 0.1, 20 with probabilities 0.95, 0.05, barrier b = −3, σ2

0 = 4.59. In eachcase, the function θ was chosen to fit the spot curve (this will be discussed in Section 8.2.1).

7.6 Distribution of term hazard rate

As in Section 5.7, we compute the distribution of the term hazard rate (recall thatthis exists only conditional on τ > t).

Proposition 7.16. For T ≥ 0 and t < T , let g(x) = 1 − e−x(T−t), x ≥ 0. ByEquation (4.2.4), λ(t, T ) is such that P(τ ∈ (t, T ]|Xt, σ

2t ) = g(λ(t, T )). Then, for

x ≥ 0,

P(λ ≤ x|τ > t)

= E (N(h1,σt(g(x),Λt))−N(h2,σt(g(x),Λt)))(

1− 2EN(

b√Λt

))−1

, (7.6.1)

with h1,σt , h2,σt as in Proposition 7.14.

The proof is the same as in Proposition 5.11. An example is given in Figure 7.1.

74 Dynamics

Chapter 8

Implementation, calibration and examples

The valuation of financial claims in the LOU variance model is done by a com-bination of Monte Carlo simulation and numerical computation. Conditional onXt, σ

2t , default probabilities P (t, T ) = P(τ ≤ T |Xt, σ

2t ), T > t, can be computed

numerically, so that Monte Carlo simulation reduces to simulating Xt and σ2t . The

advantage of such an algorithm is that valuation of a product involving P (t, T ) (orthe credit spread s(t, T )) requires simulation only until time t instead of T . For ex-ample, valuation of a default swaption requires simulation until expiry of the optioninstead of simulation until maturity of the underlying CDS.

In the first part of the chapter, we state the algorithm for computing P (t, T ),T > t, conditional on Xt, σ

2t . We then show in Section 8.2 how calibration to a given

term structure of default probabilities and credit spreads is achieved. We give someexamples of different dynamics and discuss the role of each parameter involved inthe dynamics of P (t, T ), T > t, in Section 8.3. Finally, we discuss the stability ofcalibration parameters in Section 8.4.

8.1 Computation of default probabilities and credit spreads

8.1.1 Jump size distribution of time-change Λ

Assume a credit quality process model (X,σ2) where σ2 is a LOU process driven bya compound Poisson process Z. We examine numerical computation of conditionaldefault probabilities P(τ ≤ T |Xt, σt), 0 ≤ t ≤ T , according to Equation (6.2.3).Recalling that ΛT − Λt =

∫ Ttσ2u du, inspection of Equation (6.4.3’) reveals that

essentially computation of the conditional expectation (6.2.3) entails computingthe distribution of

Lt,T :=∑

t<u≤T

(1− e−a(t−u)

)∆Zu.

Let Z have intensity λ and compounding variate Y . Recall from Section 6.4, Equa-tion (6.4.5), that Lt,T follows a compound Poisson distribution,

Lt,T ∼ CPO(λ(T − t),

(1− e−a(T−S)

)Y), (8.1.1)

75

76 Implementation, calibration and examples

with S uniformly distributed on (t, T ] and independent of Y . Moreover, Lt,TL=

L0,T−t, hence it suffices to compute the distributions of LT := L0,T . The followingresult gives us the distribution of the compounding variate of LT .

Lemma 8.1. For T > 0, let S ∼ U(0, T ) and let Y be a P–a.s. strictly positiverandom variable independent of S. The distribution of

(1− ea(T−S)

)Y is given by

F (x) = E(− ln(1− x/Y )

aT1[0,1−e−aT )(x/Y )

)+ P

(Y ≤ x

1− e−aT

), x ∈ R.

(8.1.2)

Proof. Conditioning under a larger filtration yields

F (x) = P((

1− e−a(T−S))Y ≤ x

)= E

(P((

1− e−a(T−S))Y ≤ x

∣∣∣Y )) . (8.1.3)

Define gx(y) := P((

1− e−a(T−S))y ≤ x

), y > 0. By Lemma A.17 and the inde-

pendence of S and Y , a version of the conditional probability of Equation (8.1.3) isgiven by gx(Y ). Since S ∈ [0, T ],

gx(y) =

0, x ≤ 0,

1, x ≥ (1− e−aT )y,

− ln(1−x/y)aT , x ∈ (0, (1− e−aT )y),

where the last case is obtained by

P((

1− e−a(T−S))y ≤ x

)= P (ln(1− x/y) ≤ −a(T − S))

= P(T − ST

≤ − ln(1− x/y)aT

),

and making use of the fact that (T−S)/T ∼ U(0, 1). Inserting into Equation (8.1.3)yields

F (x) = E(− ln(1− x/Y )

aT1[0,1−e−aT )(x/Y ) + 1[1−e−aT ,∞)(x/Y )

)and the claim follows.

8.1.2 Panjer recursion

The distribution of LT can be computed efficiently using the method of Panjerrecursion [Panjer (1981)], which is based on a recursive evaluation formula for afamily of compound distributions (see also [McNeil et al. (2005), Chapter 10]). Inour implementation it has proven to be numerically more stable to assume a discretedistribution of the compounding variate, even though the distribution function ofthe compounding variate in Equation (8.1.2) is continuous. For the compound Pois-son case, the method works as follows: Suppose N is a Poisson distributed randomvariable with intensity λ and compounding variate Y defined on the nonnegative

8.1. Computation of default probabilities and credit spreads 77

Table 8.1 CPU time (Panjer recursion): 331.33 CPU secs.

number of simulations CPU time (seconds) MSE (at t = 5)

1000 164.28 0.93692000 333.70 0.27255000 1966.54 0.0589

10000 8554.59 0.0403

integers. Set f(i) = P(Y = i), i = 1, 2, . . .. The goal is to compute the com-pound Poisson distribution CPO(λ, Y ). For a random variable L ∼ CPO(λ, Y ), letg(i) = P(L = i), i = 1, 2, . . ., which is given by

g(i) =i∑

n=0

fn∗(i)P(N = n), i = 1, 2, . . . ,

where fn∗(i) denotes the n-fold convolution of f at i. The number of computationsrequired for determining g(i) is of the order i2. The result by Panjer states that

g(i) =λ

i

i∑j=1

j f(j) g(i− j), i = 1, 2, . . . ,

in which case the number of computations required for determining g(i) is of theorder i. By proper scaling on an equidistant grid, the method can also be used fordiscrete nonnegative compounding variates not restricted to integers.

To illustrate the pickup in computational speed using Panjer recursion, we com-pare the computation of the distributions of Λti , with ti = i/10, i = 0, . . . , 200, forpoints x = (xi)i=0,...,8000, using Monte Carlo simulation and Panjer recursion. CPUtime of several computations are given in Table 8.1. In addition to the computationusing Panjer recursion, Λt was simulated at 200 time points with 8000 grid pointseach, with 1000, 2000 and 5000 simulations. The simulation results are comparedwith the numerical computation using Panjer recursion by computing the simula-tion mean square error (MSE) relative to the value obtained by Panjer recursion.The CPU times are also given.

8.1.3 Algorithm

Suppose we wish to compute default probabilities P (uj , uj + ti), j = 1, . . . J , i =1, . . . , N .1 The full simulation algorithm, outlined below, is given in pseudo-codein Algorithm 1. For each ti, we compute the distribution of Lti on an equidistantspace grid x1, . . . , xM . Next, we simulate K paths of (σ,X), yielding (σuj

)kj=1,...,J

and (Xuj )kj=1,...,J , k = 1, . . . ,K. For each uj , we check if default has occurred.1For notational simplicity we compute an r × n matrix of default probabilities; other setups of

time points and time-to-maturities are possible.


However, simulating on a discrete time grid underestimates the occurrence of thedefault event. This is overcome by sampling an indicator variable that determineswhether default has occurred between two time points. Taking into account thatX is a Brownian motion with a continuous time-change, the indicator takes value1 with probability e−2(b−Xuj−1 )(b−Xuj

)/(Λuj−Λuj−1 ), j = 2, . . . J . (cf. [Karatzas and

Shreve (1998), p. 265] or [Glasserman (2004), p. 368]). For each time point uj andtime-to-maturity ti, we determine Λti+uj

−Λujby computing the deterministic part

and then the expectation using the distribution Lti .2 In this way we obtain a term

structure of default probabilities, which serves as the basis for computing creditspreads according to Equation (3.4.2). In the algorithm we use the credit triangleto compute credit spreads, cf. Equation (4.2.2).

2Actually, the deterministic part need not be computed for every simulation, hence for efficiencythe computation of Line 19 should take place outside the loop k = 1, . . . , K.

8.1. Computation of default probabilities and credit spreads 79

Require: t1 = 0 < . . . < tN // time gridRequire: x1 = 0 < . . . < xM // space gridRequire: u1, . . . , uJ // desired maturitiesRequire: K // number of simulationsRequire: b // default barrierRequire: a, θ, λ, σ2

0 , F // volatility process parameters, F jump size distribution1: // Panjer recursion2: for i = 1 to N do3: for j = 1 to J do4: compute P(Lti ∈ [xj−1, xj))5: end for6: end for7: // simulation step8: for k = 1 to K do9: τk ←∞ // default time of k-th simulation

10: for j = 1 to J do11: simulate σk

ujand Xk

uj

12: sample d← 1minuj−1<s≤ujXs≤b cond. on Xuj−1 and Xuj // (see text)

13: if d = 1 or Xkuj≤ b then

14: τk ← uj

15: P k(uj , uj + ti)← 1, sk(uj , ti)← 0 // for all i = 1, . . . , N16: next k // exit k-th simulation17: end if18: for i = 1 to N do19: h←

`1− e−ati

´σ2

uj/a +

R uj+ti

ujθ(r)

“1− e−a(uj+ti−r)

”dr

20: P k(uj , uj + ti)← 2PM

m=1 N

„b−Xuj√

h+xm−1/a

«P (Lti ∈ [xm−1, xm))

21: sk(uj , uj + ti)← −(1−R) ln(1− P (uj , uj + ti))/ti // credit triangle22: end for23: end for

24: end for

Algorithm 1: Computation of conditional default probabilities


8.2 Calibration

Calibration is the process of assigning the parameters of the model such that themodel reproduces market prices. One set of market prices is the term structure ofcredit spreads (or default probabilities). Further market prices, such as prices ofdefault swaptions, provided they are available and liquid, may be suitable for cali-brating the dynamics. In the absence of a liquid market for such claims, calibratingthe dynamics via historical data may be a feasible alternative.

The Overbeck-Schmidt model has no free parameters for specifying the dynam-ics; the only free parameter – the deterministic time-change – is chosen so thatthe model reproduces a given term structure. In the LOU model the deterministicfunction θ and the initial variance σ2

0 will be chosen to reproduce a given termstructure. The remaining parameters – mean reversion constant a, jump intensityλ, jump size distribution F , barrier b – are chosen to determine the dynamics. Itis important to note, however, that the deterministic function θ influences the dy-namics and that the parameters for the dynamics influence the calibration of θ. Itis also the case that calibration to a given term structure imposes some restrictionson the dynamics parameters – in other words, given a set of dynamics parametersit is not possible to calibrate to an arbitrary term structure. The overall calibrationprocess will be to assign parameters for the dynamics first and then to calibrate tothe spot curve.

The allocation of the parameters to spot curve calibration and dynamics cali-bration is justified as follows: in a model with a jump intensity of zero, the resultingtime-change process is deterministic, which corresponds to the Overbeck-Schmidtmodel. In this case, the only parameters that are relevant for calibration to a givenspot curve are the initial variance σ2

0 and the deterministic function θ, and the dy-namics are fixed by the deterministic time-change. Only when the jump intensityis greater than zero do the dynamics change, in which case all parameters allocatedto the dynamics calibration become relevant for the dynamics.

If analytic calibration is not possible, it is important to know whether theproblem is well-posed at all. See e.g. [Kirsch (1996); Engl et al. (1996);Tikhonov et al. (1995)] for the following. Loosely speaking, a mathematical modelfor a physical problem is well-posed if it has the following three properties:

(i) There exists a solution of the problem (existence).(ii) There is at most one solution of the problem (uniqueness).(iii) The solution depends continuously on the data (stability).

This is formalised as follows:

Definition 8.2. Let X and Y be normed spaces and let f : X → Y be a (linear ornonlinear) mapping. The equation f(x) = y is called properly posed or well-posedif the following holds:

8.2. Calibration 81

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

surv

ival

pro

bab

ility

maturity [yrs]

default intensity 3%calibrated (Panjer)

poss. term structure in 1 yr.

Fig. 8.1 Calibration example that leads to an inconsistent model.

(i) Existence: For every y ∈ Y there is (at least) one x ∈ X such that f(x) = y.(ii) Uniqueness: For every y ∈ Y there is at most one x ∈ X which f(x) = y.(iii) Stability: The solution x depends continuously on y, i.e., for every sequence

(xn)n≥0 xn ∈ X, with f(xn)→ f(x) as n→∞ it follows that xn → x.

Equations for which at least one of these properties does not hold are called improp-erly posed or ill-posed.

If the calibration problem is ill-posed it may be necessary to introduce furtherconstraints for calibrating. For a general overview on calibration see e.g. [Cont andTankov (2004a), Chapter 13] and [Cont (2006); Cont and Tankov (2004b)].

8.2.1 Calibration to a term structure of default probabilities

Assume given a set of default probabilities P (Ti) := P(τ ≤ Ti), T1 < · · · < Tn,derived from market-given credit spreads (together with a recovery assumption).For fixed mean reversion a, barrier b, jump intensity λ and jump size distributionF , the objective is to determine σ2

0 and θ to match the given default probabilities.Since default probabilities in the LOU-model are expectations, cf. Equation (6.2.1),there is in general no analytic method to calibrate. Moreover, requiring absenceof arbitrage in the model, it is not even guaranteed that an exact solution exists.To rule out arbitrage, we require that P (t, T ), T ≥ t, be strictly increasing, forevery t ≥ 0, P–a.s., cf. Section 4.1. Clearly, by inspection of Equation (6.2.1), thiscondition is met if the time-change Λ is strictly increasing P–a.s., or, equivalently,if σ2

t > 0, t ≥ 0, P–a.s.. That an exact solution to the calibration problem with acertain set of dynamics parameters may not exist, is demonstrated by the followingexample. (The method of calibration is explained in detail further below.)

Example 8.3. Let us choose Ti = i, i = 0, . . . , 5 and P (Ti) = 1 − e−0.03Ti . The


model parameters are chosen to be a = 3, b = −2.5; the variance process is driven bya Poisson process with intensity λ = 10. Let θ be piecewise constant in the intervals(Ti−1, Ti], i = 1, . . . , 5. The values θ(Ti), i = 1, . . . , n, are obtained successivelyby matching the corresponding default probabilities. The values obtained lie in[−3.25,−2.8] and σ2

0 = 2.5521. The survival probabilities and the fitted values areshown in Figure 8.1. The root mean square error between the input values and thecalibrated values is sufficiently small (< 10−8) to consider this as a solution to thecalibration problem. However, we have posed no constraints on the values of θ.

Additionally, in the Figure the survival probabilities 1− P (1, Ti), Ti = 2, . . . , 5,conditional on X1 = −2.49 and σ2

1 = −2.9544, are shown. Clearly, these are notdecreasing. It remains to show that a term structure with this property is realisedwith strictly positive probability. First, σ2

1 attains the above value if the drivingPoisson process has not jumped until time 1, so that P(σ2

1 = −2.9544) = P(Z1 =0) = e−λ > 0. Moreover, P (1, Ti) is continuous in X1 for every Ti, so that thereexists ε > 0 such that for any X1 ∈ (−2.49 − ε,−2.49 + ε) the term structure isinconsistent, and finally it follows easily that P(X1 ∈ (−2.49 − ε,−2.49 + ε), σ2

1 =−2.9544) > 0. On the other hand, posing the necessary constraints on the valuesof θ to ensure that any term structures possible in the model are consistent, leadsto a large calibration error.

Even though the example demonstrates that the calibration problem is essen-tially an ill-posed problem, satisfactory calibration quality to a given term-structuremay always be obtained. Indeed, a model without jump component is equivalent tothe OS-model, where analytic and exact calibration is possible. By choosing suit-ably moderate jump dynamics, an arbitrary calibration quality may be achieved, aswe shall see below.

We calibrate numerically by minimising the error between market-given andmodel-computed default probabilities. In the following, we shall always assume θto be piecewise constant,

θ(t) =n∑i=1

θ(Ti)1(Ti−1,Ti](t), t > 0, (8.2.1)

with the convention T0 = 0. We define the root mean square error (RMSE) betweenmarket default probabilities and model default probabilities as

δ(σ20 , θ;P, a, b, λ, F ) :=

√√√√ n∑i=1

Ti − Ti−1

Tn

(P (Ti)− 2EN

(b/√

ΛTi

))2

, (8.2.2)

where the expectation denotes the model-given default probability for maturity Ti,cf. Equation (6.2.4) and ΛTi

by (cf. Equation (6.4.3’))

ΛTi =(

1− e−a Ti)) σ2

0

a+

i∑j=1

θ(j)[Tj − Tj−1 −

e−a(t−Tj) − e−a(t−Tj−1)

a

]+

1a

∑0<u≤Ti

(1− e−a(Ti−u)

)∆Zu. (8.2.3)

8.2. Calibration 83

In order to rule out arbitrage, we require that (cf. Equation (6.4.2’))

σ2t = e−at σ2

0 +n∑i=1

θ(Ti) e−at(ea(t∧Ti) − ea(t∧Ti−1)

)+∑

0<u≤t

e−a(t−u) ∆Zu > 0, t ≥ 0.

Taking into account that jumps are positive, the condition is satisfied if θ satisfies

θ(Ti) > −σ2

0 +∑i−1j=1 θ(Tj)

(eaTj − eaTj−1

)eaTi − eaTi−1

, i = 1, . . . , n. (8.2.4)

Define the set

Θ = (θT1 , . . . , θTn) : (θT1 , . . . , θTn) satisfies (8.2.4) .

For the model-given probabilities to be well-defined requires additionally that λ ≥ 0,a > 0, b ≤ 0 and F (0) = 0. Under these conditions, the solution to the calibrationproblem is then given by

(σ?20 , θ?(T1), . . . , θ?(Tn)) := arg minσ2

0∈R+,(θ(T1),...,θ(Tn))∈Θδ(σ2

0 , θ;P, a, b, λ, F ). (8.2.5)

For notational convenience, we denote the resulting RMSE by δ?.To determine δ? we used the sequential quadratic programming method pro-

vided by [GNU Octave (2008)] (function sqp).3 This is an extension to quadraticprogramming for nonlinear constrained optimisation problems; see e.g. [Boggs andTolle (1995)] for an overview. To increase the accuracy of calibration we minimised1000 δ2 instead of δ.

Example 8.4. Assume given default probabilities at times Ti = i (years), i =1, . . . , 10, determined by a constant hazard rate h = 0.03, so that P (Ti) = 1−e−hTi ,i = 1, . . . , 10. For different choices of dynamics parameters, we calibrated themodel to these default probabilities. In particular, we consider several jump sizedistributions, such as a constant jump size, a continuous jump size distribution anda two-point distribution where small jumps occur with high probability and verylarge jumps with small probability. For each of these jump size distributions, wecompare the calibration quality for different choices of mean version a and jumpintensity λ.

The distribution of L0,Ti , i = 1, . . . , 10, is computed on an equidistant grid of5000 points in the interval [0, 120]. The barrier is b = −3. The mean reversion a

and the jump intensity λ are chosen from the set 1, 2, 3, 5, 10 and the followingjump size distributions are considered:3Other minimisation routines, such as the Nelder-Mean simplex algorithm, see e.g. [Press et al.

(1992), Section 10.4], were also considered, but best result were obtained with the sequentialquadratic programming method.


1e-10

1e-08

1e-06

0.0001

0.01

1

0 2 4 6 8 10

δ?

λ

a = 1.0a = 2.0a = 3.0a = 5.0

a = 10.0

1e-10

1e-08

1e-06

0.0001

0.01

1

0 2 4 6 8 10

δ?

λ

a = 1.0a = 2.0a = 3.0a = 5.0

a = 10.0

1e-10

1e-08

1e-06

0.0001

0.01

1

0 2 4 6 8 10

δ?

λ

a = 1.0a = 2.0a = 3.0a = 5.0

a = 10.0

1e-10

1e-08

1e-06

0.0001

0.01

1

0 2 4 6 8 10

δ?

