monte-carlo pricing and sensitivities of auto-callable and ... · pdf filemonte-carlo pricing...

Monte-Carlo Pricing and Sensitivitiesof Auto-Callable and Bermudan-Callable Products

Introduction, Review and some New Results

Christian P. Fries

Version 2.5

http://www.christian-fries.de/finmath

Risk Europe · Frankfurt · June 2009

1 / 105

http://www.christian-fries.de/finmath

Outline

IntroductionExample: Linear and Discontinuous Payout

Proxy Simulation Schemes: A Review

Conditional Analytic Monte-Carlo PricingDefinition of Generalized Trigger ProductDefinition of a Modified Conditional Analytic Pricing AlgorithmNumerical Results

References

Bonus/Backup: Stable Monte-Carlo Sensitivities for Bermudan CallablesBermudan Pricing Backward AlgorithmLocally Smoothed Backward AlgorithmNumerical Results

2 / 105

INTRODUCTION

Outline




References


4 / 105

Monte-Carlo MethodPricing

Monte-Carlo Pricing:Let

Y (ω) :=(X (T1,ω), . . .X (Tm,ω)

):=

Fixings of the underlying X onpath ω

and

f (Y (ω)) :=

Discounted payoff function of aderivative product on path ω.

Risk Neutral Pricing:The evaluation of the payoff f can be expressed as an expectation:

V (t0) = EQ (f (Y ) | Ft0) ≈m

∑j=1

f (Y (ωj)) · 1m︸︷︷︸

= p(ωi)

Monte-Carlo approximation:The expectation is approximated by a finite sum of weighted pathwisepayoffs f (Y (ωj)).

5 / 105

Monte-Carlo MethodSensitivities

Sensitivities: Let θ denote a parameter of the model SDE for X(e.g. its initial condition X (0), volatility σ or any other complex functionof those). Denote the dependence of the model realizations on θ by Yθ .We are interested in

∂V∂θ

=∂

∂θEQ(f (Yθ ) | FT0) =

∂

∂θ

∫Ω

f (X (T1,ω,θ), . . .X (Tm,ω,θ)) dQ(ω)

=∂

∂θ

∫IRm f (x1, . . .xm)︸︷︷︸

payoffcan be discontinuous

· φ(X(T1,ω,θ),...X(Tm,ω,θ))(x1, . . .xm)︸︷︷︸density - in general smooth in θ

d(x1, . . .xm)

Problem: Monte-Carlo approximation inherits regularity of f not of φ :

EQ(Yθ | FT0) ≈ EQ(Yθ | FT0) :=1n

n

∑i=1

f (X (T1,ωi ,θ), . . .X (Tm,ωi ,θ))︸︷︷︸payoff on path

can be discontinuous

6 / 105

INTRODUCTIONEXAMPLE: LINEAR AND DISCONTINUOUS PAYOUT

Monte-Carlo MethodSensitivities: Example: Linear and Discontinuous PayoutLinear Payout: Valuation

Payoff

UnderlyingLinear payout evaluated on three Monte-Carlo paths (red).

8 / 105

Monte-Carlo MethodSensitivities: Example: Linear and Discontinuous PayoutLinear Payout: Sensitivities: Pathwise Method

Payoff

UnderlyingLinear payout evaluated on three Monte-Carlo paths (red). If the initial data (i.e., spot) isshifted (green), and the sensitivity (slope) is calculated by finite differences, eachMonte-Carlo paths gives the exact slope.The average (the delta) is exact (zero Monte-Carlo error).

9 / 105

Monte-Carlo MethodSensitivities: Example: Linear and Discontinuous PayoutDiscontinuous Payout: Sensitivities: Pathwise Method

Payoff

UnderlyingDiscontinuous payout evaluated on there Monte-Carlo paths (red). If the initial data (i.e.,spot) is shifted (green), and the sensitivity (slope) is calculated by finite differences,almost all Monte-Carlo paths give slope zero...

10 / 105


Payoff

Underlying...in a rare case, if the path comes close to the discontinuity, we get a very large (herenegative) slope. It is easy to show that the average will converges to the true delta. Theaverage (the delta) has a very large Monte-Carlo error.This is essentially a binomial distribution with a large value occurring rarely.

11 / 105


Payoff

UnderlyingChanging model parameters (e.g., spot) in a Monte-Carlo Simulation will result indifferent realizations on each paths (before: red, after: green).But...

