cutting edge | credit derivatives market-implied

The credit derivatives market has experienced explosive growth in the past decade. Banks have created and maintained a series of index portfolios, such as CDX North America and iTraxx

Europe, to promote market liquidity. A set of standardised index tranches, with fixed attachments and detachments, is being traded daily by brokers and dealers in the over-the-counter market. At the same time, buyers and sellers of protection have traded tranches of customised portfolios, so-called ‘bespokes’.

Among the first models that produced credit portfolio loss distri-butions were the works of Vasicek (1987) and Li (2000). Both mod-els were based on asset correlations and Gaussian copulas. It soon became evident that both approaches were not particularly suitable in reproducing the prices of traded tranches in a consistent manner.

An important approach is to infer an implied copula. Hull & White (2006) have devised a method to extract an implied copula that can fit tranche prices exactly. Its main feature is that it relies on specifiying a set of hazard-rate scenarios directly. Market tranche prices often imply longer tails than a Gaussian copula is able to gener-ate. In fact, Gaussian copulas exhibit no upper and lower asymptotic tail dependencies (see Embrechts, McNeil & Straumann, 2002).

ObjectivesThe aim of this article is to find a set of Archimedean copulas (ACs) that are associated with the loss distribution of a given index port-folio that spans its life.

Each AC corresponds to a maturity chosen from the set of matu-rities used by the market to price index tranches. The method we are about to describe is, therefore, to be applied to each loss distri-bution that corresponds to a traded maturity. The implied index copulas can be used to generate credit losses for any diversified bespoke whose average spread is close to the index average spread.

An earlier article of ours, Vacca (2005), addressed the question of how to price non-standard tranches of an index based on the price information of a few liquid tranches available in the market. The article demonstrated that it is often possible to find the risk-neutral distribution of portfolio losses if we maximise the entropy

of a distribution function on the underlying portfolio while impos-ing constraints represented by standard tranche market prices. In this article, we show how the entropy-derived loss distribution can be used to extract a risk-neutral copula for every traded maturity.

This article starts with the assumption that we have already found a probability mass function by entropy maximisation at every maturity. First, exchangeable binary distributions (EBDs) are introduced. A numerical algorithm is used to compute the kth order joint default probability l(k) of an EBD from the maximum entropy distribution PT(k) of an index portfolio. Second, we estab-lish a connection between ACs, EBDs and joint-default probabili-ties by calculating the generator of an AC from the ls. Third, armed with this generator of an AC, we conduct a series of numer-ical experiments where we demonstrate how to use a Monte Carlo simulation to price index and bespoke tranches. We conclude this article with comments on limitations and suggestions for possible improvements to this approach.

Exchangeable binary distributionsLet X1, X2, ... be a sequence of exchangeable Bernoulli random vari-ables. For any finite subset Xi1, ... , Xin, we have:

P Xπ1

= x1,...,Xπ n= xn( ) = P Xi1 = x1,...,Xin = xn( )

(1)

where π1, ... , πn is a permutation of i1, ... , in.The variables can either assume the values of Xi = 1 (default event

for obligor i) or of Xi = 0 (no default event for obligor i).From now on, we assume the maximium entropy (ME) default

distribution PT(k) (k = 0, 1, ... , N) originates from an exchangeable binary distribution and N is the number of obligors in the portfolio.

Whenever we work with probabilities of the number of k defaults P(k), as in the case of entropy maximisation, the assumption of ex-changeability for a risky credit portfolio is a natural one from a mod-elling standpoint.

It can be argued that this assumption may not capture the dif-ferent degrees of dependency among obligors in different sectors, industries or geographical regions. However, for large diversified

Cutting edge | Credit derivatives

March 200854

Computations of implied copulas are a central element in producing loss distributions of bespoke portfolios and pricing their tranches. This process is made feasible by the

availability of index tranche pricing data. Luigi Vacca shows how it is possible to calculate the implied Archimedean copulas from index loss distributions. The

advantage of this method is that implied Archimedean copulas can be used to price any bespoke tranche in a consistent and easy way

Market-implied Archimedean copulaswith local volatility models

Cutting edge.indd 70 10/3/08 21:10:22

Cutting edge | Credit derivativeswww.asiarisk.com.hk

55

portfolios, such as many bespoke and index portfolios, exchange-ability is the only realistic and tractable assumption that allows us to study the higher-order interactions that produce long tails.