λ

a = 1.0a = 2.0a = 3.0a = 5.0

a = 10.0

Fig. 8.2 RMSE’s of default probabilities for different parameter sets obtained by calibrating todefault probabilities P (Ti) = 1 − e−0.03 Ti , i = 1, . . . , 10. Each figure contains six lines thatcorrespond to parameter a = 1, 2, 3, 5, 10 (ordered from top to bottom at λ = 10). Top left, topright, bottom left and bottom right: jump size distribution as in (i)-(iv), respectively, of Example8.4.

(i) The jump size is 1/4.(ii) The jump size is 1/2.(iii) The jump size distribution is exponential with parameter ν = 4, i.e., F (x) =

1− e−νx.(iv) The jump size is 0.1 with probability 0.95 and 20 with probability 0.05. Here,

we enlarged the grid for computing the distributions of (L0,Ti)i=1,...,10 to 11000points on the interval [0, 264].

The RMSE’s δ? are given in Figure 8.2. In all cases, the quality of the calibrationincreases with increasing mean reversion constant; this may be partially due to thescaling of jumps by 1/a, cf. Equation (8.2.3). Furthermore, the calibration errorincreases with increasing jump intensity – clearly, a higher jump intensity or sizeincreases the likelihood of default. Therefore, when increasing the jump intensityor size it may be harder to calibrate well as the dynamics parameters may induce ahigher default probability than wished. This is also seen by comparing RMSE’s of

8.2. Calibration 85

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10

Q(0

,T)

T

given term structureλ = 10.0, a = 1.0λ = 1.0, a = 10.0λ = 2.0, a = 3.0

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10

Q(0

,T)

T


0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10

Q(0

,T)

T


0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10

Q(0

,T)

T


Fig. 8.3 Several calibration results of good and bad fits of survival probability term structures.Top left, top right, bottom left and bottom right: jump size distribution as in (i)-(iv), respectively,of Example 8.4.

the cases (i) and (ii), where the jump size is constant. In Figure 8.3, some examplesof particularly good and particularly bad fits of term structures are shown.

Figure 8.4 contains an example of the impact of individual parameters on theterm structure. The term structures shown were obtained by varying the meanreversion a, the barrier b, the jump size distribution F , the jump intensity λ, theinitial variance σ2

0 and the deterministic function θ.


0.75

0.8

0.85

0.9

0.95

1

0 1 2 3 4 5

Q(0

,T)

T

a : 10.05.03.02.01.0

0.75

0.8

0.85

0.9

0.95

1

0 1 2 3 4 5Q

(0,T

)T

b : −5.0−3.0−2.0−1.0

0.75

0.8

0.85

0.9

0.95

1

0 1 2 3 4 5

Q(0

,T)

T

pc : 1.00, 0.000.97, 0.030.95, 0.050.93, 0.07

0.75

0.8

0.85

0.9

0.95

1

0 1 2 3 4 5

Q(0

,T)

T

λ : 1.02.05.0

10.0

0.75

0.8

0.85

0.9

0.95

1

0 1 2 3 4 5

Q(0

,T)

T

σ20 : 0.1

1.05.0

20.0

0.75

0.8

0.85

0.9

0.95

1

0 1 2 3 4 5

Q(0

,T)

T

θ : 0.001.002.003.00

Fig. 8.4 Impact of individual parameters. The basic parameter set is a = 3, b = −3, λ = 2,σ20 = 3, θ ≡ 0, jump size in 0.1, 20 with probabilities 0.95, 0.05, respectively. In each case, the

curves are ordered from top to bottom at T = 5.

8.2. Calibration 87

1e-08

1e-06

0.0001

0.01

1

100

0 2 4 6 8 10

δ? s

λ

a = 1.0a = 2.0a = 3.0a = 5.0

a = 10.0

1e-08

1e-06

0.0001

0.01

1

100

0 2 4 6 8 10

δ? s

λ

a = 1.0a = 2.0a = 3.0a = 5.0

a = 10.0

1e-08

1e-06

0.0001

0.01

1

100

0 2 4 6 8 10

δ? s

λ

a = 1.0a = 2.0a = 3.0a = 5.0

a = 10.0

1e-08

1e-06

0.0001

0.01

1

100

0 2 4 6 8 10

δ? s

λ

a = 1.0a = 2.0a = 3.0a = 5.0

a = 10.0

Fig. 8.5 RMSE’s of credit spreads corresponding to the RMSE’s of default probabilities of Figure8.2. Each figure contains six lines that correspond to parameter a = 1, 2, 3, 5, 10 (ordered fromtop to bottom at λ = 10). Top left, top right, bottom left and bottom right: jump size distributionas in (i)-(iv), respectively, of Example 8.4.

8.2.2 Shape of credit-spread term structure

Now consider the credit-spread term structure. Continuing with Example 8.4, theRMSE’s for credit spreads, denoted by δ?s , are given in Figure 8.5. Here, the spread(in basis points) was computed from market-given default probabilities by

s(Ti) = 105 (1−R)P (Ti)∑i

j=1(1− P (Tj))(Tj − Tj−1), i = 1, . . . , n,

and accordingly for model-given default probabilities. The recovery rate was R =0.4. For some parameter sets, the RMSE between market-given and model-givencredit spreads is as small as approximately 10−6 (in basis points).

Recall from Section 3.1.2 that the term structure of credit spreads may assumedifferent shapes. Typically, an investment grade company’s term structure is up-ward sloping, reflecting lower default risk in the near future compared to higheruncertainty in the long term. A speculative-grade company may have an invertedterm structure, indicating that the firm faces higher short-term default risk, but is


more likely to survive in the long-term conditional on survival in the short-term.A common observation is that credit spreads are strictly positive as time-to-

maturity tends to zero, indicating that default may happen suddenly and unex-pectedly. In Section 7.2 we established that the model is not capable of producingthis property - credit spreads vanish as time-to-maturity tends to zero, and thedefault time is predictable. However, the possibility of large jumps in the variancemay allow for “near-jump-to-default” events. We would then expect the spreadterm structure to be very steep at the short end.

We give some examples of credit spread term structures in Figure 8.6. Thesecorrespond to the default probability term structures of Figure 8.4. By choosing ex-treme values for either the barrier or the initial variance, we obtain sharply humpedterm structures that approximate inverted term structures. Both cases reflect alow credit quality: default becomes more likely as either the credit quality processapproaches the barrier or as the variance increases.

8.2. Calibration 89

0

50

100

150

200

250

300

350

400

450

0 2 4 6 8 10

s(0,

T)

T

a : 1.02.03.05.0

10.0

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 2 4 6 8 10s(

0,T

)T

b : −1.0−2.0−3.0−5.0

0

50

100

150

200

250

0 2 4 6 8 10

s(0,

T)

T

pc : 0.925, 0.0750.950, 0.0500.975, 0.0251.000, 0.000

0

100

200

300

400

500

600

700

0 2 4 6 8 10

s(0,

T)

T

λ : 10.05.02.01.0

0

500

1000

1500

2000

2500

3000

0 2 4 6 8 10

s(0,

T)

T

σ20 : 20.0

5.01.00.1

0

100

200

300

400

500

600

700

800

900

0 2 4 6 8 10

s(0,

T)

T

θ : 3.002.001.000.00

Fig. 8.6 Impact of individual parameters on credit spread term structure. The basic parameterset is a = 3, b = −3, λ = 2, σ2

0 = 3, θ ≡ 0, jump size in 0.1, 20 with probabilities 0.95, 0.05.In each case, the curves are ordered from top to bottom at T = 10.


0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2

P

P(τ ∈ (t, T ]|Xt, σ2t )

(a)(b)(c)(d)

0

0.2

0.4

0.6

0.8

1

0 0.01 0.02 0.03 0.04 0.05 0.06

P

λ(t, T )

(a)(b)(c)(d)

Fig. 8.7 Distributions of P(τ ∈ (t, T ]|Xt, σ2t ) (left) and λ(t, T ) (right) conditional on τ > t, cf.

Equation (7.5.2) and Proposition 7.16. We choose t = 1, T = 5 and an initial hazard rate of 3%.The diamonds mark the initial 5-year default probability, resp. hazard rate. The parameters aregiven in Table 8.2.

8.3 Dynamics

In general, the prices generated by the model should be consistent with any liquidmarket prices. For example, option prices, if available, provide a source of informa-tion for calibrating risk-neutral dynamics. In the absence of such information (asis currently the case in the CDS market) one may be forced to resort to informa-tion from historical time series, often called real-world dynamics. For example, theobjective may be to reproduce some statistics, such as to match a certain numberof moments of a given time series of log-returns of CDS spreads (of a particularmaturity). Clearly, the risk-neutral dynamics may differ strongly from the real-world dynamics. It may also be the case that different parameter sets calibratewell, in which case additional aspects, such as the stability of parameters, may havebe taken into account. For an in-depth discussion of calibrating dynamics in thecontext of exponential Levy models, see [Cont and Tankov (2004a), Chapter 13].

8.3.1 Examples

We have seen that different choices of dynamics parameters calibrate well to a giventerm structure. To illustrate that indeed the model is capable of generating differentdynamics, Figure 8.7 shows the distribution of the 5-year default probability andterm hazard rate after one year conditional on no default for different parametersets (the Figure is analogous to the examples given in the analysis of the OS-modeland the dynamics of the LOU model). The corresponding parameters are given inTable 8.2. In each case, the initial variance σ2

0 and the function θ are calibrated tomatch default probabilities corresponding to an initial hazard rate of 3%.

Case (a) is a model with a deterministic time-change, and hence corresponds

8.3. Dynamics 91

Table 8.2 Parameters, RMSE’s and characteristics of dynamics examples. The jump sizesare given by the first column and the corresponding probabilities in the second column ofeach matrix in the row of parameter F . Additionally, the mean, variance, skewness andexcess kurtosis of the term hazard rate distributions are given.

(a) (b) (c) (d)

Parametersa 3 3 1 1b -3 -3 -3 -2λ 0 2 1 0.0305

F

»0.1 (0.95)20 (0.05)

– »0.1 (0.95)10 (0.05)

–1(0,2.5·104]

σ20 3.16 4.59 3.25 10−6

θ ∈ [0.23, 1.32] ∈ [−0.22, 0.04] ∈ [−1.10, 0.37] 0

RMSE’sδ? < 10−8 < 10−8 < 10−9 < 10−3

δ?s < 10−5 < 10−4 < 10−5 0.56

Characteristics of term hazard rate distributionsmean 0.0521 0.0304 0.0199 0.0311

std. dev. 0.0794 0.0341 0.0315 9.45 · 10−5

skewness 3.39 6.98 9.27 n/aexc. kurtosis 16.52 83.41 129.12 n/a

The computed values of the fields marked “n/a” are numerically unstable. Skewness is µ3/σ3,with µ3 the third centered moment and σ the standard deviation; excess kurtosis is µ4/σ4 − 3,with µ4 the fourth centered moment.

to the OU-model. Case (d) was chosen such that σ20 = 10−6 and θ ≡ 0, so that

the variance is very small until the first jump occurs. The jump size was chosenvery large (25000) so that, heuristically, a single jump leads to default very quickly.Loosely speaking, case (d) can be considered an approximation of a reduced-formmodel with a deterministic and constant intensity: the credit quality process ex-hibits practically no movement, until the first jump occurs, which leads to default, cf.Section 4.4. This is also reflected in the jump intensity λ = 0.0305, which is approx-imately the initial hazard rate, and in P(τ ∈ (1, 5)|X1, σ

21) ≈ 1− e−0.03·4 = 0.11308

conditional on no default until time 1.4 The characteristics of cases (b) and (c) are“in-between” cases (a) and (d): in both cases, the variance process exhibits jumps.However, the jumps dynamics are moderate enough for the level of the varianceprocess induced by θ and σ2

0 , both of which are obtained by calibration, to be sig-4If the initial hazard rate is not constant, then a calibration where the variance moves purely

by jumps cannot be attained. This is due to the fact that the jump intensity of the variance’scompound Poisson process is constant, whereas a non-constant, deterministic hazard rate requiresthe jump intensity to be non-constant and deterministic. The former can be incorporated byspecifying the jump process as an additive process, which is a process with independent incrementsthat is stochastically continuous, but whose increments are not necessarily stationary.


0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 0.05 0.1 0.15 0.2

λ

(a)mean

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 0.05 0.1 0.15 0.2

λ

(b)mean

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 0.05 0.1 0.15 0.2

λ

(c)mean

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2

λ

(d)mean

Fig. 8.8 First differences of term hazard rate distributions of Figure 8.7.

nificantly above zero. In other words, the variance processes of both cases featurejumps and a significant mean level.

The first differences of the distributions are shown in Figure 8.8. Additionally,the mean, variance, skewness and excess kurtosis of the distributions are shown inTable 8.2. The cases (a)-(c) exhibit positive skewness. The skewness of the cases(b) and (c), i.e., those cases where the variance features jumps, but maintains asignificant mean level, is larger than the skew of the purely continuous variancebehaviour of case (a). This is what we expect: in the cases involving jumps, thehazard rate is driven by continuous movement and by positive jumps. (That theskewness of case (c), where large jumps in the variance are of size 10, is greaterthan the skewness of case (b), where large jumps in the variance are of size 20, isdue to the scaling of jumps by 1/a, cf. Equation (8.2.3).) The kurtosis is also largerfor the cases involving jumps. We can also see that the purely continuous processevolves in a more volatile behaviour.

The distributions were computed according to Equations (7.5.2) and (7.6.1). Forthe cases (b) and (c), the distributions of (L0,Ti

)i=1,...5 (for calibration) and of L1,5

were computed on an equidistant grid of 5000 points in the interval [0, 120]. For

8.3. Dynamics 93

case (d) the interval size was [0, 105]. For case (a) the interval size was [0, 1] andthe grid size was reduced to 1000 points. The values of P (τ ∈ (1, 5]|X1, σ

21) were

computed for an equispaced (∆ = 0.01) grid of X1 ∈ [−b, 5] and a log-scaled gridσ2

1 ∈ [0.1, 40] with 20 points. The required inverses were then determined from thesevalues. Finally, the distributions were computed for an equispaced (∆ = 0.001) gridon [0, 1] via 5000 simulations of (σ2

1 ,Λ1).

8.3.2 Parameters and dynamics

We discuss each parameter that influences the dynamics. These are the parametersof the variance process, i.e., mean reversion a, jump intensity λ, jump size distri-bution F , initial variance σ2

0 and deterministic function θ. Additionally, as will beoutlined below, we include the barrier b in our discussion. The initial variance σ2

0

and deterministic function θ, although determined by calibration to a given termstructure, influence the dynamics, implying that the choice of dynamics and thecalibration to the spot term structure cannot be separated.

Mean reversion a The mean reversion parameter a determines the “speed” atwhich the variance reverts to its mean, cf. Equation (6.4.1). As can be seen fromEquation (8.2.3), the initial variance and jump size are scaled by 1/a. Additionally,the factors (1 − e−aT ) and (1 − e−a(T−u)) “dampen” the values of θ and Z: theimpact of θ(u) and Zu, u < T , increases with increasing maturity T . The smaller ais chosen, the stronger this effect. It is a frequent observation that small a leads topoor calibration: Large values of θ are determined for a satisfactory calibration atthe short end, which are then too large for a proper calibration to longer maturities.

Barrier b Strictly speaking, the inclusion of the barrier as a parameter is re-dundant: For any two barriers b and b′ we may obtain the same spot curve anddynamics by proper scaling of σ2

0 , θ and the jump size. On the other hand, wehave seen in Figure 8.6 that a change in the barrier affects the shape of the curve.To obtain the desired shape, it is more straightforward to adjust one parameter,namely the barrier, instead of adjusting the set of other parameters.

Jump intensity λ and jump size distribution F The jump intensity deter-mines the average number of jumps to occur per year. By the results from Section7.1, the jump intensity is equal for the variance process and the term structuresof default probabilities and credit spreads. The jump size of Z propagates to thecredit spread jump size in a monotone way: the larger the jump of Z, the largerthe jump in the credit spread – this is easily seen by the by monotonicity of defaultprobabilities in σ2 and extended to credit spreads via the CDS valuation formula(3.4.2).

However, the jump size also depends on the level of X and σ2. For defaultprobabilities, the jump size is not monotone in X. This is easily seen as fol-


-3 -2 -1 0 1 2 3Xt 0.1

1

10

σ2t

0

0.005

0.01

0.015

0.02

0.025

-3 -2 -1 0 1 2 3Xt 0.1

1

10

σ2t

0

50

100

150

bp

-3 -2 -1 0 1 2 3Xt 0.1

1

10

σ2t

0

0.1

0.2

0.3

0.4

0.5

-3 -2 -1 0 1 2 3Xt 0.1

1

10

σ2t

0

500

1000

1500

2000

bp

Fig. 8.9 Jump size of 5-year default probability (left) and credit spread (right) when a jump ofsize 0.1 (top), resp. 20 (bottom) occurs. Parameters correspond to case (b) of Table 8.2.

lows: Assume that at time t there is a jump, ∆σ2t > 0. For any maturity

T > t, limXt↓b ∆P (t, T ) = 0, as the jump size is bounded by 1 − P (t−, T ) andlimXt↓b P (t−, T ) = 1. On the other hand, limXt↑∞ ∆P (t, T ) = 0 since ∆σ2

t < ∞P–a.s.. By Proposition 7.1, ∆P (t, T ) > 0 for any fixed Xt, so ∆P (t, T ) cannot bemonotone in Xt.

For credit spreads, the situation is not so clear. Examples indicate that thejump size of credit spreads is monotone in X; see Figure 8.9 for an example of thejump size as a function of Xt and σ2

t .It should be noted that the jump intensities of real-world dynamics and risk-

neutral dynamics can vary widely. Some properties such as whether the driving Levyprocess is of finite or infinite activity are equal under the real-world measure andany equivalent martingale measure. However, in a risk-neutral context, calibrationessentially means determining the Levy measure of a driving Levy process in orderto fit some market-given prices. For example, in the case of a compound Poissonprocess, the Levy measure is ν = λF , and the so obtained jump intensity λ neednot coincide with the real-world jump intensity.

8.4. Uniqueness and stability of calibration parameters 95

Initial variance σ20 and deterministic function θ Although σ2

0 and θ arechosen by calibration to a given term structure, their effect on the dynamics aresignificant. Intuitively, a decrease in the Levy measure of Z (i.e., decrease in jumpintensity or jump size), decreases the probability of default, which is compensatedby a higher choice of θ when calibrating. Consequently, the variance process willmaintain a higher deterministic level, causing the credit quality process to evolve ina more volatile fashion in order to hit the default barrier with a certain probability.This is illustrated by comparing cases (a) and (d) of the previous example, Section8.3.1.

8.3.3 Evolution of the term structure shape

In Section 8.2.2 it was shown that the shape of the term structure becomes inverted(more precise: sharply humped) with increasing barrier b and with increasing initialvariance σ2

0 , cf. Figure 8.6. By inspection of the formula for conditional defaultprobabilities we see that default probabilities at time t do not depend on the actuallevel of the credit quality process Xt, but rather on the distance-to-default b −Xt. Both, a decreasing distance-to-default and an increasing variance, represent adecrease in credit quality, which eventually results in an inverted credit spread termstructure.

8.4 Uniqueness and stability of calibration parameters

We are interested in the behaviour of the calibration parameters when the givencredit spread, resp. default probabilities, change. Recall Definition 8.2, which statesconditions for a problem to be well-posed. In Section 8.2.1 we established that con-dition (i), i.e., existence of a solution to the calibration problem, is not fulfilled forarbitrary dynamics parameters. However, by restricting the dynamics parametersto suitably moderate dynamics, calibration to a term structure with arbitrary pre-cision may be achieved. In the extreme case, when jumps are omitted from thevariance process, an exact (and unique) solution for calibration to a term structureexists. In the following, we examine conditions (ii) and (iii), which are concernedwith uniqueness and stability of a solution.

Uniqueness Suppose first given the parameters a, b, λ, F and consider uniquenessof the term structure with respect to σ2

0 and θ. Assume given a set of defaultprobabilities P(τ ≤ Ti), i = 1, . . . , n. Inspection of Equation (8.2.3) reveals thatwe cannot rule out that different choices of (σ2

0 , θ) lead to equal values of ΛTi ,i = 1, . . . , n. Fixing σ2

0 and posing additional constraints on θ, such as θ > 0, impliesuniqueness of the solution, but reduces the solution space. Additional constraintsmay enforce a unique solution, such as additionally minimising σ2

0 and θ. The actualconstraints chosen may depend on the intended application of the model.


0.2

0.4

0.6

0.8

1

1.2

1.4

1 2 3 4 5

T

(a) θ?

θ?

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

1 2 3 4 5

T

(b) θ?