12 / 105


Prob

abiliy

Den

sity

Payo

ff

UnderlyingChanging model parameters (e.g., spot) in a Monte-Carlo Simulation will result indifferent realizations on each paths (before: red, after: green).But the random numbers used to generate the individual path are the same. The modelpath is generated from the same path of the driving Brownian motion.

13 / 105

Monte-Carlo MethodSensitivities: Example: Linear and Discontinuous PayoutDiscontinuous Payout: Sensitivities: Likelihood Ratio Method

Prob

abiliy

Den

sity

Payo

ff

UnderlyingIf we consider the two simulations generating the same values, then these values havedifferent probability density. Using the same values and applying the weightsrepresenting the change in probability (likelihood ratio) also converges to the delta.

For discontinuous payouts, LR method has much smaller Monte-Carlo error thanpathwise method.

14 / 105


Prob

abiliy

Den

sity

Payo

ff

UnderlyingConsider the likelihood ratio method for a smooth, e.g., constant payout. At everysample path the density changes. Some LR weights are positive, some negative.Analytically their integral is zero...

15 / 105


Prob

abiliy

Den

sity

Payo

ff

Underlying

12

3

1

2 3

Consider the likelihood ratio method for a smooth, e.g., constant payout. At everysample path the density changes. Some LR weights are positive, some negative.Analytically their integral is zero. Numerically we see a Monte-Carlo error.

For smooth payouts, pathwise method has much smaller Monte-Carlo error thanlikelihood ratio method.

16 / 105

PROXY SIMULATION SCHEMES: A REVIEW

Outline




References


18 / 105

Proxy Simulation SchemesReview

Full Proxy Simulation Schemes (with Joerg Kampen)I A design pattern for a Monte-Carlo simulation using two

models/schemes:I A scheme to generate the paths (not necessarily from the true

model).I A scheme of the true model. Calculate the corrections of transition

probabilities (likelihood ratios) from the numerical scheme!

Advantage:I Results in likelihood ratio sensitivities while implementation remains

model and product independent. You only need the transitionprobability of the numerical scheme.

Disadvantage of Likelihood Ratio:I Good when payoff is discontinuous. Can be much worse than direct

simulation for smooth payoff.19 / 105


Partial Proxy Simulation Schemes (with Mark Joshi)I A design pattern for a Monte-Carlo simulation specially suited when

calculating sensitivities, consisting of:I A scheme of the true model.I A reference scheme, here, the scheme of the true model for fixed

market data.I A function, the proxy constraint, whose value under the true model

should always agree with the reference scheme.I The simulated paths are calculated from the scheme for the true

model + a correction to ensure the proxy contrain, Monte-Carloprobabilities are adjusted accordingly.

Advantage of Partial Proxy:

I Results in likelihood ratio sensitivities only on the quantity definedby the proxy constraint.

20 / 105


Partial Proxy Simulation Schemes: Example:I Consider a product swapping a digital-CMS-index with LIBOR.

I CMS-related payoff is discontinuous⇒ Use Likelihood-Ratio/ProxyScheme when calculating sensitivities (do not use pathwisedifferentiation).

I LIBOR-related payoff is smooth⇒ Use pathwise method whencalculating sensitivities (do not use Likelihood-Ratio/Proxy Scheme).

I Solution: Use partial proxy scheme and define the CMS rate as theproxy contrain.

21 / 105


Localized Partial Proxy Simulation SchemesI Same as partial proxy simulation scheme, but in addition:

I Proxy constrain is localized in time and state-space (e.g., by thedistance to the strike/trigger level).

I Result is the use of likelihood ratio sensitivities only when somestate (proxy constrain) is close to some value.

Example:

I Consider a product with a digital-index (or some other trigger).I Solution: Localize constrain around strike (discontinuities) (or

trigger level).