Furthermore, this approach implies a perfectly homogeneous portfolio that stems from the adoption of loss distributions. A gen-eralisation of the AC that allows for some level of diversification may overcome the issue of perfect homogeneity.

As a final point on this issue, we observe that exchangeability, while a rather restrictive assumption, can generate an incredibly rich variety of loss distributions.l Joint default probabilities and EBDs. In a seminal paper, George & Bowman (1995) introduced joint event probabilities 1 = l0 ≥ l1 ≥ ... ≥ lN as a representation of EBDs. Here, we define a kth order joint-default probability for a portfolio of N names as follows:

λ k = P X1 = 1,X2 = 1,...,Xk = 1( ) (2)

where k = 0, 1, ... , N and l0 ≡ 1.Under this framework, the single-name default probability is

equal for all the obligors in the portfolio p ≡ l1. The correlation is equal to (l2 − l2

1)/[l1(1 − l1)]. The joint default probabilities must verify a set of constraints:

∆N −k λ k( ) ≥ 0 k = 0,...,N

(3)

where the forward difference is defined by D(lk) = lk − lk+1 and D2(lk) = D(D(lk)) and so on. In this article, we will use the funda-mental relation:

P k( ) =

Nk

−1( ) jj=0

N −k

∑N − kj

λ k+ j , k = 0,...,N

(4)

We can rewrite this equation in matrix form:

P = AΛ (5)

where A is a (N + 1, N + 1) triangular matrix. Theoretically speaking, one should be able to calculate the ls by inverting A or, more simply, by iterative substitution given the triangularity of A.

But these operations would fail systematically when N > 102, where 102 is the order of magnitude of a typical portfolio. This size leads to the problem of accurately calculating (N

k) for a large variety of ks.

To calculate the ls, we describe a method that may solve the curse of the dimensionality problem. A binomial distribution does satisfy equation (5) for any arbitrary default probability p:

P k( ) = B N , k; p( ) ≡

Nk

pk 1− p( )N −k , k = 0,...,N

(6)

and the corresponding joint-default probabilities are also easily de-rived:

λ k( ) = pk , k = 0,...,N (7)

If we can reasonably approximate the ME P(k) by using a mixture of binomial distribution functions, then the corresponding ls are also a mixture of independent joint-default probabilities with the same mixing distribution. Therefore, assuming that:

PME k( ) = B N , k; p( )dG p( ), k = 0,...,N

0

1∫

(8)

Then, the solution is given by the following equation:

λME k( ) = pkdG p( ), k = 0,...,N

0

1∫

(9)

Equation (9) is equivalent to the famous moment theorem by Haus-dorff. The theorem states that the completely monotone sequence of ls can be expressed as the moments of a measure. See Feller (1971) for more details on this important theorem.l Numerical solution for the mixing distribution. To solve the mixing distribution in equation (8), we discretise the integral by selecting probabilities that are uniformly distributed on the domain [0, 1], as in Wood (1999).

The unknown mixing distribution is approximated by a discrete distribution Π ≡ Tr(π0, π1/L, ... , π1), where πi/L is the weight for the probability i/L for i = 0, ... , L.