θ?

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

1 2 3 4 5

T

(c) θ?

θ?

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

1 2 3 4 5

T

(d) θ?

θ?

Fig. 8.10 Calibration parameters for cases (a)-(d): (θ?(1), . . . , θ?(5)) and (θ?(1), . . . , θ?(5)) (dot-ted).

Next, consider the question of whether different parameter sets a, b, λ, F pro-duce the same dynamics. This requires consideration of the dynamics after calibra-tion to (σ2

0 , θ). We leave this question open, but conjecture that, when the jumpsize distribution is fixed, the dynamics will differ, as the behaviour of the jump partand the mean level of the variance process will be different.

Stability In our setting, stability of the calibration problem, i.e., condition (iii)of Definition 8.2 translates into the following question: How does recalibration witha changed credit spread term structure affect σ2

0 and θ?Consider the four examples (a)-(d) from Section 8.3.1. For each example,

given the calibrated model with parameters (σ?20 , θ?(1), . . . , θ?(5)), we computedthe CDS spreads using the credit triangle, cf. Equation (4.2.2), and a recovery rateof R = 0.4. We then shifted (“bumped”) the term structure upward by 1 basispoint, and determined the respective default probabilities, again using the credittriangle. Using the same dynamics parameters, we recalibrated the model, yielding(σ?20 , θ?(1), . . . , θ?(5)). The differences between σ?20 and σ?2 lie within [−0.09, 0.09].The calibrated values of θ are shown in Figure 8.10. In order to interpret the abso-

8.4. Uniqueness and stability of calibration parameters 97

Table 8.3 RMSE’s and characteristics of bumped dynamics examples.

(a) (b) (c) (d)

RMSE’sδ? < 10−8 < 10−8 < 10−9 < 10−4

δ?s < 10−5 < 10−4 < 10−5 0.4513

Characteristics of hazard rate distributionsmean 0.0523 0.0311 0.0205 0.0394std. dev. 0.0795 0.0364 0.0332 0.0140skewness 3.38 6.90 8.76 6.04exc. kurtosis 16.47 79.15 114.69 89.00

lute figures, the magnitude of the contribution of θ to the overall behaviour of thevariance process must be taken into account. For example, in case (a) the varianceis deterministic and has no jump component, hence it is driven exclusively by thecontinuous part, which is calibrated to σ2

0 and θ. The difference in the original andbumped version of the calibration parameters is comparatively small. On the otherhand, in case (d) the variance is driven mainly by the jump component with a jumpsize of 25000 and a jump intensity of 0.0305. The difference in the calibrated valuesof θ, although as large as 0.5, is relatively small, when considered with respect toits contribution to the overall variance process.

Additionally, the RMSE’s and statistics of the dynamics of the bumped cases aregiven in Table 8.3. The skewness and kurtosis have all decreased when comparedto the original cases. This is explained as follows: Recalibration changes increasesthe mean level of the variance, while the jump behaviour is unchanged, leading toa more volatile behaviour of the credit quality process.

Chapter 9

Valuation of credit derivatives

We now turn to valuation of credit derivatives. First, we answer the question ofwhether the filtration (Ft)t≥0, to which the credit quality process (X,σ2) is adapted,is suitable for pricing. Then, we consider valuation examples for leveraged credit-linked notes (Section 9.2) and default swaptions (Section 9.3). In both cases, weuse the example models from Section 8.3.1.

9.1 Information flow and the pricing filtration

So far, we have made no assumptions about the filtration (Ft)t≥0, other than thatit is rich enough for the credit quality process (X,σ2) to be (Ft)t≥0-adapted (ofcourse, we have made the technical assumption that the filtration satisfies the usualhypotheses). Recall from Section 3.2 that for a market model to be consistentwith arbitrage theory requires that the filtration used for pricing is generated byobservable information. For building trading strategies, the underlying filtrationmust even be generated by the observable prices of traded assets, cf. [Hunt andKennedy (2004), Section 7.3]. In general, a credit quality process (X,σ2) is neitherdirectly observeable nor a traded asset.

Suppose now that we wish to price financial claims derived from credit spreads(or, equivalently, default, resp. survival probabilities). Application of the risk-neutral valuation formula (3.2.1) with conditional default probabilities given bythe model via Equation (6.2.1) is justified only if (Ft)t≥0 is generated by some ob-servable information and if X and σ2 are (Ft)t≥0-adapted. Otherwise, valuationof assets requires that prices are computed using a different – possibly coarser –filtration. One may think of a coarser filtration as the inavailability in the marketof complete information about a company’s state. For example, [Duffie and Lando(2001)] attribute a lack of information to incomplete accounting information.

In Example 3.4 the value Vt at time t of a risky zero-coupon bond with maturityT , under the assumption of independence of (risk-free) interest rates and the defaultindicator process, and assuming no recovery in case of default, was shown to be

Vt = B(t, T ) P(τ > T |Ft),

99

100 Valuation of credit derivatives

where B(t, T ) denotes the price of the corresponding risk-free zero-coupon bond. Ifwe further assume that interest rates are zero, then Q(t, T ) = P(τ > T |Ft), T > t,are traded assets. Otherwise, it is not hard to see that the filtration generated byrisk-free and risky zero-coupon bonds and the filtration generated by risk-free zero-coupon bonds and survival probabilities are equivalent. Consequently, we shallassume that (Ft)t≥0 is generated by (B(t, T ))T≥t and (P (t, T ))T≥t, t ≥ 0 (andpossibly enlarged to satisfy the usual hypotheses).

The assumption that there is indeed a process that drives a company’s creditquality via the information that is available about the company is justified by thestylised facts recorded in Section 3.1.1, namely:

(i) The arrival of bad news about a company causes default probabilities andcredit spreads to jump – regardless of their maturity.

(ii) Good news tend to affect default probabilities and credit spreads gradually –regardless of their maturity.

We shall assume that the credit quality of a firm is indeed driven by a process(X,σ2) as in Proposition 6.9 with σ2 an LOU process driven by a compound Poissonprocess (with respect to the filtration (Ft)t≥0). Furthermore, we assume that theparameters of the credit quality process are known, i.e., θ, σ2

0 , a, b, c, λ, F are F0-measurable.1 In the following Proposition we show that under some assumptionson the deterministic function θ, we have σ(Xt, σ

2t ) ≡ σ((P (t, T ))T≥t), for every

t ≥ 0. It follows that the state of the credit quality process may be derived from agiven term structure, i.e., a given term structure is “invertible”. The general case– loosening the assumption on θ – is more delicate. Here, we show that, for t ≥ 0,(Xt, σ

2t ) is measurable with respect to σ((P (s, T ))T>s, 0 ≤ s ≤ t). The general case

also allows for term structures of a bounded maturity horizon.


as in Proposition 6.9, and suppose that σ2 is driven by a compound Poisson process.Suppose further that there exist u ≥ 0, ε > 0, such that θ(r) ≥ ε, for all r ≥ u.Then, σ(Xt, σ

2t ) = σ(P (t, T )T>t), t ≥ 0.

Proof. Clearly, σ(P (t, T )T>t) ⊆ σ(Xt, σ2t ), t ≥ 0, see e.g. Corollary 6.4. We have

to show that σ(Xt, σ2t ) ⊆ σ(P (t, T )T>t), t ≥ 0. The claim follows by a Functional

Representation Theorem, cf. [Kallenberg (2001), Lemma 1.13], if we show that(Xt, σ

2t ) is a function of P (t, T )T>t.

Fix t ≥ 0 and define for T > t,

gT (x, y) = 2E(

N(


)),

1These are strong assumptions. In particular, it follows from the discussion at the beginning ofSection 8.2.1 that, given a parameter set, not all term structures can be realised in this model.

9.1. Information flow and the pricing filtration 101

with

ΛT − Λt =1− e−a(T−t)

ay +

∫ T

t

θ(u)(

1− e−a(T−u))

du+1aLt,T

Lt,T ∼ CPO(λ(T − t),

(1− e−a(T−S)

)Y),

with S uniformly distributed on (t, T ] and Y ∼ F . By the Independence LemmaA.17, on τ > t, P–a.s.,

P (t, T ) = gT (Xt, σ2t ), T > t,

For (x, y) ∈ R× R+, we show that

(x, y) : gT (x, y) = gT (x, y), for all T > 0 = (x, y) .

Assume that there exist T and (x, y) ∈ R× R+, (x, y) 6= (x, y), such that

gT (x, y) = gT (x, y) (9.1.1)

(otherwise there is nothing to show). Set c := (b− x)/(b−x), and re-write Equation(9.1.1) as

EN

c(b− x)√(1− e−a(T−t))c2y/a+ c2

∫ Ttθ(r)(1− e−a(T−r)) dr + c2Lt,T /a

= EN

c(b− x)√(1− e−a(T−t))y/a+

∫ Ttθ(r)(1− e−a(T−r)) dr + Lt,T /a

. (9.1.1’)

Assume first that c > 1. Since c2Lt,T > Lt,T , Equation (9.1.1’) implies

1− e−a(T−t)

ay >

1− e−a(T−t)

ac2y + (c2 − 1)

∫ T

t

θ(r)(1− e−a(T−t)) dr.

For notational convenience assume that u = 0, i.e., θ(r) ≥ ε, for all r ≥ 0 (otherwiseT and t must be replaced with T ∧ u and t ∧ u in some places below; however, thisdoes not affect the result). Solving for y yields

y > c2y +(c2 − 1)a

1− e−a(T−t)

∫ T

t

θ(r)(1− e−a(T−t)) dr

≥ c2y +(c2 − 1)a

1− e−a(T−t)ε

((T − t)− 1− e−a(T−t)

a

)→∞ as T →∞.

Hence, there is no (x, y) 6= (x, y) such that Equation (9.1.1) holds for all T > t.Now assume that c < 1. In this case, c2Lt,T < Lt,T and Equation (9.1.1’) implies

1− e−a(T−t)

ay <

1− e−a(T−t)

ac2y + (c2 − 1)

∫ T

t

θ(r) (1− e−a(T−t)) dr,


and

y < c2y − (1− c2)a1− e−a(T−t)

∫ T

t

θ(r) (1− e−a(T−r)) dr

≤ c2y − (1− c2)ε(

a(T − t)1− e−a(T−t)

− 1)

→ −∞ as T →∞.

It follows that there is no (x, y) 6= (x, y) such that Equation (9.1.1) holds for allT > t. Since (x, y) was arbitrary, the claim holds for all (x, y) ∈ R× R+.

In the general case, we have the following Proposition:

Proposition 9.2. Let (X,σ2) be a credit quality process as in Proposition 6.9. LetFPt = σ((P (s, T )T>s), 0 ≤ s ≤ t), i.e., FPt is the filtration generated by the defaultprobability term structures up to time t. Then, σ(Xt, σ

2t ) ⊆ FPt .

Proof. Fix t ≥ 0. For any T > t, let gT (x, y) be as in the previous proof. Observethat for fixed x, y 7→ gT (x, y) is invertible, and that for fixed y, x 7→ gT (x, y) isinvertible (both mappings being strictly monotone). By Proposition 7.1, ∆σ2

t > 0if and only if ∆P (t, T ) > 0, T > t. Moreover, the only “random” movements ofσ2 occur in jumps, any movement between jumps conditional on the value attainedby the last jump is deterministic. Consequently, if ∆P (t, T ) = 0, T > t, assumingthat the value of σ2 at the last jump time is known, then σ2

t is deterministic, andXt may be determined by inversion. On the other hand, if ∆P (t, T ) > 0, T > t,then by Equation (7.1.1),

∆P (t, T ) = gT (Xt, σ2t )− gT (Xt, σ

2t−), t < T, P–a.s.,

which can again be inferred by inversion. Hence, (X2u, σ

2u)u≤t is FPt -measurable,

and the claim follows.

9.2 Leveraged credit-linked note

As an example we value a leveraged credit-linked note using the pricing formula(3.5.1), with pricing done via Algorithm 1. The note has a maturity of 5 years anda nominal of e 100. The leverage factor is k = 5, so that the payoff amount andtime are linked to the mark-to-market value of a CDS position with nominal e 500on CDS with a maturity of 5 years at inception. The trigger level is K = e 60.The initial CDS spread term structure is flat at 180 basis point, the recovery rateis 40%. The note is monitored weekly, i.e., at time points t1 < t2 < · · · < t260, with∆ti = 1/52. Denote by V kt the mark-to-market value at time t of the CDS position.The note is unwound at S = infti : −V kti ≥ K, i = 1, . . . , 260. At S, the investorreceives max(e 100 + V kS , 0) and the issuer pays max(−V kS − e 100, 0) (this is thepayoff of the gap option incorporated into the note). The premium earned for thegap option is (k − k)s(0, T ). The risk-free interest rate is constant at 5%.

9.2. Leveraged credit-linked note 103

Table 9.1 Valuation examples of leveraged credit-linked note. The cases (a)-(d) correspondto the models of Section 8.3.1.

(a) (b) (c) (d)

k 5.000 (0.005) 4.674 (0.058) 4.865 (0.036) 1.943 (0.203)sissuer (bp) 0.60 (0.90) 58.60 (10.44) 24.31 (6.52) 550.19 (36.54)sinv (bp) 899.40 (0.90) 841.40 (10.44) 875.69 (6.52) 349.81 (36.54)E(Linv) (e ) 26.51 (1.57) 27.60 (0.73) 29.95 (10.66) 11.93 (0.75)P(Linv) 0.40 (0.024) 0.38 (0.01) 0.44 (0.015) 0.134 (0.008)P(Linv,tot) 0.002 (0.002) 0.067 (0.006) 0.030 (0.006) 0.134 (0.008)E(Einv) (e ) 26.88 (0.76) 27.28 (0.56) 25.20 (0.51) 14.44 (1.58)ES (yrs) 0.88 (0.062) 1.46 (0.09) 0.93 (0.054) 2.41 (0.08)

We computed the fair factor k, the running spread (k− k)s(0, T ) (the premiumof the gap option) and the spread on the note sinv = ks(0, T ) for each of the fourexample models exhibiting different dynamics from Section 8.3.1. Additionally,denoting by Linv the discounted loss to the investor, we computed the expected lossE(Linv), the probability that a loss occurs, the probability of a total loss Linv,tot,the expected discounted earnings from the spread payments (excluding the default-free interest of the coupon payment), E(Einv), and the expected trigger time S

conditional on a trigger event. For each dynamics examples we computed 10 runs of1000 simulations. The values obtained are given in Table 9.1; here, each table entryconsists of the mean value taken over all runs and (in parentheses) the standarddeviation with respect to the 10 runs.

Recall that in Section 3.5.1 we already determined the fair factor k for somemodels via no-arbitrage arguments. Specifically, in the case where the mark-to-market value of a CDS evolves continuously, and when there is no jump-to-defaultrisk, the fair factor is k = k, as there is no gap risk involved. This correspondsto case (a). Now consider the case where the mark-to-market value is constantand the note is exposed to default risk only by a jump-to-default event. Thenk = 1/(1−R) as the investor’s payoff is equivalent to selling protection on 1/(1−R)CDS. This corresponds to case (d). Here, k is slightly larger than 1/(1−R) = 1.67as there is still some, albeit small, volatility that drives the credit quality process,and consequently the underlying CDS’s mark-to-market value is not constant. Alsonote that in this case the probabilities of a loss and of a total loss to the investor areapproximately equal and correspond to the 1-year default probability 1−e−0.03·5 =0.139. Finally, note that in this case, the expected trigger time conditional on atrigger event is roughly half of the note’s maturity. In the other cases the expectedtrigger time is significantly earlier. This can be explained informally as follows.Calibration to the short end of the term structure requires rather high volatilityin the credit quality process; this is a consequence of the fact that in the model


short-term credit spreads vanish, whereas in reality they do not. This leads to highvalues of σ2

0 and θ for short maturities. The resulting high volatility in the creditquality process affects the whole term structure, so that not only short-term, butalso long-term credit spreads exhibit higher volatility in the short term.

The efficiency of the simulation can be increased significantly as follows. Mostof the mark-to-market value computations are used only to check whether the trig-ger level has been hit. The actual mark-to-market value is needed only when thetrigger level has been hit, which is a rare event compared with the number of totalobservations. Now, observe that the mark-to-market value is monotone in both X

and σ2. For each time step, and for a set of five values of the variance process σ2t

we computed the corresponding value of Xt for the mark-to-market value to be atthe trigger level.2 In each simulation, we then first checked against these computedvalued whether the mark-to-market value needs to be computed at all. It turned outthat in more than two thirds of evaluation step the computation of credit spreadsand mark-to-market value could be skipped.

9.3 Default swaptions

We consider the valuation of options on CDS; see [O’Kane (2008), Chapter 9] for adetailed description.

A (default) swaption features the right to enter into a CDS at a later point intime at a spread that is fixed today, the so-called strike spread. The maturity ofthe option is called the expiry date. We distinguish receiver swaptions, giving theoption holder the right to sell protection and receive the strike spread, and payerswaptions, giving the holder the right to buy protection and pay the strike spread.

The underlying of a swaption is a CDS that starts at the option expiry; this is aforward starting CDS . For a protection seller of a forward starting CDS with expiryt and maturity T , the payoff on the protection leg is equal to selling CDS protectionwith maturity T and buying CDS protection with maturity t. In particular, if thereis a default event before option expiry, then the contract cancels at no cost to eithercounterparty.

Denote by A(t, T ) =∫ TtB(t, u)Q(t, u) du the risky present value of a basis point

at t until T . When t = 0, then A(0, T ) =∫ T0B(0, u)Q(0, u) du ist just the risky

present value of a basis point until T . The present value of the position in the twoCDS satisfies

s(0, T )A(0, T )− s(0, t)A(0, t) = (1−R)∫ T

t

B(0, u) P(τ ∈ du),

The forward CDS spread s(0, t, T ) is the spread at time 0 for a CDS contract startingat t and maturing at T (by definition, s(t, t, T ) = s(t, T )). The present value of the

2Actually, to be on the safe side, we computed X for a slightly smaller mark-to-market value.

9.3. Default swaptions 105

premium leg is then

s(0, t, T ) E

(β(0, t)−11τ>t

∫ T

t

β(t, u)−1P(τ > u|Ft) du

)= s(0, t, T )(A(0, T )−A(0, t)).

Solving for the fair forward spread yields,

s(0, t, T ) =s(0, T )A(0, T )− s(0, t)A(0, t)

A(0, T )−A(0, t). (9.3.1)

Turning back to swaptions, the payoff at expiry of a payer (resp. receiver) swaptionwith expiry t and strike spread K on a CDS with maturity T is

(s(t, T )−K)+A(t, T ) (resp. (K − s(t, T ))+A(t, T )). (9.3.2)

Valuation of default swaptions is conveniently done using as numeraire N(s) =1τ>s(A(s, T )−A(s, t)), s ≥ 0.3 Under the corresponding martingale measure Q,called forward survival measure or forward risky annuity measure, the values of thepayer and receiver swaptions at time 0 are

V0,payer = (A(0, T )−A(0, t)) EQ

((s(t, T )−K)+

)V0,receiver = (A(0, T )−A(0, t)) EQ

((K − s(t, T ))+

).

Observe that the spread s(t, T ) is the price of a claim expressed in units of N(t),hence a martingale under the forward survival measure, so that

EQ(s(t, T )) = s(0, t, T ).

A put-call parity4 that relates the value of a forward starting CDS with the pricesof payer and receiver swaptions follows easily. It is given by

V0,payer − V0,receiver = (s(0, t, T )−K) (A(0, T )−A(0, t)).

Finally, assuming a lognormal model for the forward spread under the forwardsurvival measure yields a Black formula for option prices, where

V0,payer = (A(0, T )−A(0, t)) (s(0, t, T ) N(d1)−K N(d2))

V0,receiver = (A(0, T )−A(0, t)) (K N(−d2)− s(0, t, T ) N(−d1)),

with

d1 =ln(s(0, t, T )/K) + 1/2σ2t

σ√t

d2 = d1 − σ√t.

There is a one-to-one relationship between the price and the volatility σ of optionsin the Black model. Instead of quoting prices, it is custom in the market to quotethe implied volatility , which is the volatility σ that yields the option price.3Technically, it is required that the numeraire be strictly positive, which is not the case for N .

However, in the case when N(s) = 0, then the option cancels and its value is 0.4Put ≡ payer, call ≡ receiver


Table 9.2 Valuation examples of default swaptions; payer option prices (standarderror).