22 / 105


Numerical ResultsDelta of CMS TARN Swap (5000 paths)

0,0 5,0 10,0 15,0 20,0 25,0

shift in basis points

-6,00%

-5,00%

-4,00%

-3,00%

-2,00%

de

lta

Gamma of CMS TARN Swap (5000 paths)

0,0 5,0 10,0 15,0 20,0 25,0


-50 ,00

-25,00

0,00

25,00

50,00

ga

mm

a

Delta and Gamma of a target redemption note (the coupon is a reverse CMS rate)calculated by finite difference applied to direct simulation (red), to a partial proxyscheme simulation (yellow) and to a localized proxy simulation scheme (green). Directsimulation produces enormous Monte-Carlo variances for small shift sizes. The methodis useless. The partial proxy simulation scheme shows an increase in Monte-Carlovariance if the shift size is large. The localized proxy simulation scheme is animprovement on the partial proxy simulation scheme and shows only small Monte-Carlovariance for large shifts. Note: The localizer used is not the optimal one. 23 / 105


Further Reading: For details on the proxy simulation scheme see:I Quantitative Methods in Finance 2005, Sydney:

Proxy Simulation Scheme.

I Risk Quant Congress Europe 2007, London:Partial and Localized Proxy Scheme.

I Global Derivatives 2008, Paris:Partial and Localized Proxy Scheme.

I And the references [FriesKampen2005] (with Joerg Kampen),[FriesJoshi2006] (with Mark Joshi), [FriesLocalizedProxy2007]

Other works:I Kampen, J.; Kolodko, A.; Schoenmakers, J.: Monte Carlo Greeks for

financial products via approximative transition densities.Siam J. Sc. Comp., vol. 31 , p. 1-22, 2008.

I Kienitz, Joerg: A Note on Monte Carlo Greeks for Jump Diffusion andOther Levy Processes. SSRN, 2008.

24 / 105

CONDITIONAL ANALYTIC MONTE-CARLO PRICING

Outline




References


26 / 105

CONDITIONAL ANALYTIC MONTE-CARLO PRICINGDEFINITION OF GENERALIZED TRIGGER PRODUCT

Generalized Trigger ProductDefinition

Definition: Given a tenor structure T1 < T2 < · · ·Tn+1 we consider a(generalized) trigger product paying

X (Tj+1) =

Cj if Ij < Hj and ∀k < j : Ik < Hk ,Rj if Ij ≥ Hj and ∀k < j : Ik < Hk ,0 else

in Tj+1 for j = 1,2, . . . ,n. Where:I Ij is the trigger index with fixing in Tj , i.e. it is FTj -measurable.I Hj is the trigger level, an FTj−1-measurable random variable.I Cj is the coupon, FTj+1-measurable and paid in Tj+1

I Rj is the redemption (including last coupon), FTj+1-measurable andpaid in Tj+1

28 / 105


Definition: We can rewrite the payoff. Let

Aj := Ij < Hj and ∀ k < j : Ik < Hk survivalBj := Ij ≥ Hj and ∀ k < j : Ik < Hk trigger hit

denote the survival and the trigger hit regime, respectively, then thepayout can be written as

X (Tj+1) = Cj 1Aj + Rj 1Bj .

29 / 105


Assumption: Conditional Analyticity of the Redemption Payment.We assume that conditional to FTi−1 we have an analytic pricing formula(or approximation) for the next period’s redemption payment, i.e., weanalytically have

Rj(Tj−1) := N(Tj−1)EQ(

Rj

N(Tj+1)1Bj |FTj−1

).

This allows us to equivalently reformulate the payoff in the followingsense:

30 / 105

Generalized Trigger ProductProperties

Lemma: Define

X (Tj+1) =Rj(Tj−1)

P(Ti+1;Ti−1)+

Cj if Ij < Hj and ∀k < j : Ik < Hk ,0 otherwise,

then at Tk ≤ Tj−1, the risk-neutral value of the payoffs X (Tj+1) andX (Tj+1) agree, i.e.,

EQ(

X (Tj+1)

N(Tj+1)| FTj−1

)= EQ

(X (Tj+1)

N(Tj+1)| FTj−1

).

Note: Modified product pays 0 and terminates when trigger is hit.

31 / 105

Generalized Trigger ProductProperties

Proof: Let Aj and Bj as above. Then

X (Tj+1) = Cj 1Aj + Rj 1Bj .

Let Q denote the pricing measure corresponding to the numéraire N.

EQ(

X (Tj+1)

N(Tj+1)| FTj−1

)= EQ

(Cj

N(Tj+1)1Aj +

Rj

N(Tj+1)1Bj | FTj−1

)= EQ

(Cj

N(Tj+1)1Aj | FTj−1

)+ EQ

(Rj

N(Tj+1)1Bj | FTj−1

)= EQ

(Cj

N(Tj+1)1Aj | FTj−1

)+

Rj(Tj−1)

N(Tj−1)= EQ

(X (Tj+1)

N(Tj+1)| FTj−1

).