The discretised version of equation (8) becomes an undetermined system of linear equations:

PME ,1( ) = B,1( )Π

(10)

where:

PME ,1( ) ≡PME 0( )

M

PME N( )1

and:

B,1( ) ≡

B N , 0;0( ) B N , 0;1 / L( ) L B N , 0;1( )M M M M

B N ,N ;0( ) B N ,N ;1 / L( ) L B N ,N ;1( )1 1 1 1

where L > N + 1.Since the solutions are also probabilities, we impose positive con-

straints on the solution: πi/L ≥ 0 for i = 0, ... , L.To solve this under-determined system, we used a sophisticated

least-squares algorithm developed by Lawson & Hanson (1995), also called non-negative least-squares (NNLS). Explaining NNLS is outside the scope of this article; we refer the reader to the origi-nal source to learn about this powerful tool. Once the discrete mixing distribution Π is obtained, joint-default probabilities are easily calculated by solving the discretised equation (9) directly, where the solution is reduced to a weighted sum of powers of uni-formly distributed probabilities:

λME k( ) = π i /L i / L( )k

i=0

L

∑

(11)

To achieve a high degree of quality, we recommend a number of columns L ≥ 105 >> N ≈ 102, which is much larger than the typical number of unknown joint-default probabilities.

EBDs and ACsWe now address the existence of a theoretical connection between EBDs and ACs. We refer to two important theorems on this subject. The first one is attributed to Frey & McNeil (2001). They essen-tially state that if an n-dimensional random vector Y is distributed



March 200856

as an AC, the indicators 1(Y ≤ y) are distributed like an EBD.The second theorem was proved by Muller & Scarsini (2005).

They showed that, for every binary infinite exchangeable sequence Y, there exists a generator whose corresponding AC is a copula for Y. From now on, we make the formal assumption that our finite portfolio is a sample that was extracted from an ideally infinite portfolio whose obligor default events are distributed as an exchangeable binary process.

Multivariate ACsMultivariate ACs can be easily constructed and simulated. The func-tional form of an AC with n dimensions is:

Cn u( ) = φ−1 φ u1( ) + φ u2( ) + ...+ φ un( )( )

(12)

where u is the n-dimensional uniform random vector.In financial literature, the function φ is often called the genera-

tor. A generic AC can include many popular parametric distribu-tions, such as the families of Frank, Clayton and Gumbel copulas.

However, a parametrised AC family may not offer enough flexi-bility to faithfully replicate the dependencies implied by market-loss distributions because it relies on too few parameters. On the basis of this consideration, we propose that we work with genera-tors and their respective inverses instead of a copula with few parameters. Indeed, to know the functional form of a specific gen-erator is tantamount to finding its corresponding AC as well.l Properties of ACs and generators. A function φ (x) : [0, 1] → [0, ∞] is called a generator of a two-dimensional AC if and only if it is strictly convex, decreasing and φ (1) = 0. If φ (0) = ∞, the generator is called strict. Strict generators can be inverted according to the standard definition of the inverse of a function.

Most importantly, for φ to be a generator of an AC in arbitrary dimension n, φ −1 must be a completely monotonic (CM) function. This additional fundamental condition is attributed to Kimberling (1974).

A function is said to be completely monotonic if its derivatives exist at all orders and if formula (13) applies for all x > 0:

−1( )k d k

dxkφ−1 x( ) ≥ 0, k = 0,1,2,...

(13)

Strict generators produce ACs that exhibit only positive depend-ence. While this is an obvious limitation, it does not constitute a significant problem, because default event dependencies are, on average, positive. See Nelsen (1999) and Joe (1997) for further details on ACs and on copulas in general. To construct a generic CM function for numerical investigation, we make use of a theo-rem (see Feller, 1971) that states the Laplace transform of a generic positive random variable with cumulative distribution function F is a CM function:

φ−1 x( ) = e− xydF y( ), s ≥ 0

0

∞∫

(14)

The mixing distribution F can be either continuous or discrete. In this article, we use the hyperexponential distribution to construct CM functions as follows:

φ−1 x( ) = pie

−ai x

i=1

M

∑

(15)

where the coefficients are discrete probabilities:

pi = 1, pi ≥ 0

i=1

M

∑

(16)

The hyperexponential distribution is often derived as a Laplace transform of a discrete distribution:

dF y( ) = piδ y − ai( )dy, ai ≥ 0

i=1

M

∑

(17)

where d(x) is the Dirac delta function.Identifying an AC whose inverse generator is a hyperexponential

distribution involves finding the value of 2M–1 parameters. The number M of exponentials is also arbitrary and needs to be deter-mined.