K (a) (b) (c) (d)

0 556.81 (16.43) 554.06 (5.55) 553.52 (7.89) 557.20 (1.95)50 459.38 (16.18) 407.55 (5.62) 412.84 (7.87) 405.39 (1.66)100 395.41 (15.66) 290.10 (5.28) 323.85 (7.78) 253.59 (1.42)150 346.91 (14.83) 208.94 (5.08) 270.77 (7.42) 101.79 (1.24)182.73 320.72 (14.43) 171.73 (4.93) 245.56 (7.19) 2.44 (1.18)200 308.23 (14.30) 171.73 (4.88) 234.18 (7.10) 1.24 (1.17)250 276.39 (13.75) 156.03 (4.78) 206.42 (6.92) 1.21 (1.16)300 249.74 (13.00) 122.16 (4.69) 184.36 (6.65) 1.18 (1.14)

As an example, we consider valuation of default swaptions in the example models(a)-(d) of Section 8.3.1. We assume options of different strike spreads with expiryt = 1 (year) to enter into a CDS of 4 year maturity (i.e., T = 5). Suppose givensurvival probabilities Q(0, ti) = e−0.03·ti , ti = i, i = 1, . . . , 5, a recovery rate ofR = 0.4 and a constant interest rate of r = 0.05. We compute CDS spreads by

s(0, ti) =(1−R)

∑ij=1 e−r tj ∆P (0, tj)∑i

j=1 e−rtj Q(0, tj)∆tj,

with ∆P (0, tj) = P (0, tj)− P (0, tj−1) and ∆tj = tj − tj−1. We obtain a flat creditterm structure of 182.73 basis points. By Equation (9.3.1), the forward CDS rate iss(0, t, T ) = 182.73 and the present value of the forward CDS is 554.62 basis points.

For each of the example models (a)-(d), we computed 10 runs of 5000 simulationsto value payer and receiver swaptions of different strikes. We simulated (Xt1 , σ

2t1)

and then computed the term structure of default probabilities and the 4-year CDSspread as described in Section 8.1.3, with CDS spreads computed according to

s(t1, ti) =(1−R)

∑ij=2 e−r (tj−t1) ∆P (t1, tj)∑i

j=2 e−r (tj−t1)Q(t1, tj)∆tj, i = 2, . . . 5,

and the risky present value of a basis point according to

A(t1, ti) =i∑

j=2

e−r (tj−t1)Q(t1, tj)∆tj , i = 2, . . . 5.

The resulting prices for payer swaptions, i.e., the discounted means of the optionpayoff, cf. Equation (9.3.2), are given in Table 9.2, with the corresponding standarddeviations in parentheses. Figure 9.1 shows the prices (left) and implied volatilities(right) of receiver and payer swaptions in the cases (a)-(d) (top to bottom). Notethat the payer option price with strike spread 0 is just the forward CDS price.

The implied volatility for case (a), which corresponds to the model with thecontinuous, deterministic variance process, is very high. On the other hand, for case

9.3. Default swaptions 107

(d), where the credit quality process’ variance is very small unless a very large jumpoccurs, there is approximately no time value in the option prices and volatility iscomparably low. Theoretically, implied volatilities for payer and receiver swaptionsare equal by put-call parity. In our simulation we computed receiver prices byput-call parity from the forward spread s(0, t, T ) and payer prices, hence there is asimulation error present in computing implied volatilities.


0

100

200

300

400

500

600

700

0 50 100 150 200 250 300

pric

e

K

(a) payerreceiver

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 50 100 150 200 250 300

impl

ied

vola

tilit

y

K

(a) payerreceiver

0

100

200

300

400

500

600

700

0 50 100 150 200 250 300

pric

e

K

(b) payerreceiver

0

0.2

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Chapter 10

Summary and conclusion

10.1 Summary

We have developed a first-passage time model for the dynamics of credit spreadsthat incorporates jumps in default probabilities and credit spreads. The objectiveis to price credit derivatives whose price is not given by the market.

Many studies provide evidence that credit spreads exhibit jumps as they evolvethrough time. These jumps are mostly positive and affect the whole maturity spec-trum of CDS. The economic rationale is that jumps are triggered by the arrival ofbad news about a risky entity. Good news on the other hand tend to propagate grad-ually. The payoff of some credit derivatives, such as leveraged credit-linked notes,CPDO’s and CPPI structures, is particularly sensitive to jumps in credit spreads.We treat the leveraged credit-linked note in detail and illustrate the sensitivity ofits payoff to jumps.

The principal modelling idea is to extend the Overbeck-Schmidt model. In theOS-model, a credit quality process X = (Xt)t≥0 is modelled as a time-changedBrownian motion X = BΛ, where the time-change Λ = (Λt)t≥0 is deterministic,continuous and strictly increasing. The default time is τ = inft ≥ 0 : Xt ≤ b,with b < X0 a constant barrier. Analytic calibration to a given term structure ofdefault probabilities is straightforward. However, the dynamics of the OS-modelare limited, as they are fully determined by calibration to a given term structure.Moreover, it is not possible to include jumps in credit spreads at random pointsin time. We determine the dynamics of default probabilities and the term hazardrate (as a proxy for credit spreads) using the Ito formula, and we compute thedistributions P(P (t, T ) ≤ x|τ > t), x ≥ 0, where P (t, T ) is the probability ofdefault until time T at time t, and P(λ(t, T ) ≤ x|τ > t), x ≥ 0, with λ(t, T ) thecorresponding term hazard rate. Examples indicate that the behaviour of defaultprobabilities and the term hazard rate is very volatile. This is due to the continuousnature of the model: the credit quality process must move in a volatile manner inorder to match the calibration target.

The extension of the OS-model is as follows: we model a credit quality processas an Ito integral with respect to Brownian motion, Xt =

∫ t0σu dWu, with the

109

110 Summary and conclusion

integrand σ an adapted, cadlag, strictly positive process independent of W . Asbefore, the time of default is τ = inft ≥ 0 : Xt ≤ b, b ≤ X0.

By the Theorem of Dambis, Dubins-Schwarz, the credit quality process, being acontinuous local martingale, can be represented as a time-changed Brownian motion,BΛ, where Λ =

∫ ·0σ2u du. The time-change is now stochastic, continuous and strictly

increasing. The processes B and Λ are independent (this is proved in Chapter 2 onOcone martingales), which is the key for deriving a formula for conditional defaultprobabilities,

P (t, T ) = 2E(

N(


) ∣∣∣Ft) , T > t, on τ > t.

The continuous nature of the time-change allows to draw on the hitting-time dis-tribution of Brownian motion.

Our standard example throughout is a variance process σ2 that is a Levy-drivenOrnstein-Uhlenbeck process, i.e., the solution of

dσ2t = a(θ(t)− σ2

t−) dt+ ∆Zt,

with Z a compound Poisson process (independent of W ) and θ a deterministicbounded function.

We analyse the properties of default probabilities and credit spreads, showingthat jumps in the variance process translate into jumps in default probabilitiesand credit spreads. The short-term credit spread vanishes, as it does in the OS-model and many other credit models of structural type, since the default event isa predictable stopping time. This contradicts empirical observation: credit spreadsare strictly positive for any maturity, which is attributed to the possibility of ajump-to-default at any time. We show later how the model can approximate thisbehaviour by including the possibility of large jumps in the variance process. Us-ing the Ito formula, we determine the dynamics of default probabilities and creditspreads. As for the OS-model, we compute the distributions P(P (t, T ) ≤ x|τ > t)and P(λ(t, T ) ≤ x|τ > t). Considering the same examples as in the analysis of theOS-model, we see that default probabilities and the term hazard rate are much lessvolatile in the extended model.

The model is implemented as follows: For each time point t where a term struc-ture is needed, simulate (Xt, σ

2t ). Then, the term structures of default probabilities

(P (t, T )T>t) and credit spreads (s(t, T )T>t) can be computed efficiently using atechnique called Panjer recursion. Many computations need only be carried outonce when the model is initialised. An advantage of the implementation is thatsimulation need only be done for time points where a term structure is required,instead of required credit spread maturities.

Calibration is a two-step process: after assigning parameters for the dynam-ics, such as jump size distribution and jump intensity of the driving compoundPoisson process, the model is calibrated to a given term structure by assigning σ2

0

and the deterministic function θ. Contrary to the OS-model, analytic calibration

10.2. Model requirements revisited 111

is not possible. Instead calibration is done numerically by minimising the rootmean square error between market-given and model-implied default probabilities.A solution to the calibration problem does not exist for arbitrary dynamics, butdynamics can always be chosen such that calibration with arbitrarily small error ispossible. We consider examples of different parameters, and show how the shapeof the term structure is affected by the individual parameters. In particular, themodel can generate normal and hump-shaped term structures. Since short-termcredit spreads vanish, the model cannot produce inverted term structures. How-ever, if the jump size distribution of the variance process includes very large jumps,then such an event leads to default in a very short time; hence, the model includes“near-jump-to-default” events. This is also manifested in sharply humped termstructures.

We study four examples of different dynamics parameters, and we compare thedynamics of default probabilities and term hazard rates. The first example is amodel without jumps, which corresponds to the OS-model. Next, we consider twomodels whose variance process is driven by rather moderate jumps. Here, thedynamics of default probabilities and the term hazard rate are less volatile than inthe first example. The last example features only very large jumps (jump size 25000,barrier −2, θ = 0, σ2 ≈ 0). The dynamics are such that the credit quality processexhibits virtually no movement until a jump occurs, which leads to default veryquickly. This is an approximation of a simple reduced-form model with a constanthazard rate.

We discuss uniqueness and stability of calibration parameters. To assess thestability of (σ2

0 , θ), we calibrate the model, then “bump” the credit spread termstructure by 1 basis point, and compare with the parameters obtained after re-calibration.

In order to apply the formula for conditional default probabilities for pricingrisky financial claims, we show that (Xt, σ

2t ) is measurable with respect to the

filtration generated by default probability term structures, σ((P (s, T )T>t), 0 ≤ s ≤t) (the latter corresponding to information generated by observable and tradedassets). We provide valuation examples for leveraged credit-linked notes and defaultswaptions.

10.2 Model requirements revisited

In Section 4.1, we stated a list of twelve criteria considered important in modelling.Let us review the model capabilities with regard to this list.

(i) Model fit: The model fits the term structure of CDS spreads under someconstraints on the dynamics parameters; the calibration error can be madearbitrarily small by choosing appropriate dynamics parameters.

(ii) Model consistency: Conditional default probability distributions are always


strictly increasing in the model, hence term structures obtained in the modelare arbitrage-free. The probability of a gap event is strictly positive in themodel, which is consistent with real-world observation. The issue of time-inconsistency requires further analysis of the need to re-calibrate the modelwhen market prices change.

(iii) Model produces valid quantities: Spreads are nonnegative; default probabilitiesare in [0, 1].

(iv) Model captures spread risk: Yes.(v) Model produces desired dynamics: Due to the lack of liquid market instru-

ments, risk-neutral dynamics were not calibrated. There are some constraintson the choice of dynamics parameters, in order to produce a well-calibratedmodel, but a set of four examples shows that valid dynamics can vary widely.

(vi) Model captures any default timing: The default time in the model is pre-dictable; sudden, totally unexpected default is not possible. On the otherhand, jumps in the variance process are totally inaccessible – a large jump inthe variance, which occurs totally unexpected, leads to default in a very shorttime. Heuristically, the model may thus approximate unexpected default atany time, even for short maturities.

(vii) Model captures joint risks: We have assumed a constant interest rate through-out, but valuation examples may be extended accordingly.

(viii) Model captures uncertain recovery payment: We followed the frequent assump-tion of a constant recovery rate.

(ix) Model allows fast, stable, accurate pricing: Pricing is done by simulating thestate of the credit quality process and computing term structures of conditionaldefault probabilities and credit spreads numerically using Panjer recursion.The procedure is fast, especially as the Panjer recursion needs only be carriedout once, even when term structures at different points in time or conditionalon different states of the credit quality process are involved.

(x) Calibration parameters are stable: We give examples, where a credit spreadterm structure is “bumped” by one basis point, and the calibration parametersare compared before and after re-calibration. For a precise answer, a criterionfor what constitutes stable parameters is needed.

(xi) Hedging strategies are stable: Not analysed.(xii) The model is as simple as possible: This depends of course on what constitutes

a simple model. The model may not be considered simple as a hidden quan-tity – the credit quality process – is modelled, instead of directly modellingcredit spreads. On the other hand, the model focuses on the stylized facts oncredit spreads, and is computationally tractable. For a further analysis of thiscriterion, a systematic comparison with existing models is helpful.

10.3. Outlook 113

10.3 Outlook

Many topics that are interesting in their own right have not been covered:

• Building a hedging strategy for a product is often as important as pricing.Mathematically, a hedging strategy requires a martingale representation the-orem. If such a theorem is not available, for example due to market incom-pleteness, other methods of hedging, such as minimum variance hedging, maybe considered. Building hedging strategies also involves examining issues oftime-consistency in the model (see Section 4.1).

• A systematic comparison with existing models for valuing spread risky prod-ucts may be insightful, for example by employing the list of model requirementsof Section 4.1.

• A multivariate extension of the model can serve to value multi-name creditderivatives that depend on the joint dynamics of a portfolio of credit riskyunderlyings, such as leveraged first-to-default credit baskets. There are dif-ferent ways to specify the dependence between credit quality processes, eitherby correlating the driving Brownian motions or by correlating the varianceprocesses. [Kammer (2007)] studies multivariate extensions of the OS-modelwith stochastic time-changes driven by Brownian motions.

• Pricing under a coarser filtration may be worth investigating. The rationaleis that investors typically do not have full information about a company’scredit quality. In such a model, the default time may be a totally inaccessiblestopping time, avoiding that credit spreads vanish as time-to-maturity decays.

• Other variance processes may be considered, such as a variance driven by anadditive process instead of a Levy process (a stochastic process is an additiveprocess if it has independent increments and if it is stochastically continuous).Instead of calibrating a given term structure of default probabilities by choosingthe deterministic function θ, the Levy measure of the additive process, whichnow depends on time, may be used for calibration. In this way, the dynamicsof credit spreads for different maturities may be captured more appropriately.

• In the valuation examples of the LCLN, the expected trigger time lies farbelow half of the note’s maturity. This is an indication that the model doesnot separate the dynamics of short-term credit spreads and mid- to longer-term credit spreads appropriately. In an extension, one could try to set up amodel where short-term and long-term credit spreads although correlated, aremodelled separately.

• One could also start with the formula for conditional default probabilities, i.e.,define conditional default probabilities, on τ > t, to be

P (t, T ) = 2E(

N(


) ∣∣∣Ft) , T > t,

and investigate more general models of the time-change Λ.

PART 2

Latin hypercube sampling

with dependence

115

Chapter 11

Latin hypercube sampling with dependence

11.1 Introduction

Consider the problem of reducing the variance of a Monte Carlo estimator targetedat a vector of dependent random variables. Many existing variance reduction tech-niques are powerful, but exploit particular properties of the problem at hand; see[Glasserman (2004), Section 4.7] for a comparison of variance reduction techniquestaking into account their complexity and effectiveness. The method proposed here,Latin hypercube sampling with dependence (LHSD), is generally applicable, is par-ticularly simple, and achieves an effective variance reduction for many estimationproblems, including problems with rare events and high-dimensional problems. It isoften effective even for low sample sizes, and it may easily be combined with othervariance reduction techniques.

LHSD is a generalisation of a multivariate variance reduction technique knownas Latin hypercube sampling (LHS), introduced by [McKay et al. (1979)] and fur-ther studied by [Stein (1987)] and [Owen (1992)], amongst others. LHS relies onindependence of the components of the random vector involved. Essentially, LHSDextends LHS to random vectors with dependent components. The method is men-tioned by [Stein (1987)], but, to the best of our knowledge, it has not been analysedin detail and no results about its effectiveness have been derived yet.

On a probability space (Ω,F ,P), let (U1, . . . , Ud) be a random vector with uni-form marginals and with copula1 C. Suppose the goal is to estimate Eg(U1, . . . , Ud)with g : [0, 1]d → R Borel-measurable and C-integrable.

The usual Monte Carlo estimator based on n independent samples (U1i , . . . , U

di ),

i = 1, . . . , n, is 1/n∑ni=1 g(U1

i , . . . , Udi ). It is a strongly consistent esti-

mator, i.e., 1/n∑ni=1 g(U1

i , . . . , Udi ) P–a.s.−→ Eg(U1, . . . Ud) as n → ∞. The

Central Limit Theorem for sums of independent random variables statesthat the scaled estimator converges in distribution to a Normal distribu-tion, i.e., 1/

√n∑ni=1

[g(U1

i , . . . , Udi )− Eg(U1, . . . , Ud)

] L→ N(0, σ2), with σ2 =

1A copula C is the distribution function of a random vector with uniform marginals, see e.g. [Joe(1997)] and [Nelsen (1999)]. We also associate with C the measure induced by the copula C.

117

118 Latin hypercube sampling with dependence

Var(g(U1, . . . , Ud)). The Central Limit Theorem serves as an indicator of the speedof convergence via the approximation 1/n

∑ni=1

[g(U1

i , . . . , Udi )− Eg(U1, . . . Ud)

]≈

X, for some X ∼ N(0, σ2/n), from which we may derive confidence intervals andother statistics. In general, the variance of an estimator is a key figure for assessingthe quality of an estimation.

LHSD transforms n independent samples (U1i , . . . , U

di ), i = 1, . . . , n, in such a

way that for each dimension j, the marginals U ji , i = 1, . . . , n, are uniformly spreadover [0, 1]. At the same time, the transformation aims to preserve the copula. Weshow that the LHSD estimator of Eg(U1, . . . , Ud) is strongly consistent for boundedand continuous g, and consistent for bounded and C-a.e. continuous g. In thebivariate case, under some moderate conditions on the copula C of the underlyingrandom vector, we derive a Central Limit Theorem, which states that the LHSDestimator converges to a Normal distribution. The Central Limit Theorem is derivedby applying a result from [Fermanian et al. (2004)]. We show that, under somemonotonicity conditions on g, the limit variance of the LHSD estimator is nevergreater than the respective Monte Carlo limit variance.

Monte Carlo simulation is widely used for the valuation of financial claims.The general approach to value a financial claim is to generate sample paths of theunderlying financial securities. The discounted expectation of the claim’s payoffunder a risk-neutral measure is then an estimator of the claim’s fair value. Fora comprehensive overview of Monte Carlo simulation in financial applications, werefer to [Glasserman (2004)].

We consider two examples of financial claims that depend on the joint distri-bution of several underlying assets. A first-to-default credit basket is valued basedon random numbers and Sobol sequences, both with and without LHSD. The vari-ance (resp. mean square error) of the LHSD estimators is between 2.25 and 4 timessmaller compared to the corresponding estimators without LHSD. An interestingfeature of the LHSD estimator is that, even though defaults are rare events, it guar-antees that a fixed number of default events are sampled. The second example isconcerned with the valuation of an Asian basket option, which may be formulatedas a high-dimensional estimation problem (dimension 2500 in the example). Thevariance reduction achieved depends on the strike of the option and lies betweenfactors of 6 and 200.

11.2 Preliminaries

11.2.1 Stratified sampling

Stratified sampling is a variance reduction technique in a univariate setting thatconstrains the fraction of samples drawn from specific subsets, so-called strata. Fora detailed exposition we refer to [Glasserman (2004), Chapter 4.3].

Suppose the goal is to estimate Eg(U) with U ∼ U(0, 1) (i.e., a uniform ran-

11.2. Preliminaries 119

dom variable on [0, 1]), and with g : [0, 1] → R a Borel-measurable and integrablefunction. Let A1, . . . , An be a partition of [0, 1]. Then,

Eg(U) =n∑i=1

E(g(U)|U ∈ Ai)P(U ∈ Ai),

and a corresponding estimator of Eg(U) is derived from sampling U conditionalon U ∈ Ai, i = 1, . . . , n. In the simplest case, the strata are chosen to be theequiprobable intervals Ai = ((i− 1)/n, i/n], i = 1, . . . , n, and one sample is drawnfrom each stratum. This is achieved for example by drawing independent U(0, 1)samples, U1, . . . , Un, and setting

Vi :=i− 1n

+Uin, i = 1, . . . , n. (11.2.1)

The resulting estimator of Eg(U), given by 1/n∑ni=1 g(Vi), is consistent, and by a

Central Limit Theorem for the stratified estimator it follows that the limit varianceis smaller than the Monte Carlo variance, cf. [Glasserman (2004), Section 4.3.1].