32 / 105

Generalized Trigger ProductExample

Example: Target Redemption Note.For a target redemption note the trigger criteria is

j

∑k=1

Ck ≥ C∗,

where C∗ is the target coupon. The redemption usually consists of anotional payment (assumed to be 1) and a coupon filling the gap for thetarget coupon. Within the notation above, the target redemption notehas

Ij = Cj , Hj = C∗−j−1

∑k=1

Ck , Rj = 1 + Hj .

For the case where the redemption is paid at Tj+1 then Rj(Tj−1) is thevalue of a digital option with the underlying index Ij (fixing in Tj , paymentin Tj+1).

33 / 105

CONDITIONAL ANALYTIC MONTE-CARLO PRICINGDEFINITION OF A MODIFIED CONDITIONAL ANALYTIC PRICING

ALGORITHM

Conditional Analytic PricingDefinition

Pricing AlgorithmThe equivalent reformulation of the payout allows us to develop anadapted / improved pricing algorithm:

Main idea:I Generate a (Monte-Carlo) simulation restricted to the domain ∪iAi .

This allows the numerical evaluation of the complex coupon partCi1Ai , as usual.

I The conditional analytic part Ri1Bi will be treated in every time stepusing the conditional analytic formula Ri .

With this reformulation, the Monte-Carlo simulation will not suffer fromthe Monte-Carlo error induced by the discontinuity at the border of ∪iAi .If Ci is smooth, then the Monte-Carlo simulation will effectively beapplied to a smooth product. The discontinuous part is handledanalytically. The result is a sizeable reduction of Monte-Carlo variancefor price and particularly sensitivities.

35 / 105

Conditional Analytic PricingDefinition of the Numerical Scheme

Modification for an Monte-Carlo Euler Scheme: For illustrativepurposes we consider a model given by an Itô stochastic process:

dK = µ(t) dt + Σ(t) ·Γ(t) ·dW (t),

where W = (W1, . . . ,Wm) and Wi are Brownian motions with

dWi dWj = δi ,j dt ,

and Σ and Γ denote the volatility and the factor matrix, respectively,determining the instantaneous covariance of the model.

36 / 105


Original Numerical SchemeLet K (ti) be an approximation of K (ti) given by a numerical scheme,e.g., an Euler-like discretization of our model given by

∆K (ti) = µ(ti)∆ti + Σ(ti) ·Γ(ti) ·∆W (ti), K (0) = K (0).

Let ∆Wk (ti) be generated by drawings from independent equidistributedrandom variables Zi ,k using

∆Wk (ti) = Φ−1(Zi ,k )√

∆ti ,

where Φ−1 denotes the inverse of the cumulative standard normaldistribution function.

37 / 105


Numerical Scheme adapted to the Trigger ProductIdea:

I generate only those paths that do not hit the triggerI calculate the corresponding probability measureI and semi-analytically calculate the value given by a trigger hit.

To do so, we define the gradient of the trigger criteria (i.e. I−H) andcalculate the location of the trigger.

38 / 105


Assumption on the TriggerWe assume that the trigger index Ij of the trigger product is a function ofthe model’s state variables K (Tj), i.e.,

Ij = f (Tj ,K (Tj)).

In other words, we assume that the trigger index Ij itself is notpath-dependent in terms of the model primitives.

However, since we allow that the trigger level Hj is an FTj−1-measurablerandom variable, most products with path-dependent triggers can berewritten in the above form, e.g., as for the target redemption note in theprevious example.

39 / 105


Numerical Scheme adapted to the Trigger ProductInduction Start: Let K ∗(t0) := K (t0).Induction Step: Given K ∗(ti) let

g(x) = f (K ∗(ti) + µ(ti ,K ∗)∆ti + x) .

Distance to trigger as function ofdiffusion term (random vector) x .