Identifying the inverse generator from joint-default probabilities of an EBDThe final step is to find the generator or, equivalently, its inverse from a finite series of joint-default probabilities l(k) for k = 0, 1, ..., N, where N is the number of obligors in the original portfolio. Us-ing the definition of AC, its exchangeability and the Muller-Scarsini theorem, we can write:

λ k( ) = φ−1 kφ p( )( ) for 1 ≤ k ≤ N

(18)

where p is a default probability that is identical for every single ob-ligor.

We now apply the operator φ to both sides of equation (18):

φ λ k( )( ) = kφ p( ) for 1 ≤ k ≤ N

(19)

By definition, p ≡ l1. In addition, we use the multiplicative property of CM functions to arbitrarily set:

φ p( ) = 1 (20)

Finally, inserting equation (20) into equation (19) and applying the inverse generator to both sides of the equation, we obtain:

φ−1 k( ) = λ k( ) for 1 ≤ k ≤ N

(21)

This is the fundamental result that links an AC to an exchangeable binary distribution from a numerical standpoint. The ultimate goal of this article is therefore to find a CM function that, sampled at inte-gral values k, is equal to a given joint-default probability l(k). Here, we adopt the hyperexponential distribution with M exponentials. Upon substitution of equation (15) into equation (21), we obtain:

pie

−aik

i=1

M

∑ = λ k( ) for 1 ≤ k ≤ N

(22)

The case k = 0 in equation (22) is trivially satisfied due to relation (16).

To solve equation (22) is equivalent to finding the coefficients pis and the frequencies ais in the sum of exponentials that satisfy equa-tion (22). The exponents ais must be positive because we require the exponentials to be damped exponentials. Luckily, the tools needed to find the solution of equation (22) are well known in numerical analysis. We solve equation (22) using a modified and improved Prony method from Osborne & Smyth (1995). Their



57

article also contains a long list of references on Prony’s method and its variations. A final observation is that the number of exponentials M can be found by trying different Ms, and by identifying the value of M that yields the smallest numerical fit error.

Generating loss distributions from Archimedean copulasOnce the inverse of a generator φ−1 is found, the corresponding AC is also known. Several methods can be applied to ACs to simulate loss distributions. For instance, Schonbucher (2002) derived a semi-analytic formula for loss distributions with ACs assuming a large and homogeneous portfolio, and Whelan (2004) discovered an efficient method to sample random variates from an AC.

We will use a basic Monte Carlo method in this article. There is a simple algorithm attributed to Marshall & Olkin (1988) that can produce a random vector distributed like an assigned AC. It is implemented as follows:l Generate N independent and uniformly distributed random numbers Ui on [0, 1], where N is the number of obligors in the portfolio.l Draw a random value aj with probability pj from the discrete mass distribution in formula (17).l Calculate Xi = φ−1(−lnUi/aj), where 1 ≥ i ≥ N and aj is the same for all Uis. Notice that, while we used the assumption of a homo-geneous portfolio to find the generator, we can use different sin-gle-obligor default probabilities in applications.l Compare every Xi with its corresponding obligor i with uncon-ditional default probability pi.If Xi ≤ pi, then obligor i defaults.l Repeat all steps above until a determined large number of sam-ples is reached.

A numerical exampleOur numerical example is based on a randomly selected sample of standard tranche prices on the CDX.IG8 index. The standard maturities available for this index are the three-year, five-year,

seven-year and 10-year maturities. The three-year index is not very liquid, so we used only the index tranche price to produce the three-year portfolio loss distribution. The price for a 0–3% equity tranche always refers to its upfront value, as market convention fixes the running premium at 5%. We also added extra 30–100% tranches that we artificially priced at a 0% spread. This addition avoids a flat loss distribution in the 30–100% section of the capital structure due to an absence of pricing information in that area. The data is summarised in table A.