11.2.2 Latin hypercube sampling

Simply extending stratified sampling to d-dimensional random vectors by strati-fying each dimension with n samples is unfeasible even for moderately small di-mensions, since to have one sample in each stratum requires at least nd samples.Latin hypercube sampling (LHS) efficiently extends stratified sampling to randomvectors (U1, . . . , Ud) whose components are independent (i.e., they are linked bythe independence copula). It was introduced in [McKay et al. (1979)] and furtherdeveloped by [Stein (1987)] and [Owen (1992)]. For an in-depth treatment of LHSsee [Glasserman (2004), Section 4.4].

Assume that the goal is to estimate Eg(U1, . . . , Ud) with g : [0, 1]d → R Borel-measurable and integrable. Fixing a sample size n, generate n independent samples(U1

i , . . . , Udi ), i = 1, . . . , n, and generate d independent permutations π1, . . . , πd of

1, . . . , n drawn from the distribution that makes all permutations equally prob-able. Denoting by πji the value to which i is mapped by the j-th permutation, aLatin hypercube sample is given by

V ji :=πji − 1n

+U jin, j = 1, . . . , d, i = 1, . . . , n.

An example of a Latin hypercube sample is shown in Figure 11.1. Observe thatin each dimension j, (V j1 , . . . , V

jn ) is a stratified sample. Furthermore, each point

(V 1i , . . . , V

di ), is uniformly distributed on [0, 1]d, 1 ≤ i ≤ n. The LHS estimator

1/n∑ni=1 g(V 1

i , . . . , Vdi ) is consistent. [Stein (1987)] shows that, for functions g with

finite second moment, the variance of the LHS estimator is smaller compared to thestandard Monte Carlo estimator as long as the number of samples is sufficientlylarge. For bounded g, [Owen (1992)] derives a Central Limit Theorem for the LHSestimator.


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Fig. 11.1 Example of Latin hypercube sampling. Left: Original sample (U11 , U2

1 ), . . . , (U110, U2

10),with (U1

1 , U21 ) marked by a circle. Right: Corresponding Latin hypercube sample, with

(V 11 , V 2

1 ) marked by a circle. The permutations are π1 = 5, 9, 7, 8, 1, 10, 4, 2, 3, 6 and π2 =1, 7, 9, 6, 3, 2, 5, 10, 4, 8.

Requiring independence of the components of the random vector is fundamental:Applying LHS to a sample of a random vector whose components are dependentdestroys the dependence by application of random and independent permutations ineach dimension. Conversely, applying first LHS to a sample of a random vector withindependent components, and then applying a transform to introduce dependencebreaks, in general, the stratification of the marginals, thereby losing much of theappeal of LHS.

11.3 LHSD method and LHSD estimator

We now describe an extension of LHS for random vectors with dependence. Thegeneral idea is to generate a Latin hypercube sample, albeit with the following mod-ification: Instead of choosing a random permutation in each dimension, a particularpermutation that depends on the samples of that dimension is chosen. For this weneed the notion of a rank statistic.

Definition 11.1 (Rank statistic). Let X1, . . . , Xn be i.i.d. random variableswith continuous distribution function. Reorder them such that X(1) < . . . < X(n)

P–a.s.. The index of Xi within X(1), . . . , X(n) is the i-th rank statistic, given by

ri,n(X1, . . . , Xn) :=n∑k=1

1Xk≤Xi. (11.3.1)

11.3. LHSD method and LHSD estimator 121

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1 , U21 ), . . . , (U1

10, U210) linked with a Gaussian copula with correlation ρ = 1/2; (U1

1 , U21 ) is

marked by a circle. Right: Corresponding LHSD sample, with (V 11,10, V 2

1,10) marked by a cir-

cle. The rank statistics are r1 = 8, 6, 1, 4, 3, 7, 5, 2, 9, 10 and r2 = 7, 9, 6, 4, 3, 2, 5, 1, 8, 10, andηj

i,10 := 1/2, j = 1, 2, i = 1, . . . , 10.

That such an ordering exists P–a.s. follows from the continuity of the distributionfunction. For ease of notation, we write just ri,n instead of ri,n(X1, . . . , Xn).

Consider a random vector (U1, . . . , Ud), U j ∼ U(0, 1), j = 1, . . . , d, whosecomponents are linked by an arbitrary copula C, and let (U1

i , . . . , Udi ), i = 1, . . . , n,

be n independent samples of (U1, . . . , Ud). For 1 ≤ i ≤ n and 1 ≤ j ≤ d denoteby rji,n the i-th rank statistic of (U j1 , . . . , U

jn). A Latin hypercube sample with

dependence is given by

V ji,n :=rji,n − 1

n+ηji,nn, i = 1, . . . , n, j = 1, . . . , d, (11.3.2)

where ηji,n are random variables taking values in [0, 1], which we specify below.Figure 11.2 shows an example with 10 samples drawn from a bivariate Gaussiancopula with correlation 1/2 and the corresponding LHSD samples.

Just as in regular LHS, (V j1 , . . . , Vjn ) is a stratified sample in each dimension

j. Recall that each sample from the stratified sample of Equation (11.2.1) is uni-formly distributed within its stratum. If ηji,n := U ji this property is lost by appli-cation of the rank statistic: in each dimension, the smallest sample is allocated tothe first stratum, the second smallest to the second stratum, and so on. Condi-tional on rji,n = k, U ji follows a beta distribution with parameters k and n, i.e.,P(U ji ≤ x|rji,n = k) = Bnk (x), which is the distribution of the k-th order statisticof n independent uniform random variables, see e.g. [Feller (1971), Ch. I.7]. Thefollowing choices produce a LHSD sample with uniform marginals:


(i) ηji,n := Bnrj

i,n

(U ji ), i = 1, . . . , n, j = 1, . . . , d,

(ii) (ηji,n)i=1,...,n;j=1,...,d is a sample of independent U(0, 1) random variables in-dependent of (U ji )i=1,...,n;j=1,...d.

If the primary goal is to capture the joint distribution, the following choices arecomputationally more efficient:

(iii) ηji,n := 1/2, which places each sample in the middle of its stratum,(iv) ηji,n := 1, in which case V ji,n is just the empirical distribution function of

(U j1 , . . . , Ujn) at U ji ,

i = 1, . . . , n, j = 1, . . . , d.

Remark 11.2. LHS is a special case of LHSD: Let (U1, . . . , Ud) be inde-pendent, and let (ηji )i=1,...n;j=1,...d be chosen according to choice (ii). Then(U ji )i=1,...,n;j=1,...,d determine independent and equiprobable permutations that al-locate samples to strata, and (ηji )i=1,...n;j=1,...d determine independently the posi-tion, uniformly distributed, of each sample in its stratum.

Assume that the quantity to estimate is Eg(U1, . . . , Ud) with g : [0, 1]d → RBorel-measurable and integrable and (U1, . . . , Ud) a random vector with uniformmarginals and copula C. The LHSD estimator is given by

1n

n∑k=1

g(V 1i,n, . . . , V

di,n), (11.3.3)

with V ji,n, i = 1, . . . , n, j = 1, . . . , d, obtained from the transformation of Equation(11.3.2).

Before we analyse the estimator formally, let us reflect why it would reduce thevariance: Variance reduction over the usual Monte Carlo estimator is achieved bydrawing “favourable” samples and avoiding “unfavourable” samples (i.e., sampleswith a large contribution to the variance of the estimator). For each dimension1 ≤ j ≤ d, LHSD ensures that the samples V j1,n, . . . , V

jn,n are uniformly spread over

the unit interval, thereby deleting inter-stratum variance and leaving only intra-stratum variance. As a consequence however, in general, the original dependencestructure of the samples is broken, i.e., for fixed n, the copula of (V 1

i,n, . . . , Vdi,n),

i = 1, . . . , n, differs from the copula of (U1, . . . , Ud). On the other hand, as n→∞,each sample V ji,n converges to U ji , since the fraction of samples V jk,n, k = 1, . . . , n,such that V jk,n ≤ V ji,n, tends to U ji . This notion is captured by the rank statistic.We shall see below in Lemma 11.5 that the empirical distribution function of theLHSD samples tends to the original copula C. Summarising, an LHSD sample hasmarginals that are uniformly spread over the unit interval and, provided n is largeenough, we can expect the error between the original copula and the copula of theLHSD samples to be small.

11.4. Consistency of the LHSD estimator 123

11.4 Consistency of the LHSD estimator

We establish consistency of the LHSD estimator, provided g is bounded and C-a.e.continuous.

Observe that the usual laws of large numbers for sums of independent randomvariables do not apply, for the following reasons:

• In each dimension, by application of the rank statistic, the samples fail to beindependent.

• For any i, j, V ji,n 6= V ji,n+1, hence, when progressing from n to n + 1, we arenot just adding an (n+ 1)-th term to the existing sum (11.3.3), but all termsof the sum change.

Proposition 11.3. Let g : [0, 1]d → R be bounded and C-a.e. continuous. Thenthe LHSD estimator (11.3.3) is strongly consistent, i.e.,

1n

n∑i=1

g(V 1i,n, . . . , V

di,n) P–a.s.−→ Eg(U1, . . . , Ud), as n→∞.2

It follows immediately by Dominated Convergence that the estimator is asymptot-ically unbiased:

Corollary 11.4. Let g : [0, 1]d → R be bounded and continuous C-a.e.. Then theLHSD estimator (11.3.3) is asymptotically unbiased, i.e.,

E

(1n

n∑i=1

g(V 1i,n, . . . , V

di,n)

)−→ Eg(U1, . . . , Ud), as n→∞.

We require some preliminary results for the proof of Proposition 11.3.

Lemma 11.5. For 0 ≤ u1, . . . , ud ≤ 1, define Cn : [0, 1]d → [0, 1] by

Cn(u1, . . . , ud) :=1n

n∑k=1

1V 1k,n≤u1,...,V d

k,n≤ud.

Then Cn is a distribution function and

sup(u1,...,ud)∈[0,1]d

∣∣Cn(u1, . . . , ud)− C(u1, . . . , ud)∣∣ P–a.s.−→ 0, as n→∞.

Proof. It is straightforward to verify that Cn is a distribution function on [0, 1]d,n ∈ N. For the second statement, let

F jn(u) =1n

n∑k=1

1Ujk≤u

, u ∈ [0, 1],

2A previous version of the Proposition required g to be continuous. For C-a.e. continuous g,it was shown that the estimator is consistent by a different technique. That the Propositionmay be extended to C-a.e. continuous functions was noted by Stefanie Muller from Universityof Kaiserslautern and members of the Financial Mathematics group at Fraunhofer Institut furTechno-und Wirtschaftsmathematik in Kaiserslautern.


be the empirical distribution function based on U j1 , . . . , Ujn, j = 1, . . . , n. Define

Cn : [0, 1]d → [0, 1] as

Cn(u1, . . . , ud) :=1n

n∑k=1

1F 1n(U1

k)≤u1,...,Fdn(Ud

k )≤ud. (11.4.1)

It is a consequence of [Deheuvels (1979), Theoreme 3.1] (or [Deheuvels (1981),Lemmas 6 and 7]) that

sup(u1,...,ud)∈[0,1]d

∣∣∣Cn(u1, . . . , ud)− C(u1, . . . , ud)∣∣∣ P–a.s.−→ 0, as n→∞.

Using the fact that F jn(U jk) = rjk,n/n, the claim follows from

|Cn(u1, . . . , ud)− Cn(u1, . . . , ud)|

≤ 1n

n∑k=1

1(u1∈

»r1

k,n−1

n ,r1

k,nn

«,...,ud∈

"rd

k,n−1

n ,rd

k,nn

!) ≤ 1n,

for any (u1, . . . , ud) ∈ [0, 1]d.

Proof of Proposition 11.3. Assume first that g is continuous and observe that∫[0,1]d

g dCn =1n

n∑k=1

g(V 1k,n, . . . , V

dk,n),

which is just the LHSD estimator. It follows from Lemma 11.5 that Cn convergesweakly to C for P-almost all ω ∈ Ω, which is equivalent to∫

[0,1]dg dCn −→

∫[0,1]d

g dC = Eg, for P− a.a. ω, (11.4.2)

for every bounded, continuous function g : [0, 1]d → R. That Equation (11.4.2)holds for functions g that are C-a.e. continuous is a consequence of [Billingsley(1968), Theorem 5.2].

Remark 11.6. The boundedness condition on g ensures existence of the expecta-tions E|g(V 1

i,n, . . . , Vdi,n)−g(U1

i , . . . , Udi )|, i = 1, . . . , n, n ∈ N. Uniform integrability

of the family (V 1i,n, . . . , V

di,n), i = 1, . . . , n, n ∈ N, would be sufficient for establishing

the claim. However, we have no means of establishing uniform integrability otherthan requiring boundedness, as in general the distribution of (V 1

i,n, . . . , Vdi,n) is not

known. On the other hand, boundedness is an acceptable limitation when doingMonte Carlo simulation.

11.5 Central Limit Theorem for LHSD and variance reduction

It is natural to investigate the speed of convergence of the LHSD estimator andcompare this to the rate of convergence of the standard Monte Carlo estimator.

11.5. Central Limit Theorem for LHSD and variance reduction 125

Assuming the bivariate case and posing some conditions on the copula, we statea Central Limit Theorem for the LHSD estimator and we establish that the limitdistribution is Normal. We derive a closed-form expression for the LHSD estimator’slimit variance, and we compare it to the corresponding Monte Carlo limit variance.Finally, we show that if the copula fulfills a certain positive dependence propertyand if the function to be estimated is nondecreasing in each argument, then theLHSD limit variance is always less or equal to the corresponding MC limit variance.

The empirical distribution function of the LHSD samples bears close resem-blance to the empirical copula of the original sample, and it turns out that theLHSD estimator is a special case of some multivariate rank-order statistics. Forthe study of empirical processes and empirical copulas, see e.g. [Deheuvels (1979)],[Deheuvels (1981)], [Gaenssler and Stute (1987)] and [Vaart and Wellner (1996)],[Fermanian et al. (2004)]. For results on multivariate rank-order statistics we re-fer to [Ruymgaart et al. (1972)], [Ruschendorf (1976)], [Genest et al. (1995)] and[Fermanian et al. (2004)]. The Central Limit Theorem stated below is derived fromTheorem 6 of [Fermanian et al. (2004)]. Although the following analysis is restrictedto the bivariate case, we presume that it can be extended to the multivariate case.

Definition 11.7. A function g : [0, 1]2 → R is of bounded variation (in the senseof Hardy-Krause), if there exists a constant K such that

(i) for every bounded rectangle [a, b] × [c, d] ⊆ [0, 1]2, for all m,n and pointsa = x0 < x1 < · · ·xm = b, c = y0 < y1 < · · · < yn = d,

m−1∑i=0

n−1∑j=0

|g(xi, yj) + g(xi+1, yj+1)− g(xi, yj+1)− g(xi+1, yj)| ≤ K,

(ii) for u ∈ [0, 1], v 7→ g(u, v) is a function whose variation is bounded by K,(iii) for v ∈ [0, 1], u 7→ g(u, v) is a function whose variation is bounded by K.

Note that there are different definitions of bounded variation in the bivariate case,see [Clarkson and Adams (1933)]. We use the term “bounded variation” as a syn-onym of “bounded variation in the sense of Hardy-Krause”. For illustration welist some properties of bounded variation functions. It is a consequence of [Hobson(1921), §308] that if g : [0, 1]2 → R is of bounded variation, then limn→∞ g(u1

n, u2n)

exists for any sequence (u1n, u

2n)n≥1, with (ujn)n≥1 monotone, j = 1, 2. By [Adams

and Clarkson (1934), Corollary to Theorem 13], the discontinuities of a function ofbounded variation are located on a denumerable number of parallels to the axes.Finally, note that a function of bounded variation is bounded [Clarkson and Adams(1933), p. 827].

Definition 11.8. A function g : [0, 1]2 → R is right-continuous if for any sequence(u1n, u

2n)n≥1, with ujn ↓ uj, j = 1, 2, limn→∞ g(u1

n, u2n) = g(u1, u2).

See [Kallenberg (2001), Theorem 4.28] or [Jacod and Protter (2003), Theorem18.8] for the following Lemma:


Lemma 11.9. Let (Xn)n≥1 and (Yn)n≥1 be sequences of R-valued random variables,

with XnL→ X and |Xn − Yn|

P→ 0. Then YnL→ X.

In the following, all integrals are Lebesgue-Stieltjes integrals and integrals areover (0, 1] if not stated otherwise. Throughout U, V are U(0, 1)-distributed randomvariables.

Theorem 11.10 (Central Limit Theorem for LHSD). Let the copula C of(U, V ) have continuous partial derivatives and let g : [0, 1]2 → R be of boundedvariation and right-continuous. Then

1√n

n∑i=1

(g(V 1

i,n, V2i,n)− Eg(U1, U2)

) L−→ N(0, σ2LHSD),

where, setting ∂1C(u, v) = ∂C(u, v)/∂u and ∂2C(u, v) = ∂C(u, v)/∂v,

σ2LHSD =

ZZZZC(u ∧ u′, v ∧ v′) dg(u, v) dg(u′, v′)−

„ZZC(u, v) dg(u, v)

«2

+

ZZZZ n∂1C(u′, v′)(C(u, v)u′ − C(u ∧ u′, v)) + ∂1C(u, v)(C(u′, v′)u− C(u ∧ u′, v′))

+ ∂2C(u′, v′)(C(u, v)v′ − C(u, v ∧ v′)) + ∂2C(u, v)(C(u′, v′)v − C(u′, v ∧ v′))

+ ∂1C(u, v) ∂1C(u′, v′)(u ∧ u′ − uu′) + ∂2C(u, v) ∂2C(u′, v′)(v ∧ v′ − vv′)

+ ∂1C(u, v) ∂2C(u′, v′)(C(u, v′)− uv′) + ∂1C(u′, v′) ∂2C(u, v)(C(u′, v)− u′v)o

dg(u, v) dg(u′, v′).

(11.5.1)

Proof. Theorem 6 of [Fermanian et al. (2004)] states that, under the above condi-tions on g and C,

1√n

n∑i=1

(g(F 1

n(U1i ), F 2

n(U2i ))− Eg(U1, U2)

) L−→∫

[0,1]2GC(u, v) dg(u, v),

where F jn is the empirical distribution function based on the sample U j1 , . . . , Ujn,

j = 1, 2, and

GC(u, v) = BC(u, v)− ∂1C(u, v)B

C(u, 1)− ∂2C(u, v)BC(1, v) ,

with BC a Brownian bridge on [0, 1]2, i.e., a Gaussian family (B

C(u, v))(u,v)∈[0,1]2 ,with mean zero and covariance function

E(BC(u, v)B

C(u′, v′)) = C(u ∧ u′, v ∧ v′)− C(u, v)C(u′, v′), 0 ≤ u, u′, v, v′ ≤ 1.

In particular, the limit distribution is Gaussian.


Recall that V ji,n = (rji,n− 1 + ηji,n)/n and F jn(U ji ) = rji,n/n, j = 1, 2. Fix n, and,for notational convenience, set vji := V ji,n and uji := F jn(U ji ), j = 1, 2. Assume thatthe variation of g is bounded by K. Then,∣∣∣∣∣

n∑i=1

[g(V 1

i,n, V2i,n)− g(F 1

n(U1i ), F 2

n(U2i ))]∣∣∣∣∣ =

∣∣∣∣∣n∑i=1

[g(v1

i , v2i )− g(u1

i , u2i )]∣∣∣∣∣

=

∣∣∣∣∣n∑i=1

[g(v1

i , v2i ) + g(u1

i , u2i )− g(v1

i , u2i )− g(u1

i , v2i )− 2g(u1

i , u2i )

+ g(v1i , u

2i ) + g(u1

i , v2i )− g(v1

i , 0) + g(u1i , 0) + g(0, u2

i )− g(0, v2i )

+ g(v1i , 0)− g(u1

i , 0)− g(0, u2i ) + g(0, v2

i )]∣∣∣∣∣

≤n∑i=1

∣∣g(v1i , v

2i ) + g(u1

i , u2i )− g(v1

i , u2i )− g(u1

i , v2i )∣∣

+n∑i=1

∣∣g(v1i , u

2i ) + g(u1

i , 0)− g(u1i , u

2i )− g(v1

i , 0)∣∣

+n∑i=1

∣∣g(u1i , v

2i ) + g(0, u2

i )− g(u1i , u

2i )− g(0, v2

i )∣∣

+n∑i=1

∣∣g(v1i , 0)− g(u1

i , 0)∣∣+

n∑i=1

∣∣g(0, v2i )− g(0, u2

i )∣∣ ≤ 4K,

since each sum consists of terms that refer to non-overlapping intervals. Hence,

1√n

∣∣∣∣∣n∑i=1

[g(V 1

i,n, V2i,n)− g(F 1

n(U1i ), F 2

n(U2i ))]∣∣∣∣∣ −→ 0, as n→∞,

and the first statement follows by Lemma 11.9.The expression for σ2

LHSD is obtained by taking the second moment of the limitdistribution, E

(∫∫GC(u, v) dg(u, v)

)2, and applying Fubini’s Theorem, which isjustified as follows: g as a function of bounded variation is the difference of twoquasi-monotone functions (see e.g. [Adams and Clarkson (1934), Theorem 5]) andmay be written as the difference of two integrals with respect to positive measures.Since g is bounded, the conditions for Fubini’s Theorem are satisfied by observingthat E|XY | <∞ for two jointly Normal random variables X and Y .