Definev = ∇g(0)/‖∇g(0)‖

and let q be the solution of the linearization of

g(qv) = Hi+1,

i.e.,

g(qv) = g(0) + ∇g(0) ·qv , i.e., define q :=g(0)−Hi+1

‖∇g(0)‖. (1)

Then (to first order, i.e., if g is linear)

Ii+1 < Hi+1 ⇔ g(ΣΓ∆W ) < Hi+1 ⇔ ΣΓ∆W < qv ⇔ 〈v ,ΣΓ∆W 〉< q

40 / 105


Induction Step (continued):Let

X := 〈v ,Γ∆W 〉.

Scalar, normal distributedrandom number.

We wish to replace the sampling of X with a sampling Y such thatY < q. Clearly, X is a normal distributed random variable with mean 0.Let σX denote the standard deviation of X . Then x = Φ(X/σX ) isuniform distributed. Let b := Φ(q) and Y := Φ−1(bx). Then we have thatbx < b, thus Y < q. Furthermore,

P(X < K ) = bP(Y < K )

for all K < q. i.e., the distribution function of Y and X differ on (−∞,q)only by the constant factor b.In other words: sampling Y is equivalent to sampling X on the restricteddomain (−∞,q), with a Monte-Carlo weight b. For Γ∆W + (Y −X )v wehave

〈v ,Γ∆W + (Y −X )v〉 = X + Y −X = Y ≤ q41 / 105


Induction Step (continued):In place of K we consider the numerical scheme K ∗ defined by

K ∗(ti+1) := K ∗(ti) + µ(ti)∆ti + Σ ·Γ(∆W +(Y −X )v︸︷︷︸

adjustment

)

Interpretation:I We do not draw a new random number / brownian increment, we

calculate an adjustment such that the trigger is not hit.I Idea bears some similarity to a partial proxy simulation scheme.

New: The (proxy) constrain is an inequality!

42 / 105


Properties: This scheme has the property that (to first order)

f (K ∗(ti+1)) = f (K ∗(ti) + µ(ti)∆ti) + ∇g ·(

∆W + (Φ−1(qZ )−Φ−1(Z ))v1

)= g(0) + ∇g ·qv ≤ Hi+1

Thus, for linear triggers we have that this scheme generates realizationsthat sample the non-trigger hit region. For the original increment we had

Q(f (K (ti+1)) ≤ Hi+1 | K ∗(ti)) = b,

for the adapted scheme we have

Q(f (K ∗(ti+1)) ≤ Hi+1 | K ∗(ti)) = 1,

i.e., the Monte-Carlo weight of the corresponding sample path will bemultiplied with a factor of b.Note that this is applied conditionally to ti in each time step, i.e., x ,y ,bare processes.

43 / 105

Conditional Analytic PricingDefinition of the Pricing Algorithm

Reformulation of the PricingIf our numerical scheme samples only the survival region, then we mayrewrite the product such that it may be evaluated purely on the paths ofK ∗. On each path ωk we calculate the value

X (Tj+1,ωk ) = Cj(ωk ) ·Qj(ωk ) +Rj(Tj−1)

P(Ti+1;Ti−1), (2)

where Qj(ωk ) is the likelihood ratio given by the importance samplingK ∗ versus K . The probability Qj may be calculated directly from theconditional probabilities of not hitting the barrier, provided by the modelK ∗:

Qj = ∏i:Tj≤ti≤Tj+1

bi .

Note: In the payout (2) the discontinuity of the trigger has beenremoved.

44 / 105

Relation to Existing Work

Other Works:I [GlassermanStaum2001] discussed the “Smoothing by

Conditioning”.I [Piterbarg2003] discussed its application to LIBOR TARNs.I For linear triggers the above is largely the same.

Improvements here:I We calculate a correction on the level of the numerical scheme (like

for the proxy simulation scheme) (no re-sampling, rotation).I We present the method for generalized trigger products (including

non-linear triggers).I Non-Linear Triggers can be handled. Note:

I [GlassermanStaum2001] proposed a sampling for non-linear triggerswhich improves the pricing, but likely leads to noisy sensitivities.

I [Piterbarg2003] is formulated specifically for LIBOR TARNS in aLIBOR Market Model (→ linear trigger).

I Much simpler, more model independent implementation.45 / 105

Conditional Analytic PricingRemarks / Generalizations

Generalizations: Non-Linear TriggersIf the trigger is not a linear function of the model primitives, there arethree options:

I Transform the trigger equation, such that it is linear in the modelprimitives. Example:

I Consider a trigger criteria L > H where L follows a lognormalprocess.