Using the ME method and the tranche-pricing equation, we cal-culated the loss distributions for every listed maturity. The ME loss distributions for each maturity are shown in figure 1. We calculated the prices of standard tranches using the ME distributions. It can be easily verified, by comparing table A and table B, that the ME prices and the original tranche prices are virtually identical.

The next step is to calculate the joint-default probabilities ls from the ME loss distributions using NNLS. The results are shown in figure 2. Notice that, except for the three-year case, the ls are closely positioned. It is important to keep in mind that the three-year maturity case is not as reliable as the other maturities due to limited pricing information. We proceeded to extract the inverse-generator functions from the joint-default probabilities for each

2. ME produced loss distributions

0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0 20 40 60 80 100 120Number of defaults

Pro

bab

ility

Three-yearFive-yearSeven-year10-year

1. AC produced loss distributions

0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0 20 40 60 80 100 120Number of defaults

Pro

bab

ility


Attachment Detachment 3y 5y 7y 10y 0 3 22.7500 39.3750 50.8750

3 7 0.9075 2.1425 4.7800

7 10 0.2025 0.4625 1.1875

10 15 0.0863 0.2213 0.5475

15 30 0.0338 0.0850 0.1675

30 100 0.0001 0.0001 0.0001

0 100 0.2040

A. CDX.IG8 standard tranche prices (%)

Attachment Detachment 3y 5y 7y 10y 0 3 7.35000 22.75000 39.37500 50.87500

3 7 0.14753 0.90750 2.14250 4.78000

7 10 0.00166 0.20250 0.46250 1.18750

10 15 0.00003 0.08630 0.22130 0.54750

15 30 0.00000 0.03375 0.08500 0.16750

B. Prices generated by ME (%)



March 200858

maturity using the modified Prony method. The Prony algorithm looks for more than 100 separate exponential functions, although only very few are needed to obtain a good fit.

The generator inverses are plotted in figure 3. We also included the plot of the independent inverse generator φ−1 = exp(−x). The inverse loss generators decay much faster than the independent inverse generator, suggesting that the index portfolio exhibits sig-nificant default dependence over time. To get a better sense of the different degrees of dependence for the four copulas, we used the Kendall tau coefficient or, more simply, t. t behaves very much like the linear correlation coefficient, as its value lies between minus one and one, where an increasing value implies a higher level of concordance between random variables. However, t is a much more meaningful measure of association (concordance) than the stand-ard correlation coefficient when working with copulas.

Joe (1997) gives a very useful expression for t for an AC:

τ = 1− 4 dφ−1 u( ) / du

2

0

∞⌠⌡ udu

(22)

The results of such a calculation applied to our four inverses are illustrated in table C. The seven-year AC has the highest level of concordance (0.23), followed by the 10-year, three-year and five-year copulas.

We easily generated loss distributions using the Monte Carlo ver-sion of the AC with 250,000 runs for each maturity, where the marginal distributions were calculated by the premium informa-tion of each obligor in the portfolio. The four distributions are shown in figure 4. Using these loss distributions and the tranche pricing equation, we were able to calculate the prices of selected standard tranches, which are listed in table D.

If we compare the original standard tranche prices in table A with their corresponding prices in table D, we observe that the prices are rather close in value, where the error is often smaller than the typical bid-ask spread for these tranches.

To price a bespoke tranche with a standard maturity using an implied AC, one would have to follow the same steps we followed to calculate the tranche prices of an index, except that the single-obligor default probabilities and possibly the average recovery

3. Inverse of generators

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

y

x

Three-yearFive-yearSeven-year10-yearexp(–x)

4. Log-linear joint default probabilities

–120

–100

–80

–60

–40

–20

00 20 40 60 80 100 120

Order

Log

of j

oin

t d

efau

lt p

rob

abili

ty


Embrechts P, A McNeil and D Straumann, 2002Correlation and dependence in risk management: properties and pitfallsIn Risk Management: Value at Risk and Beyond, Cambridge University Press