We now examine the relationship between σ2LHSD and the limit variance of the

standard Monte Carlo estimator, denoted by σ2MC. By the usual Central Limit

Theorem for sums of i.i.d. random variables,

σ2MC = Var(g(U, V )) =

∫∫g(u, v)2 dC(u, v)−

(∫∫g(u, v) dC(u, v)

)2

.


We first derive an expression for σ2LHSD when C is the independence copula. Recall

that LHSD is a generalisation of Latin hypercube sampling (cf. Remark 11.2), sothat σ2

LHSD is a different way of writing the LHS limit variance derived in [Stein(1987)] and [Owen (1992)], where by a different argument, the LHS variance isderived as the “residual from additivity” of g.

We need the following Lemma:

Lemma 11.11. Let C be a copula and let h : [0, 1]4 → R be bounded. Then∫∫∫∫h(u, v, u′, v′) dC(u ∧ u′, v ∧ v′) =

∫∫h(u, v, u, v) dC(u, v).

Proof. Observe that C(u ∧ u′, v ∧ v′) is a copula, since by

C(u ∧ u′, v ∧ v′) = P(U ≤ u ∧ u′, V ≤ v ∧ v′)= P(U ≤ u, U ≤ u′, V ≤ v, V ≤ v′), (11.5.2)

it is a joint probability distribution with uniform marginals. By Equation (11.5.2),

Eh(U, V, U, V ) =∫∫∫∫

h(u, v, u′, v′) dC(u ∧ u′, v ∧ v′),

and the statement follows.

Proposition 11.12. Let g : [0, 1]d → R be of bounded variation and right-continuous, and let C be the independence copula, i.e., C(u, v) = uv, u, v ∈ [0, 1].Then for independent and U(0, 1)-distributed U1, U2, U3,

σ2LHSD = σ2

MC + 2(Eg(U1, U2)

)2 − E(g(U1, U2)g(U1, U3))− E(g(U1, U3)g(U2, U3))

≤ σ2MC.

Proof. For the first statement, by Equation (11.5.1), after some computations,

σ2LHSD =

∫∫∫∫ (u∧u′)(v∧v′)+uvu′v′−(u∧u′)vv′−uu′(v∧v′)


By integration by parts (see Appendix 12.2) and Lemma 11.11, after some calcula-tions,

σ2LHSD =

ZZ(g(1, 1) + g(u, v)− g(u, 1)− g(1, v))2 du dv

+

„ZZ(g(1, 1) + g(u, v)− g(u, 1)− g(1, v)) du dv

«2

−ZZZ

(g(1, 1) + g(u, v)− g(u, 1)− g(1, v))`g(1, 1) + g(u, v′)− g(u, 1)− g(1, v′)

´du dv dv′

−ZZZ

(g(1, 1) + g(u, v)− g(u, 1)− g(1, v))`g(1, 1) + g(u′, v)− g(u′, 1)− g(1, v)

´du du′ dv

=

ZZZZ(g(1, 1) + g(u, v)− g(u, 1)− g(1, v))

`g(u, v) + g(u′, v′)− g(u, v′)− g(u′, v)

´| z (?)

du du′ dv dv′


Observe that∫∫∫∫

(g(1, 1)− g(u, 1)− g(1, v))(?) dudu′dvdv′ = 0, so that

σ2LHSD =

∫∫∫∫g(u, v) (g(u, v) + g(u′, v′)− g(u, v′)− g(u′, v)) du du′ dv dv′

=∫∫

g(u, v)2 du dv +(∫∫

g(u, v) du dv)2

−∫∫∫

g(u, v)g(u, v′) du dv dv′ −∫∫∫

g(u, v)g(u′, v) du du′ dv,

which establishes the first statement.For the second statement, we show that E(g(U1, U2)g(U1, U3)) ≥

Eg(U1, U2)Eg(U1, U3). For the left-hand side we obtain by the Tower Law forconditional expectations and conditional independence of U2 and U3 given U1,

E(g(U1, U2)g(U1, U3)) = E(E(g(U1, U2)g(U1, U3)|U1

)= E

(E(g(U1, U2)|U1) E(g(U1, U3)|U1)

)= E(

(h(U1)2

),

with h(u) = Eg(u, U), U ∼ U(0, 1). By Jensen’s inequality

E(h(U1)2

)≥(Eh(U1)

)2= E

(E(g(U1, U2)|U1)

)E(E(g(U1, U3)|U1)

)= Eg(U1, U2) Eg(U1, U3).

By establishing E(g(U1, U3)g(U2, U3)) ≥ Eg(U1, U3)Eg(U2, U3) in the same way,the second statement follows.

The following Proposition gives us a means of comparing σ2LHSD and σ2

MC.

Proposition 11.13. Let the copula C of (U, V ) have continuous partial derivativesand let g : [0, 1]2 → R be of bounded variation and right-continuous. Then,

σ2LHSD = σ2

MC − 2Cov(g(U, V ), g(U, 0))− 2Cov(g(U, V ), g(0, V ))

+ Var(g(U, 0) + g(0, V ))− Cg= Var(g(U, V )− g(U, 0)− g(0, V ))− Cg,


where

Cg =∫∫∫∫

(1− ∂1C(u′, v′))(C(u, v)u′ − C(u ∧ u′, v))

+ (1− ∂1C(u, v))(C(u′, v′)u− C(u ∧ u′, v′))

+ (1− ∂2C(u′, v′))(C(u, v)v′ − C(u, v ∧ v′))

+ (1− ∂2C(u, v))(C(u′, v′)v − C(u′, v ∧ v′))

+ (1− ∂1C(u, v)∂1C(u′, v′))(u ∧ u′ − uu′)

+ (1− ∂2C(u, v)∂2C(u′, v′))(v ∧ v′ − vv′)

+ (1− ∂1C(u, v)∂2C(u′, v′))(C(u, v′)− uv′)

+ (1− ∂1C(u′, v′)∂2C(u, v))(C(u′, v)− u′v)


(11.5.3)

Proof. By Lemma 11.11,

σ2MC = Var(g(U, V ))

=∫∫

g(u, v)2 dC(u, v)−∫∫∫∫

g(u, v)g(u′, v′) dC(u, v) dC(u′, v′)

=∫∫∫∫

g(u, v)g(u′, v′) dC(u ∧ u′, v ∧ v′)

−∫∫∫∫

g(u, v)g(u′, v′) dC(u, v) dC(u′, v′).

Observe that the conditions required for integration by parts (see Appendix 12.2)are satisfied; in particular every copula is continuous [Nelsen (1999), Theorem 2.2.4].Integration by parts yields

σ2MC =

∫∫∫∫C(u ∧ u′, v ∧ v′) dg(u, v) dg(u′, v′)−

(∫∫C(u, v) dg(u, v)

)2

+∫∫∫∫

(C(u, v)u′ − C(u ∧ u′, v)) + (C(u′, v′)u− C(u ∧ u′, v′))

+ (C(u, v)v′ − C(u, v ∧ v′)) + (C(u′, v′)v − C(u′, v ∧ v′))

+ (u ∧ u′ − uu′) + (v ∧ v′ − vv′)

+ (C(u, v′)− uv′) + (C(u′, v)− u′v)

dg(u, v) dg(u′, v′)

+ 2Cov(g(U, V ), g(U, 0)) + 2Cov(g(U, V ), g(0, V ))−Var(g(U, 0) + g(0, V )).


The first statement follows by combination with Equation (11.5.1). The secondstatement follows from

2Cov(g(U, V ), g(U, 0)) + 2Cov(g(U, V ), g(0, V ))−Var(g(U, 0) + g(0, V ))

= Var(g(U, V ))−Var(g(U, V )− g(U, 0)− g(0, V )).

For copulas with a specific dependence property and assuming that g is nonde-creasing in each argument, σ2

LHSD is never greater than σ2MC as we now show. For

a comprehensive treatment of dependence properties of copulas, see [Nelsen (1999),Section 5.2] and [Joe (1997), Section 2.1].

Let X and Y be two random variables. We say that Y is right-tail increasing inX if, for all y, x 7→ P(Y > y|X > x) is nondecreasing. If X and Y are continuousrandom variables whose copula C has continuous partial derivatives, then Y isright-tail increasing in X if and only if

∂1C(u, v) ≥ v − C(u, v)1− u

, u, v ∈ [0, 1],

cf. [Nelsen (1999), Corollary 5.2.6]. We say that C is RTI if X is right-tail increasingin Y and Y is right-tail increasing in X. An example of a copula that is RTI and thathas continuous partial derivatives is the bivariate Normal copula with parameterρ ∈ (0, 1); see [Joe (1997), Secion 5.1] for a comprehensive list of one- and two-parameter copulas that are RTI.

Proposition 11.14. Let the copula C be RTI and have continuous partial deriva-tives and let g : [0, 1]2 → R be right-continuous, of bounded variation and monotonenondecreasing in each argument. Then σ2

LHSD ≤ σ2MC.

Proof. First note that if C is RTI then C(u, v) ≥ uv, for all u, v ∈ [0, 1] (thisproperty is called positive quadrant dependence).

Under the conditions stated, Var(g(U, V )) ≥ Var(g(U, V ) − g(U, 0) − g(0, V )),which can be verified for example by integration by parts. It remains to be estab-lished that Cg given by Equation (11.5.3) is nonnegative. Consider first the caseu ≤ u′ and the first, second, fifth and seventh term of the integral of Equation(11.5.3):


(1− ∂1C(u′, v′))(C(u, v)u′ − C(u, v)) + (1− ∂1C(u, v))(C(u′, v′)u− C(u, v′))

+ (1− ∂1C(u, v)∂1C(u′, v′))(u− uu′) + (1− ∂1C(u, v)∂2C(u′, v′))(C(u, v′)− uv′)

= (1− ∂1C(u′, v′))(1− u′)(u− C(u, v))− (1− ∂1C(u, v))u(v′ − C(u′, v′))

+ ∂1C(u′, v′)(1− ∂1C(u, v))u(1− u′) + ∂1C(u, v)(1− ∂2C(u′, v′))(C(u, v′)− uv′)

= (1− ∂1C(u′, v′))(1− u′)(u− C(u, v))− (1− ∂1C(u, v))uv′ − C(u′, v′)

1− u′(1− u′)


RTI

≥ (1− ∂1C(u′, v′))(1− u′)(u− C(u, v))− (1− ∂1C(u, v))u∂1C(u′, v′)(1− u′)


= (1− ∂1C(u′, v′))(1− u′)(u− C(u, v)) + ∂1C(u, v)(1− ∂2C(u′, v′))(C(u, v′)− uv′)

≥ 0,

since all partial derivatives are in [0, 1], u ≥ C(u, v) and C(u, v′) ≥ uv′. In the casev ≤ v′, the same computation may be applied for the remaining terms of the integralof Equation (11.5.3). In the same way nonnegativity for the case u′ ≤ u, v′ ≤ v isobtained. Finally, consider the cases u ≤ u′, v′ ≤ v and u′ ≤ u, v ≤ v′. Observethat we may regroup the integrand of Equation (11.5.3), taking into account thatg(u, v) and g(u′, v′) may be exchanged appropriately. In the case u ≤ u′, v′ ≤ v,write the last two terms of the integrand of Equation (11.5.3) as∫∫∫∫

2(1− ∂1C(u, v)∂2C(u′, v′))(C(u, v′)− uv′) dg(u, v) dg(u′, v′)

and in the case u′ ≤ u, v ≤ v′ as∫∫∫∫2(1− ∂1C(u′, v′)∂2C(u, v))(C(u′, v)− u′v) dg(u, v) dg(u′, v′),

and repeat the computation above accordingly.

Example 11.15. Let g(u, v) = ln(ln(uv + 1) + 1) and let (U1, U2) be a randomvector with uniform marginals and Normal copula with parameter ρ = 0.5. Numer-ical integration yields σ2

MC = 0.022756 and σ2LHSD = 0.001101. We estimated σ2

MC

and σ2LHSD by running 1000 batches of n independent simulations of the respective

estimators, for n ∈ 200, 400, 600, 800, 1000. The deviations to the numbers fromnumerical integration are within 0.003 for MC and 4 · 10−5 for LHSD.

Numerical examples indicate that the classes of functions and copulas for which theLHSD limit variance is bounded from above by the respective MC limit varianceare much larger than the ones stated in Proposition 11.14.

11.6. LHSD on random vectors with nonuniform marginals 133

11.6 LHSD on random vectors with nonuniform marginals

So far, we have restricted our analysis to vectors of uniform random variables on[0, 1]. We now provide the link to random vectors with nonuniform marginals. It isalways possible to generate a random variable of arbitrary distribution from a uni-form random variable on [0, 1] by applying the so-called inverse transform method.The association of a joint distribution function with a copula (a distribution func-tion with uniform marginals on [0, 1]) leads to methods for constructing randomvectors (X1, . . . , Xd) with arbitrary marginals from random vectors (U1, . . . , Ud),where U j ∼ U(0, 1), j = 1, . . . , d. We discuss this in more detail.

The inverse transform method is explained for example in [Glasserman (2004),Section 2.2.1] and [Nelsen (1999), Sections 2.3, 2.9]. Let X be a random variablewith distribution function F . We shall assume F to be continuous, which impliesP(X = x) = 0, x ∈ R. The right-inverse of F is defined as the function F (−1) :[0, 1]→ R ∪ ±∞ with

F (−1)(u) := infx : F (x) > u, u ∈ [0, 1].

The right-inverse is right-continuous, strictly increasing and has at most countablymany discontinuities. If F is strictly increasing, then F (−1) is just the inverse ofF . From the monotonicity of distribution functions, F (−1)(u) < x if and only ifu < F (x). It follows that if U ∼ U(0, 1), then X

L= F (−1)(U), since

P(X < x) = F (x) = P(U < F (x)) = P(F (−1)(U) < x).

Accordingly, for a Borel-measurable function h : R → R, h(X) L= g(U), withg := h F (−1).

Now consider the multivariate case. Recall that a copula is a multivariatedistribution function whose margins are U(0, 1) distributions. By Sklar’s Theo-rem [Nelsen (1999), Theorem 2.10.9], the copula associated with a d-dimensionaldistribution function F and univariate marginal distribution functions F1, . . . , Fdis the distribution function C : [0, 1]d → [0, 1] that satisfies F (x1, . . . , xd) =C(F1(x1), . . . , Fd(xd)). Conversely, for any (u1, . . . , ud) ∈ [0, 1]d,

C(u1, . . . , ud) = F (F (−1)1 (u1), . . . , F (−1)

d (ud)),

cf. [Nelsen (1999), Corollary 2.10.10]. If F is continuous, then C is unique, otherwiseC is unique on RanF1× · · ·×RanFd, where RanFj ⊆ [0, 1] denotes the range of Fj ,j = 1, . . . , d. The copula provides the link between the marginal distributions andthe joint distribution of a random vector.

Now consider a random vector (X1, . . . , Xd) with marginal distribution functionsF1, . . . , Fd and joint distribution function F . Then, for a Borel-measurable functionh : Rd → R,

h(X1, . . . , Xd) L= h(F (−1)1 (U1), . . . , F (−1)

d (Ud)) =: g(U1, . . . , Ud), (11.6.1)

where the joint distribution of (U1, . . . , Ud) is determined by the copula correspond-ing to F and F1, . . . , Fd. The following properties are immediate:


(i) If Fj , j = 1, . . . , d are continuous, and if h is F -a.e. continuous, then g isC-a.e. continuous.

(ii) If h is right-continuous, and F(−1)j , j = 1, . . . , d, are the right-inverses of Fj ,

j = 1, . . . , d, then g is right-continuous. Moreover, if h is of bounded variation,then so is g; this follows from the the strict monotonicity of the right-inverses.

Now, assuming that h is F -integrable, the LHSD estimator of Eh(X1, . . . , Xd) isgiven by

1n

n∑i=1

h(F (−1)1 (V 1

i,n), . . . , F (−1)d (V di,n)), (11.6.2)

with V ji,n, i = 1, . . . , n, j = 1, . . . , d as in Equation (11.3.2).By the following Lemma, the ranks may be computed without first transforming

the marginals X1, . . . , Xd into uniforms.

Lemma 11.16. Let X1, . . . , Xn be i.i.d. random variables whose distribution F iscontinuous. Then, for any i = 1, . . . , n,

ri,n(X1, . . . , Xn) = ri,n(F (X1), . . . , F (Xn)) P–a.s..

Proof. If F is strictly increasing, the statement is clear by Equation (11.3.1). ByEquation (11.3.1) it suffices to show that P–a.s. Xi ≤ Xj if and only if F (Xi) ≤F (Xj), for any i, j = 1, . . . , n. By monotonicity of F , Xi ≤ Xj implies F (Xi) ≤F (Xj). For the reverse statement consider

P(F (Xi) = F (Xj), Xi > Xj) = P(Xi ∈ (Xj , F(−1)(F (Xj))])

=∫

P(Xi ∈ (y, F (−1)(F (y))])F (dy) =∫ [

F (F (−1)(F (y)))− F (y)]F (dy) = 0,

where the last equality follows from F (F (−1)(z)) = z, because of the continuity ofF .

Chapter 12

Applications in finance

We demonstrate the effectiveness of LHSD with two examples. First, we valuea first-to-default credit basket (FTD) - a contract that insures the loss incurredby the first default event in a basket of underlying securities. The value of anFTD depends crucially on the joint default probability distribution of the basketcomponents. The example demonstrates that LHSD is an effective technique whensampling rare events; in fact, LHSD guarantees that a certain number of rare eventsis sampled. We also combine Quasi-Monte Carlo (QMC) and LHSD by feeding ouralgorithm with Sobol sequences instead of random numbers. The combination ofthese techniques leads to a further pickup in efficiency.

In the second example we value an Asian basket call option. Here, a call optionis written on the weighted sum of a basket of securities monitored at several timepoints. The example is taken from [Imai and Tan (2007)], where a basket of 10assets is monitored at 250 time points. [Imai and Tan (2007)] show that eachsimulation entails generating a correlated random vector of size 2500. This exampledemonstrates that LHSD can be used for high-dimensional problems. It is knownthat low discrepancy sequences lose their effectiveness in high dimensions, hencewe do not test the combination of QMC and LHSD. There are techniques to useQMC in a high-dimensional setting, see e.g. [Owen (1998)]; a combination of thesetechniques with LHSD may again improve results.

12.1 Valuing a first-to-default credit basket

An FTD is a contract between two counterparties, a protection buyer and a protec-tion seller, that insures the protection buyer against the loss incurred by the firstdefault event in a portfolio of some underlying risky entities over a fixed time hori-zon. The protection buyer regularly pays a constant premium s, called the spread,as a fraction of the notional until the first default event in the underlying portfoliotakes place or until maturity of the FTD, whichever occurs first. This stream ofpayments is termed the premium leg of the FTD. In turn, the protection buyercompensates the protection buyer for the loss incurred by the first default event at

135

136 Applications in finance

the time of default. This side of the contract is called the default leg .For the valuation of an FTD we follow [Schmidt and Ward (2002)]. With each

credit j = 1, . . . , d of the underlying portfolio we associate the random default timeτj and the recovery rate Rj . We assume Rj to be constant and known. Furthermore,we assume the default distributions P(τj ≤ t), t ≥ 0, j = 1, . . . , d, to be given. Thesecan be derived from the credit default swap (CDS) market; as an approximation,assuming a constant CDS spread sj for credit j, we determine the default intensityλj , of credit j from the so-called credit triangle, λj := sj/(1−Rj), and we set

Fj(t) := P(τj ≤ t) = 1− e−λjt, t ≥ 0. (12.1.1)For t ≥ 0, denote by Bt today’s default-free zero bond price with maturity t.

Let t0 = 0 and let t1 < t2 < . . . < tK = T be the spread payment dates of an FTDwith maturity T , and set ∆tk := tk − tk−1, k = 1, . . . ,K. Denote the time of theFTD’s default event by τ := min(τ1, . . . , τd). The discounted payoffs of the defaultleg and the premium leg are given by

hd(τ1, . . . , τd) =d∑j=1

(1−Rj)B(τ)1(0,T ](τ) 1τ=τj (12.1.2)

hp(τ1, . . . , τd) = s

K∑k=1

∆tkB(tk)1τ>tk. (12.1.3)

The fair spread s of the FTD is then obtained by equating the expected value (undera risk-neutral measure) of the premium and the default leg,

s

K∑k=1

∆tkBtkP(τ > tk) =d∑j=1

(1−Rj)∫ T

0

BuP(τ ∈ du, τ = τj). (12.1.4)

From this equation and from P(τ ≤ t) = 1−P(τ1 > t, τ2 > t, . . . , τd > t) it is clearthat the value of the FTD depends on the joint distribution of τ1, . . . , τd. Settings = 1, the left-hand side of Equation (12.1.4) can be interpreted as the risky presentvalue of a basis point.

In our example we assume that the joint distribution of the default timesτ1, . . . , τd is driven by a Normal copula (Gaussian copula),

P(τ1 ≤ t, . . . , τj ≤ t) = NΣ

(N(−1)(F1(t)), . . . ,N(−1)(Fj(t))

),

with NΣ the multivariate standard normal distribution function with correlationmatrix Σ and N(−1) the inverse of the univariate standard normal distributionfunction.

The valuation algorithm for the fair FTD spread is given by Algorithm 2. Theinput parameters for an example involving 5 homogeneous credits are given in Table12.1. The fair FTD spread was computed from simulations using random numbersand using low discrepancy sequences, both “as is” and adding a LHSD step (fordetailed exposition on Quasi-Monte Carlo methods and low discrepancy sequences,see [Niederreiter (1992)] and [Glasserman (2004), Chapter 5]). This leads to thefollowing four simulation cases:

12.1. Valuing a first-to-default credit basket 137

1: // n: number of simulations, d : number of credits2: for j = 1 to d do3: λj ← sj/(1−Rj) // default intensities; credit triangle4: end for5: Compute A such that AAT = Σ // e.g. Cholesky factorisation6: for i = 1 to n do7: for j = 1 to d do8: generate Xj

i ∼ N(0, 1) // independent of Xmk , k = 1, . . . i− 1, m =

1, . . . , j − 19: end for

10: (Z1i , . . . , Z

di )T ← A · (X1

i , . . . , Xdi )T // vector of correlated standard

normal samples11: end for12: for j = 1 to d do13: compute rj1,n, . . . , r

jn,n // ranks from (Zj1 , . . . , Z

jn), cf. Lemma 11.16

14: for i = 1 to n do15: V ji,n ← (rji,n − 1/2)/n // Equation (11.3.2)

16: τ ji ← F(−1)j (V ji,n) // default times; F (−1)

j (t) := − ln(1−t)/λj , Equa-tion (12.1.1)

17: end for18: end for19: s← 120: for i = 1 to n do21: Li ← hd(τ1

i , . . . , τdi ) // discounted default leg, Equation (12.1.2)

22: Pi ← hp(τ1i , . . . , τ

di ) // Equation (12.1.3)

23: end for24: L← (L1 + · · ·+Ln)/n // present value of expected loss, RHS of Equation

(12.1.4)25: P ← (P1 + · · · + Pn)/n // PV of a risky basis point, left-hand side of

Equation (12.1.4)26: return s← L/P // fair spread of FTD

Algorithm 2: FTD valuation

(i) Standard Monte Carlo simulation(ii) LHSD based on random numbers(iii) Simulation with low discrepancy sequence(iv) LHSD based on low discrepancy sequence

The implementation was done in C++ with the QuantLib library [QuantLib (2008)]using the Mersenne twister algorithm for random number generation and Sobolsequences for low discrepancy sequences. Root mean square error (RMSE) estimateswere obtained by simulating each estimator 100 times. The RMSE estimates and


Table 12.1 Parameters of FTD example; the fair spread of theFTD is 417.88bp.Parameter Value

Maturity T = 5 (years)spread payment dates (frequency) (tk)k=1,...,K (quarterly)Default-free zero bond prices Bt = e−.05t, t ≥ 0Number of underlying credits d = 55yr.-CDS spread of each credit sj = 1%, j = 1, . . . , dRecovery rate of each credit Rj = 0.3, j = 1, . . . , dCorrelation between any two credits ρ = 30%

RMSE ratios for various samples sizes are given in Table 12.2. The ratios of CPUtime consumed for generating samples with and without LHSD is also shown forvarious sample sizes. The CPU time ratios do not include the CPU time requiredfor computing the FTD payoff; consequently the efficiency of LHSD increases withthe CPU time required for computing the payoff function. The LHSD step involvessorting a sequence of random numbers (see e.g. [Press et al. (1992)], Chapter 8.4for sorting algorithms), hence the computational overhead of the LHSD step is ofthe complexity of the sorting algorithm. On the other hand, by Lemma 11.16,the rank statistics can be computed from samples of any distribution (cf. Line13 in Algorithm 2), whereas in a typical Monte Carlo simulation, the generatedsamples may additionally need to be transformed to uniforms. Observe that overall simulations, LHSD samples a fixed number of default events of the individualcredits, but the occurrence of joint defaults is random. The rather moderate pick-up in variance may explained by the fact that most samples in FTD valuationcorrespond to no-default events, both when applying naive Monte Carlo simulationand when adding the LHSD step. We presume that the combination with othervariance reduction techniques, such as importance sampling, may prove to be moreeffective.

12.2 Valuing an Asian basket option

We now consider pricing an Asian basket option1, whose payoff depends on thesum of several underlying assets monitored at various points in time. As this isa path-dependent option in a high-dimensional setting, simulation is a standardvaluation approach. Following [Imai and Tan (2007)], the payoff may be formulatedas a function of a matrix product whose dimensions depend on the number of assetsand time points monitored.

Assume a basket of m assets, with Sit the price of the i-th asset at time t,i = 1, . . . ,m. Fixing a maturity T , a strike K, a set of n monitoring time points0 < t1 < t2 < . . . < tn = T and weights wij , i = 1, . . .m, j = 1, . . . n,

∑i,j w

ij = 1,1This is also known as an arithmetic average Asian option.

12.2. Valuing an Asian basket option 139

Table 12.2 Root mean square error of estimation in basis points andCPU time ratios for various sample sizes (100 simulations of estimator).Comparable ratios were obtained for smaller simulation sizes. The fairFTD spread is 417.88bp.No. of sim. (×103) 200 400 600 800 1000

MC 2.02 1.47 1.10 0.89 0.80MC + LHSD 1.00 0.61 0.53 0.45 0.39Sobol 0.30 0.20 0.16 0.14 0.11Sobol + LHSD 0.21 0.12 0.11 0.09 0.08

MC/(MC + LHSD) 2.02 2.41 2.08 1.98 2.05Sobol/(Sobol + LHSD) 1.43 1.67 1.45 1.56 1.38

CPU time (MC + LHSD)/MC 1.66 1.71 1.75 1.78 1.82CPU (Sobol + LHSD)/Sobol 1.47 1.52 1.54 1.55 1.56

CPU time ratios involve the generation of random samples only. Adding theCPU time required for computing the payoff decreases the ratios accordingly.

the payoff of the Asian basket call option on the m-asset basket is

max

m∑i=1

n∑j=1

wijSitj −K, 0

. (12.2.1)

We assume that asset prices follow a Geometric Brownian motion, i.e., S1, . . . , Sm

is the solution of the stochastic differential equation (SDE)

dSit = rSit dt+ σiSit dW it , i = 1, . . . ,m,

where r is the risk-free interest rate, σi is the volatility of the i-th asset and(W 1, . . . ,Wm) is an m-dimensional Brownian motion, whose components W i, W k

are correlated with ρik, 1 ≤ i, k ≤ m. The solution of the SDE is given by

Sit = Si0e(r−(σi)2/2)t+σiW i

t , i = 1, . . . ,m. (12.2.2)

Pricing the option requires simulating the paths of each asset at the monitoringtime points. Assume that the time points t1, . . . , tn are equidistant and let ∆t =T/n so that tj = j∆t. Let Σ be an m × m covariance matrix given by Σ =(ρikσiσk ∆t)i,k=1,...,m. Let Σ be the nm× nm-matrix generated from Σ via

Σ =

Σ Σ · · · ΣΣ 2Σ · · · 2Σ...

.... . .

...Σ 2Σ · · · nΣ

.

The asset prices may be simulated according to Equation (12.2.2) with W =(σ1W 1

t1 , . . . , σmWm

t1 , σ1W 1

t2 , . . . , σmWm

tn )′ derived via

W = CZ,

where C is such that CC ′ = Σ and Z is a vector of nm independent standardNormal random variables. The payoff at time T of the Asian basket option canthen be written as

max(g(W )−K, 0)


Table 12.3 Parameters of Asian basket optionParameter Value

Maturity T = 1 (years)Number of assets m = 10Number of time steps n = 250Weights wij = 1/(nm), i = 1, . . . , n, j = 1, . . . , m

Initial asset value Sj0 = 100, j = 1, . . . , m

Asset volatility σj = 0.1 + (j − 1)/(m− 1) · 0.4, j = 1, . . . , mCorrelation ρij = 0.4, 1 ≤ i < j ≤ mInterest rate r = 0.04Strike K = 90, 100, 110

with

g(W ) =mn∑k=1

eµk+Wk

µk = ln(wk1 k2Sk1(0)) +(r − (σk1)2/2

)tk2 , where

k1 = (k − 1) mod m+ 1

k2 = b(k − 1)/mc+ 1, k = 1, . . . ,mn.

In this approach, simulation of option payoffs involves the computation of prod-ucts of high-dimensional matrices. For C we choose the Cholesky decompositionof Σ (i.e., C is lower triangular with CC ′ = Σ). Typical choices of C other thanthe Cholesky decomposition yield a reduction of the dimension of the matrix mul-tiplication, while keeping the error introduced small. For example, the simulationtechnique of [Imai and Tan (2007)] reduces the dimension of the problem by deter-mining C as the solution of an optimisation problem.

Based on the data set of [Imai and Tan (2007)], we simulate W using a Choleskydecomposition and introducing an LHSD step in each dimension over all simulations.The parameters of the option are given in Table 12.3. As in [Imai and Tan (2007)],we computed 10 runs of 4096 simulations and 10 runs of 8192 simulations. Theresulting prices and standard deviations for Monte Carlo and LHSD estimates aregiven in Table 12.4. The implementation was done in C++ using QuantLib, see[QuantLib (2008)], and using matrix multiplication routines from the Fortran codeof GNU Octave, see [GNU Octave (2008)]. The results show that LHSD outperformsthe standard Monte Carlo simulator by factors of 2.5 to 15 based on standarddeviations (resp. 9 and 200 in terms of variance); the computing time consumed byLHSD increases by a factor of approximately 1.4. The pickup in accuracy dependsstrongly on the strike of the option and decreases with increasing strike. The sameobservation is made in an example from [Glasserman (2004), p. 242-243], where anAsian call option is priced using standard LHS. There, this behaviour is attributedto the fact that LHS is more effective the more the function to be estimated is

12.2. Valuing an Asian basket option 141

Table 12.4 Simulated prices of an Asian basket option (parameters in Table 12.3) forstrikes K ∈ 90, 100, 110. The results are based on 10 runs with 4096 and 8192 simulationseach. The numbers in parentheses denote the sample standard deviation based on the 10runs. The CPU time ratios of LHSD versus MC were 1.40 CPU seconds (4096 simulations)and 1.44 CPU seconds (8192 simulations).

sim. size K = 90 K = 100 K = 110

MC 4096 12.3045 (0.1930) 5.6726 (0.1402) 2.0574 (0.0916)MC+LHSD 4096 12.3283 (0.0130) 5.6567 (0.0187) 2.0288 (0.0316)MC 8192 12.3481 (0.1602) 5.6697 (0.1041) 2.0413 (0.0633)MC+LHSD 8192 12.3253 (0.0150) 5.6535 (0.0166) 2.0302 (0.0261)

MC/(MC+LHSD) 4096 14.8462 7.4973 2.8987MC/(MC+LHSD) 8192 10.6800 6.2711 2.4253

”additive”; this is resembled at lower strikes, where the option payoffs are morelinear.

To benchmark their method, [Imai and Tan (2007)] simulated the Asian basketoption using a Quasi-Monte Carlo method together with a technique called Latinsupercube sampling. The latter method avoids sampling low discrepancy sequencesin high dimensions, see [Owen (1998)]. The standard error of LHSD is comparableto that of the QMC technique, the latter being between 0.00905 and 0.0144. Recallthat LHSD is a very simple and practicable technique. Finally, it should be notedthat our results do not keep up with standard errors obtained from the dimensionreduction technique of [Imai and Tan (2007)], but we conjecture that combinationsof LHSD together with dimension-reduction techniques may be effective.

Appendix A

Probability Theory andStochastic Calculus

A.1 Regular conditional probabilities

We recall here the definitions of a regular conditional probability and regular condi-tional distribution, some facts and some conditions for their existence; see [Shiryaev(1996), Chapter II.§7.7] for details and proofs. Let (Ω,F ,P) be a probability spaceand let G be a sub-σ-algebra of F .

Definition A.1. A function P(ω;B), defined for all ω ∈ Ω and B ∈ F , is a regularconditional probability with respect to G if

(i) P(ω; ·) is a probability measure on F for every ω ∈ Ω;(ii) For each B ∈ F the function P(ω;B), as a function of ω, is a version of the

conditional probability P(B|G), i.e., P(ω;B) = P(B|G)(ω) for P–a.a.ω.

Theorem A.2. Let P(ω;B) be a regular conditional probability with respect to Gand let ξ be an integrable random variable. Then,

E(ξ|G)(ω) =∫

Ω

ξ(ω)P(ω; dω) for P–a.a.ω.

Definition A.3. Let (E,E ) be a measurable space, X = X(ω) a random elementwith values in E. A function Q(ω;B) defined for all ω ∈ Ω and B ∈ E is a regularconditional distribution of X with respect to G if

(i) for each ω ∈ Ω the function Q(ω;B) is a probability measure on (E,E );(ii) for each B ∈ E , the function Q(ω;B), as a function of ω, is a version of the

conditional probability P(X ∈ B|G)(ω), i.e., Q(ω;B) = P(X ∈ B|G)(ω) forP–a.a.ω.

Definition A.4. Let ξ be a random variable. The function F (ω;x), ω ∈ Ω, x ∈ R,is a regular conditional distribution function for ξ with respect to G if

(i) F (ω;x) is, for each ω ∈ Ω, a distribution function on R;(ii) F (ω;x) = P(ξ ≤ x|G)(ω) for P–a.a.ω, for each x ∈ R.

143

144 Probability Theory and Stochastic Calculus

We also say that F is a regular version of the conditional probability P(ξ ≤ ·|G).

Theorem A.5. A regular conditional distribution and a regular conditional distri-bution function always exist for the random variable ξ with respect to G.

Theorem A.6. Let X = X(ω) be a random element with values in a completeseparable metric space (E,E ). Then there is a regular conditional distribution of Xwith respect to G.

A.2 Ito formula for semimartingales

The following material is from [Protter (2005), Sections 2.6, 2.7]. For a semimartin-gale X, the process [X,X]c denotes the path-by-path continuous part of [X,X], andwe can write

[X,X]t = [X,X]ct +∑

0<s≤t

(∆Xs)2.

The following statement is sometimes helpful to determine [X,X]c.

Theorem A.7. If X is adapted, cadlag, with paths of finite variation on compacts,then [X,X]c = 0.

Theorem A.8 (Ito formula for semimartingales). Let X = (X1, . . . , Xn) bean n-tuple of semimartingales, and let f : Rn → R have continuous second orderpartial derivatives. Then f(X) is a semimartingale and the following formula holds:

f(Xt)− f(X0) =n∑i=1

∫ t

0+

∂f

∂xi(Xs−) dXi

s +12

∑1≤i,j≤n

∫ t

0+

∂2f

∂xi∂xj(Xs−) d[Xi, Xj ]cs

+∑

0<s≤t

f(Xs)− f(Xs−)−

n∑i=1

∂f

∂xi(Xs−) ∆Xi

s

.

A.3 Classification of stopping times

See e.g. [Protter (2005), Section III.2] for the following definitions:

Definition A.9. A stopping time T is predictable if there exists a sequence ofstopping times (Tn)n≥1 such that Tn is increasing, Tn < T on T > 0, for all n,and limn→∞ Tn = T P–a.s.. Such a sequence (Tn) is said to announce T .

Definition A.10. A stopping time T is totally inaccessible if for every predictablestopping time S

P(ω : T (ω) = S(ω) <∞) = 0.

A.4. Martingale decomposition 145

A.4 Martingale decomposition

For the following see e.g. [Jacod and Shiryaev (2003), Section 1.4].

Definition A.11. The local martingales M and N are orthogonal if MN is a localmartingale. A local martingale M is a purely discontinuous local martingale ifM0 = 0 and if it is orthogonal to all continuous local martingales.

Theorem A.12. Any local martingale M admits a unique decomposition

M = M0 +M c +Md,

where M c0 = Md

0 = 0, M c is a continuous local martingale and Md is a purelydiscontinuous local martingale.

Lemma A.13. A martingale M of finite variation with M0 = 0 is a purely discon-tinuous martingale.

Recall the definition of a compensator, see e.g. [Protter (2005), Section III.5] or[Klebaner (2005), Section 8.9].

Definition A.14. Let N = (Nt)t≥0 be a finite variation process with N0 = 0,with locally integrable variation.2 The unique predictable process of finite variation,A = (At)t≥0 such that N −A is a local martingale is called the compensator of N .

See [Klebaner (2005), Theorem 8.45] for the following statement.

Lemma A.15. The compensator of N is continuous if and only if the jump timesof the process N are totally inaccessible.

A.5 Stochastic exponential

The formula of the stochastic exponential for semimartingales is given for examplein [Jacod and Shiryaev (2003), Section I.4].

Theorem A.16. Let X be a real-valued semimartingale. Then the cadlag , adaptedprocess

Y = 1 + Y−·X (or: dY = Y− dX and Y0 = 1),

also denoted by E (X) and called the stochastic exponential or Doleans-Dade expo-nential of X is the unique semimartingale given by

E (X)t = eXt−X0− 12 [X,X]ct

∏0<s≤t

(1 + ∆Xs)e−∆Xs ,

where [X,X]c is the quadratic variation of the continuous part of X.2An increasing process N is called integrable if supt≥0 ENt < ∞. A finite variation process N

is of integrable variation if its variation process is integrable, i.e., supt≥0 EV Nt < ∞. A finite

variation process N is of locally integrable variation if there is a sequence of stopping times τn,n ≥ 0, such that τn ↑ ∞ so that Nt∧τn is of integrable variation, i.e., supt≥0 EV N

t∧τn< ∞.

146 Probability Theory and Stochastic Calculus

A.6 Other results

We use the Independence Lemma many times. See e.g. [Elliott and Kopp (2005),Lemma 6.6.5] for a proof.

Lemma A.17 (Independence Lemma). Let G ⊂ F be a σ-algebra and let X,Y be random elements with values in the Borel space (E,E ) and such that X isG-measurable and Y is independent of G. Let h : E ×E → R be a bounded E × E -measurable function. Define g(x) := Eh(x, Y ). Then g(X) is a version of theconditional expectation E(h(X,Y )|G).

The following Theorem is from [Nielsen (1999), Appendix B]; see [Ethier and Kurtz(1986), Proposition 2.4.6] for a proof.

Theorem A.18 (Conditional Fubini Theorem). Let (Ω,F ,P) be a probabil-ity space, T a subset of the real line, with B(T ) the Borel σ-algebra on T , µ asigma-finite measure on (T ,B(T )),

X : Ω× T → R

a function that is measurable with respect to B(T )⊗F and such that∫E|Xt|µ(dt) <∞,

and let G ⊂ F be a σ-algebra. Then, there exists a function

Y : Ω× T → R,

measurable with respect to G ⊗ B(T ), and such that

•∫

E|Yt|µ(dt) <∞,•∫Yt µ(dt) is a version of E

(∫Xt µ(dt)

∣∣G),• for all t ∈ T , Yt is integrable with respect to P and is a version of E(Xt|G).

Under the conditions of Theorem A.18, we have P–a.s.∫E(Xt|G)µ(dt) = E

(∫Xt µ(dt)

∣∣∣G) .

Appendix B

Levy processes

B.1 Levy processes

We review the definition and some properties of Levy processes, subordinators (i.e.,Levy processes with nondecreasing paths), and we introduce the compound Poissonprocess. We state conditions for the solution of a Levy-driven stochastic differentialequation to be a Markov process.