Feller W, 1971An introduction to probability theory and its applications, vol IIWiley

Frey R and A McNeil, 2001Modelling dependent defaultsWorking paper, Department of Mathematics, ETH Zurich

George O and D Bowman, 1995A full likelihood procedure for analyzing exchangeable binary dataBiometrics 34, pages 69–76

Hull J and A White, 2006Valuing credit derivatives using an implied copula approachJournal of Derivatives 14(2), pages 8–28

Joe H, 1997Multivariate models and dependence conceptsChapman and Hall, London

Kimberling C, 1974A probabilistic interpretation of complete monotonicityAequationes Mathematicae 10(2–3)

Lawson C and R Hanson, 1995Solving least squares problemsPrentice Hall

Li D, 2000On default correlation: a copula function approachWorking paper 99-07, RiskMetrics Group

Marshall A and I Olkin, 1988Families of multivariate distributionsJournal of the American Statistical Association 83, pages 834–841

Muller A and M Scarsini, 2005Archimedean copulae and positive dependenceJournal of Multivariate Analysis 93(2), pages 434–445

Nelsen R, 1999An introduction to copulasLecture Notes in Statistics, Springer, Berlin

Osborne M and G Smyth, 1995A modified Prony algorithm for exponential function fittingSIAM Journal of Scientific Computing 16, pages 119–138

Schönbucher P, 2002Taken to the limit: simple and not-so-simple loan loss distributionsAvailable at www.gloriamundi.org

Vasicek O, 1987Probability of loss on loan portfolioWorking paper, KMV Corporation

Vacca L, 2005Unbiased risk-neutral loss distributionsRisk November, pages 97–101

Whelan N, 2004Sampling from Archimedean copulasQuantitative Finance 4, pages 339–352

Wood G, 1999Binomial mixtures: geometric estimation of the mixing distributionAnnals of Statistics 27(5), pages 1,706–1,721

REFERENCES


59


would have to be modified due to the different composition of a bespoke portfolio.

ConclusionThis article presents a new approach to deriving implied ACs from tranche prices of marketed indexes that complements the tech-niques developed in an earlier publication. The implied copulas can be used to produce loss distributions of bespoke portfolios and, consequently, price their tranches.

To relax the assumption of exchangeability is desirable because portfolio-default dependencies may imply hidden structures that are waiting to be uncovered. ACs are clearly not the right copulas if one wants to capture different levels of dependence within a portfolio. Hence, a more sophisticated approach to the extraction of an implied copula is predicated on the existence of a much more structured copula whose random variates can be easily simulated. There is a scarcity of flexible multivariate copulas that can be used to fit realistic loss distributions. Therefore, we expect any improve-ment on this front to be accompanied by a parallel breakthrough in the theory of copulas.

Another much needed improvement is a method such that the ME probability mass distribution incorporates the information contained in the default probabilities of the portfolio obligors. This advancement would certainly translate into a more precise identifi-cation of implied copulas and more accurate tranche pricing. l

Luigi Vacca is head of quantitative analytics at Radian Asset As-

surance in New York. He would like to thank the referees for their

useful feedback. The views and opinions expressed herein are

solely those of the author and do not reflect the views, opinions

or policies of Radian Group or its affiliates. Neither Radian Group

nor any of its affiliates assumes any legal liability or responsibility

for the accuracy, completeness or functionality of any information

disclosed or described herein. Email: [email protected]

Attachment Detachment 3y 5y 7y 10y 0 3 7.34000 22.42000 38.96200 50.22400

3 7 0.14715 0.90749 2.14153 4.73475

7 10 0.00174 0.20109 0.45248 1.15684

10 15 0.00000 0.08173 0.21423 0.49886

15 30 0.00000 0.03380 0.08500 0.16750

D. Prices generated by implied Archimedean copulas (%)

3y 5y 7y 10y 0.0962 0.0955 0.2328 0.1345

C. Taus for the four Archimedean copulas

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