Monographs on Levy processes are [Sato (1999); Bertoin (1998); Applebaum(2004b)]. For overviews we refer to [Protter (2005), Section I.4] and [Applebaum(2004a)]. Levy processes in financial applications are treated by [Cont and Tankov(2004a); Schoutens (2003)].

Definition B.19. An (Ft)t≥0-adapted stochastic process X = (Xt)t≥0 on Rd is aLevy process if the following conditions are satisfied:

(i) X0 = 0 P–a.s..(ii) (Independent increments) For any 0 ≤ s < t, Xt −Xs is independent of Fs.(iii) (Stationary increments) For any s ≤ t, Xt+s −Xs

L= Xt.(iv) (Stochastic continuity) For every t ≥ 0 and for every ε > 0,

lims→t

P(|Xs −Xt| > ε) = 0.

A stochastic process X = (Xt)t≥0 that satisfies (i), (ii) and (iv) is called an additiveprocess. We shall henceforth consider only Levy processes on R.

Every Levy process has a unique cadlag modification [Protter (2005), TheoremI.30], and we shall henceforth always assume that we are working with the cadlagmodification. Consequently, the only type of discontinuities a Levy process canhave are jump discontinuities. For a Levy process X, letting Xt− = lims↑tXt, wedefine

∆Xt = Xt −Xt−,

to be the jump at t. Moreover, for any fixed t > 0,

Xt = Xt− P–a.s., (B.1.1)

147

148 Levy processes

since by the cadlag property lims↑tXs = Xt− P–a.s., by stochastic continuity XsP→

Xt as s ↑ t, and the two limits agree, see also [Sato (1999), p. 6]. A cadlag processhas at most countably many jumps in any interval [s, t], which justifies the notation∑s<u≤t ∆Xu to denote the sum of jumps of Xt −Xs.3 The jump times of a Levy

process are totally inaccessible, cf. [Protter (2005), p. 105].An important characterisation of Levy processes is via the Levy-Khintchine

representation, which states that the characteristic function of a Levy processX = (Xt)t≥0 is given by

E[eiuXt

]= etψ(u), u ∈ R, t ≥ 0, (B.1.2)

with

ψ(u) = iγu− 12η2u2 +

∫ ∞

−∞

[eiux − 1− iux1|x|<1

]ν(dx), (B.1.3)

where γ ∈ R, η2 ≥ 0 and ν is a measure on R \ 0 with∫ ∞

−∞

(1 ∧ x2

)ν(dx) <∞.

The function ψ is called the characteristic exponent, and the triplet (γ, η2, ν) iscalled the Levy triplet.4 Conversely, given ψ as in Equation (B.1.3) (satisfying theconstraints on η and ν), there exists a Levy process whose characteristic exponentis ψ, cf. [Bertoin (1998), Theorem 1.1].

The parameter γ describes the linear deterministic behaviour of a Levy process,η2 parameterises a Brownian motion part. The measure ν is called the Levy measure.The Levy measure describes how jumps occur, via

ν(A) = E [# t ∈ [0, 1] : ∆Xt 6= 0,∆Xt ∈ A] , A ∈ B(R).

B.2 Subordinators

A subordinator is a Levy process taking values in [0,∞), which implies that its sam-ples paths are nondecreasing, see e.g. [Bertoin (1998), Chapter 3].5 Subordinatorsfind application for example as time-changes of other processes. A subordinatorhas no Brownian part, γ > 0, and its Levy measure ν satisfies ν((−∞, 0]) = 0 and∫∞0

(1 ∧ x) ν(dx) < ∞. Moreover, it is of finite variation, and its characteristicexponent is given by

ψ(u) = iγu+∫ ∞

0

(eiux − 1

)ν(dx). (B.2.1)

3This does not imply thatP

s<u≤t ∆Xu < ∞.4Strictly speaking, the parameter γ of the Levy triplet depends on the choice of the cutoff function

in the integral of Equation (B.1.3). The function 1|x|<1 is a common choice. The reason for thecutoff is that possibly the sum of jumps near 0 does not converge and needs to be compensatedaccordingly.5The definition of a subordinator is not standardised in the literature: Sometimes, a subordinator

refers more generally to a semimartingale with P–a.s. nonnegative and nondecreasing paths.

B.3. Compound Poisson processes 149

Conversely, a Levy process with η2 = 0, γ > 0, ν((−∞, 0]) = 0,∫∞0

(1 ∧ x) ν(dx) <∞ and characteristic exponent (B.2.1) is a subordinator. For X a subordinator,limt→∞Xt =∞ P–a.s.. (See [Bertoin (1998), Section 3.1] for both statements.)

Examples of subordinators are the Poisson process, compound Poisson processeswith strictly positive jumps, α-stable subordinators, inverse-Gaussian subordina-tors, Gamma subordinators, see [Applebaum (2004b), Section 1.3.2] or [Schoutens(2003), Chapter 5]. An extensive review of subordinators is given by [Bertoin(1997)].

B.3 Compound Poisson processes

A Levy process is of finite activity if ν(R) < ∞, in which case there are finitelymany jumps in any finite interval. Conversely, if ν(R) =∞, then a Levy process isof infinite activity, in which case there infinitely many (small) jumps in any finiteinterval. Informally, one can say that jumps are rare events for a process of finiteactivity, whereas a process is mainly driven by jumps if it is of infinite activity.

A pure jump Levy process of finite activity is a compound Poisson process. Sucha process can also be characterised as a Levy process with piecewise constant paths,cf. [Cont and Tankov (2004a), Proposition 3.3].

Definition B.20. Let λ > 0 and let F be a probability distribution on R withF (0) = 0. A stochastic process X = (Xt)t≥0 on R is a compound Poisson processassociated with λ and F if it is a Levy process and for any t > 0 its characteristicexponent is given by

ψ(u) = λ

∫R

(eiux − 1

)F (dx), u ∈ R. (B.3.1)

The parameter λ is called the (jump) intensity and F is the jump size distribution.

Inspection of Equation (B.2.1) reveals that ν = λF . We also use the followingterminology: Let λ > 0, and let Y be a random variable with distribution F ,F (0) = 0, then for any t ≥ 0, Zt has a compound Poisson distribution withintensity λt and compounding variate Y , and we write Zt ∼ CPO(λt, Y ).

A standard construction for compound Poisson processes is as follows: Let N =(Nt)t≥0 a Poisson process with intensity λ and let Y1, Y2, . . . , i.i.d. random variables,independent of N , with distribution F , F (0) = 0. Define Z = (Zt)t≥0 via

Zt =Nt∑i=1

Yi, t ≥ 0.

The process Z so constructed is a compound Poisson process with jump intensityλ and jump size distribution F ; see [Sato (1999), Theorem I.4.3] for a proof.

The following result generalises the construction above.

150 Levy processes

Lemma B.21. Let Nt be a Poisson process with intensity λ, and let Un be then-th jumping time of N . Let Yn be i.i.d. random variables on R, independentof N , with distribution F , F (0) = 0. Let h(s, y) be a measurable function from(0,∞) × R to R. Define X = (Xt)t≥0 by X0 = 0, Xt =

∑Nt

i=1 h(Ui, Yi), t > 0.Then, for any t > 0, the characteristic function of Xt is

E[eiuXt

]= exp

[λ

∫ t

0

∫R

(eiuh(s,y) − 1

)F (dy) ds

], u ∈ R.

Moreover, Xt ∼ CPO(λt, h(S, Yn)), where S ∼ U(0, t) and independent of Yn.

The first part, i.e., the characteristic function of the construction is from [Sato(1999), Exercise 22.4]. The second part follows easily by writing down the charac-teristic exponent of the CPO(λt, h(S, Yn)) distribution.

B.4 SDEs driven by Levy processes as Markov processes

Theorem B.23 below gives us a means of constructing Markov processes from Levy-driven stochastic differential equations. It is a weaker than [Protter (2005), TheoremV.32], which may be consulted for a proof.

Definition B.22. A function f : R+×Rn → R is said to be Lipschitz if there existsa finite constant k such that

(i) |f(t, x)− f(t, y)| ≤ k|x− y|, for each t ∈ R+, and(ii) t 7→ f(t, x) is cadlag for each x ∈ Rn.

Theorem B.23. Let (Z1, . . . , Zd) be a vector of independent Levy processes (wrt.(Ft)t≥0), Z

j0 = 0, 1 ≤ j ≤ d, and let (f ij), 1 ≤ j ≤ d, 1 ≤ i ≤ n, be Lipschitz

functions. Let X0 ∈ F0 and let X be the solution of

Xit = X0 +

d∑j=1

∫ t

0

f ij(s−, Xs−) dZjs .

Then X is a Markov process (wrt. to (Ft)t≥0).

Appendix C

Integration by parts formula

We derive the integration by parts formula as it is used in Part 2. An integrationby parts formula for two dimensions is given in [Gill et al. (1995)]; a version for Rkis found in [Gill and Johansen (1990), p. 1530].

Let us recall some well-known concepts and facts, see e.g. [von Neumann (1950),Chapter X.5]. Let H : [0, 1]d → R be a right-continuous function. For rectanglesB = (a1, b1]× · · · × (ad, bd] ⊂ [0, 1]d, define

VH(B) :=∑

c vertex of B

sgn(c)H(c),

where

sgn(c) =

1, if ck = ak for an even number of k’s,

−1, if ck = ak for an odd number of k’s.

If in addition VH(B) ≥ 0 for all rectangles B, then H is called quasi-monotone. IfH is quasi-monotone and right-continuous, it determines a σ-additive, nonnegativemeasure, which we also denote by H, via

∫B

dH = VH(B), (C.0.1)

for all rectangles B. If H is of bounded variation and right-continuous, then it is thedifference of two quasi-monotone, right-continuous functions, and hence determinesa σ-additive, signed measure via the relationship (C.0.1).

Proposition C.24. Let H,G : [0, 1]4 → R be of bounded variation and right-continuous, with at least one of H,G continuous and such that

∫∫∫∫H dG exists.

151

152 Integration by parts formula

Then,∫∫∫∫H(u1, u2, u3, u4) dG(u1, u2, u3, u4)

=∫∫∫∫

VG((u1, 1]× · · · × (u4, 1]) dH(u1, u2, u3, u4)

+∫∫∫∫

H(0, u2, u3, u4) +H(u1, 0, u3, u4) +H(u1, u2, 0, u4) +H(u1, u2, u3, 0)

−H(0, 0, u3, u4)−H(0, u2, 0, u4)−H(0, u2, u3, 0)

−H(u1, 0, 0, u4)−H(u1, 0, u3, 0)−H(u1, u2, 0, 0)

+H(0, 0, 0, u4) +H(0, 0, u3, 0) +H(0, u2, 0, 0) +H(u1, 0, 0, 0)

−H(0, 0, 0, 0)

dG(u1, u2, u3, u4).

(C.0.2)

Proof. By Equation (C.0.1),

H(u1, . . . , u4) =∫

(0,u1]×···×(0,u4]

dH(x1, . . . , x4)

+H(0, u2, u3, u4) +H(u1, 0, u3, u4) +H(u1, u2, 0, u4) +H(u1, u2, u3, 0)

−H(0, 0, u3, u4)−H(0, u2, 0, u4)−H(0, u2, u3, 0)

−H(u1, 0, 0, u4)−H(u1, 0, u3, 0)−H(u1, u2, 0, 0)

+H(0, 0, 0, u4) +H(0, 0, u3, 0) +H(0, u2, 0, 0) +H(u1, 0, 0, 0)

−H(0, 0, 0, 0).

Insert this expression into Equation (C.0.2) and apply Fubini’s theorem to the firstterm, for which we then obtain∫∫∫∫

VG([u1, 1]× · · · × [u4, 1]) dH(u1, u2, u3, u4).

The statement follows by observing that from the continuity of one of H and G,∫(0,1]8

4∑i=1

1ui=xi

4∏j=1,j 6=i

1uj≥xjdG(u1, . . . , u4) dH(x1, . . . , x4) = 0.

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List of symbols

1τ≤t, 1τ>t default (resp. survival) indicator∧ minimum, i.e., u ∧ v = min(u, v)a mean reversion of LOU processb default barrier of credit quality processB = (Bt)t≥0 Brownian motion (possibly DDS-Brownian motion)βt money market account, which accrues at the short rateβ(t, T ) money market account accrual between t and T

B(t, T ) default-free zero-coupon bond priceC copula (Part 2, LHSD)(C,B(C)) measurable space of real-valued continuous function with σ-

algebra generated by the (finite-dimensional) cylinder sets ofC

Cn(u1, . . . , ud) d-dimensional empirical copula based on LHSD sample ofsize n

fn∗ n-fold convolution of function f

CPO(λ, Y ) Compound Poisson distribution with intensity λ and com-pounding variate Y

(D,B(D)) measurable space of real-valued cadlag functions with σ-algebra generated by the (finite-dimensional) cylinder setsof D

δ(x; y) Mean square error of a quantity with respect to x subject toparameters y

δ?, δ?s calibration error (RMSE) with respect to default probabili-ties, resp. credit spreads

E expectation operator (Part 1: poss. wrt. risk-neutral mea-sure)

F jump size distribution of compound Poisson process ZFn(u) empirical distribution function based on n samples(Ft)t≥0, (Gt)t≥0 filtrations(FXt )t≥0 filtration generated by process X

159

160 List of symbols

L= equality in distribution(Λt)t≥0 quadratic variation of credit quality process X; time-change

of BΛt= Xt, t ≥ 0

λ jump intensity of compound Poisson process Zλ(t, T ) hazard rate (forward default rate) for time T at time tλ(t, T ) term hazard rateLt,T , LT random part of ΛT − Λt, resp. ΛTM = (Mt)t≥0 (local) martingalen Normal density functionN,N(−1) Normal distribution function and its inverseν Levy measure(Ω,F , (Ft)t≥0,P) filtered probability spaceP probability measure (Part 1: mostly risk-neutral measure)P (t, T ) conditional probability of default until T at t; P (t, T ) =

P(τ ∈ (t, T )|Ft)Q(t, T ) Probability of survival until T at t; Q(t, T ) = P(τ > T |Ft)r = (rt)t≥0 short rate processri,n(X1, . . . , Xn), ri,n i-th rank statistics(t, T ), s (fair) credit spread at t of a CDS with maturity T(σt)t≥0, (σ2

t )t≥0 volatility process, resp. variance processσ2

LHSD Variance of LHSD estimatorσ2

MC Variance of Monte Carlo estimatorτ random default time(τt)t≥0 family of stopping timesθ : R+ → R+ mean level of LOU processU uniform random variable on [0, 1]U(0, 1) uniform distribution on [0, 1](U1, . . . , Ud) d-dimensional random vector with uniform marginalsV = (Vt)t≥0 price process(V ji,n)i=1,...,n,j=1,...d LHSD sample of dimension d and n samplesW = (Wt)t≥0 Brownian motion∆Xt jump of cadlag process X at t, i.e., ∆Xt = Xt −Xt−X = (Xt)t≥0 credit quality processXt−, Xt+ left (resp. right limit) of a process X at t[X,X] quadratic variation of stochastic process XY Y ∼ F ; distribution corresponds to jump size distribution of

compound Poisson process ZZ = (Zt)t≥0 subordinator; often compound Poisson process

Index

additive process, 147Asian basket option, 138

bounded variation (in the sense ofHardy-Krause), 125

Brownian motionDDS, 8

CDS, see credit default swapCDS spread, see credit spreadCentral Limit Theorem for LHSD,

126characteristic exponent, 148compensator, 145complete market, 2compound Poisson distribution, 149compound Poisson process, 149conditional survival (default) proba-

bility, 21consistent model, 29copula, 5, 117, 133

right-tail increasing (RTI), 131credit default swap, 15

premium leg, 16protection leg, 16

credit derivative, 15credit event, 15, 16credit quality process, 2, 34, 35, 48credit spread, 15, 24

term structure, 16

credit spread term structure, 24credit triangle, 32

Dambis, Dubins-Schwarz Theorem, 8DDS-Theorem, see Dambis, Dubins-

Schwarz Theoremdefault intensity, 34default swaption, 104defaultable zero-coupon bond, 21discounted price process, 19dividend process, 20Doleans-Dade exponential, see

stochastic exponential

estimator(asymptotically) unbiased, 123strongly consistent, 123

fair credit spread, see credit spreadfiltration, 19

pricing, 99usual hypotheses, 19

first-passage time model, 33First-to-default credit basket, 135forward CDS spread, 104forward default rate, see hazard rateforward starting CDS, 104forward surival measure, 105

gap option, 2, 28

161

162 INDEX

gap risk, 1, 26

hazard rate, 31term, 32

hedging strategy, 2

ill-posed problem, 80implied volatility, 105Independence Lemma, 146instantaneous volatility, 71integration by parts formula (bivari-

ate), 151inverse transform method, 133Ito formula, 144

Latin hypercube sampling, 119Latin hypercube sampling with de-

pendence, 120LCLN, see leveraged credit-linked

noteleveraged credit-linked note, 1, 25,

102Levy-Khintchine representation, 148Levy measure, 148Levy process, 147

(in)finite activity, 149Levy triplet, 148Levy-driven Ornstein-Uhlenbeck pro-

cess, 51LHS, see Latin hypercube samplingLHSD, see Latin hypercube sampling

with dependenceLipschitz function, 150LOU process, see Levy-driven

Ornstein-Uhlenbeck processlow discrepancy sequence, 137

mark-to-market value, 25Merton model, 32money market account, 19

no-arbitrage price, 2

Ocone martingale, 8OS-model,

see Overbeck-Schmidt model, seeOverbeck-Schmidt model

Overbeck-Schmidt model, 2, 35

Panjer recursion, 76payer swaption, see default swaptionpositive quadrant dependent

copula, 131price process, 20promised cumulative cash flow pro-

cess, 20purely discontinuous (local) martin-

gale, 145

quasi-monotone function, 151

rank statistic, 120receiver swaption, see default swap-

tionrecovery process, 20reduced-form model, 34regular conditional distribution, 143regular conditional distribution func-

tion, 143regular conditional probability, 143right-continuous (bivariate function),

125right-tail increasing (RTI)

copula, 131risk-neutral conditional survival (de-

fault) probability, 21risk-neutral measure, 19risk-neutral valuation formula, 20risky present value of a basis point,

104, 136RMSE, see Root mean square errorRoot mean square error, 82

semimartingale, 22short rate, 19

INDEX 163

Sklar’s Theorem, 133Sobol sequence, 137spread

credit, see credit spreadFirst-to-default credit basket, 135

spread curve, see credit spread termstructure

stochastic exponential, 145stochastic process

cadlag , 7indistinguishable, 22modification, 22

stopping time, 20predictable, 144totally inaccessible, 144

stratified sampling, 118structural model, 32stylised fact, 17subordinator, 148

term hazard rate, see hazard ratetime-change, 7

well-posed problem, 80

yield spread, 17

164 INDEX

Short CV

Natalie Packham was born in Munich in 1972. She graduated with a Master degree(German Diplom) in Computer Science from the University of Bonn in 2000. Inaddition, she received a Master degree in Banking and Finance from FrankfurtSchool of Finance & Management in 2005. Between 1997 and 2001 she held severalpositions as a software engineer in the IT industry. From 2001 to 2005 she wasa senior software engineer and project manager at Dresdner Kleinwort, where shedeveloped part of the inhouse bond trading system. Since 2005 she has been aresearch assistant at the Centre for Practical Quantitative Finance at FrankfurtSchool of Finance & Management.

166 INDEX

Eidesstattliche Versicherung

Ich versichere hiermit an Eides statt, daß ich die vorliegende Arbeit selbstandig undohne Benutzung anderer als der angegebenen Hilfsmittel angefertigt habe. Fur alleInhalte, die wortlich oder sinngemaß aus veroffentlichten oder nicht veroffentlichtenQuellen gleich welcher Art entnommen sind, habe ich die fremde Urheberschaftkenntlich gemacht. Die Arbeit ist in gleicher oder ahnlicher Form weder im Inlandnoch im Ausland als Prufungsarbeit eingereicht worden.

Frankfurt am Main, Dezember 2008

Declaration of authorship

I hereby certify that unless otherwise indicated in the text or references, or acknowl-edged, this thesis is entirely the product of my own scholarly work. Appropriatecredit is given where I have used language, ideas, expressions, or writings of anotherfrom public or non-public sources. This thesis has not been submitted, either inwhole or part, for a degree at this or any other university or institution.

Frankfurt am Main, December 2008

credit dynamics in a ﬁrst-passage time model with jumps

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