I Transform the trigger criteria to log(L) > log(H), andI define an Euler scheme for K := log(L)

I Model the trigger. Example:I If the trigger is a CMS swap-rate it can be written as a linear trigger if

we are using a swap-rate market model instead of a LIBOR marketmodel. See, for example, [6] or [8].

Effectively, this procedure represents a subtle linearization of thetrigger, because the underlying state variable K is linearized withinthe time-step ∆t through the numerical scheme.

46 / 105


Generalizations: Non-Linear TriggersIf this is not possible, we may linearize g. If g is smooth, thelinearization error will tend to 0 as ∆t → 0. We will then work with alinearization of (1):

g(qv1) ≈ g(0) + ∇g ·qv1, (3)

the scheme then has the property that

f (K ∗(ti+1)) ≈ f (K ∗(ti) + µ(ti)∆ti) + ∇g ·

(qv1z1 +

n

∑j=2

vjzj

)(4)

= g(0) + ∇g · (qv1)z1 ≤ Hi+1

So in first order we have that the scheme generates realizations that donot hit the trigger. In the limit we have obviously

P(f (K ∗(ti+1)) ≤ Hi+1) → 1 as ∆t → 0.

In addition we have

Q(f (K (ti+1)) ≤ Hi+1 | K ∗(ti)) ≈ q.

Due to the time discretization error it is not guaranteed that the schemedoes not generate paths for which the trigger is hit. However, in the limit∆t → 0 this is the case. We can cope with this by modifying the payoutin such a way that the product priced under the scheme is no longer atrigger.

47 / 105


Other Transition Probabilities / Other Models

I Conditional analytic numerical scheme may be generalized to othertransition probabilities.

I Similar to generalizing the proxy simulation to other schemes withother transition probabilities.

I See [Kienitz2008] for a generalization of proxy simulation to Levyprocesses.

48 / 105

CONDITIONAL ANALYTIC MONTE-CARLO PRICINGNUMERICAL RESULTS

Conditional Analytic PricingNumerical Results

Pricing of Digitals and TaRNs

Product Direct Simulation Conditional AnalyticDigital Caplet, T = 0.5 21.40% ±0.31% 21.40% ±0.00%Digital Caplet, T = 2.0 17.38% ±0.27% 17.39% ±0.19%Digital Caplet, T = 5.0 12.04% ±0.19% 12.03% ±0.15%

LIBOR TaRN Swap 1, T = 6.0 3.56% ±0.07% 3.56% ±0.06%LIBOR TaRN Swap 2, T = 6.05 2.511% ±0.012% 2.511% ±0.005%

Table 1: Prices and standard deviation of a Monte-Carlo pricing using directsimulation and conditional analytic simulation, both with 5000 paths. TheLIBOR TaRN Swap 2 has a short first period of length 0.05.

Compared to direct simulation, the conditional analytic simulation reduces theMonte-Carlo error. The reduction is small for product with long maturity, because herethe Monte-Carlo error induced by the discontinuity is not the prominent part. For shortmaturities the reduction gets significant. The digital caplet with maturity t = 0.5 is a limitcase, where the pricing under a conditional analytic simulation becomes analytic. 50 / 105


Sensitivities of Digitals and TaRNs

I In the following we will presents delta, gamma and vega calculatedby finite differences applied to the respective pricing algorithm.

I In the figures we draw mean (line) and standard deviation(transparent corridor) for

I direct simulation (red),I partial proxy simulation scheme (yellow) and theI conditional analytic scheme (green).

The scaling of the sensitivities is as follows: Delta and gamma arenormalized as price change per 100 bp shift. Vega is normalized asprice change per 1% volatility change times 100.

51 / 105


Digital Caplet: Delta

Delta of Digital Caplet, exercise at t=5.0 (5000 paths)

0.0 2.5 5.0 7.5 10.0


1.00%

1.25%

1.50%

1.75%

2.00%

2.25%

2.50%

2.75%

del

ta

Delta of Digital Caplet, exercise at t=5.0 (5000 paths)

0.0 10.0 20.0 30.0 40.0 50.0


1.25%

1.50%

1.75%

2.00%

2.25%

2.50%

2.75%

del

taFigure 1: Delta of a 5Y Digital Caplet: Finite difference is applied to a directsimulation (red), to a partial proxy simulation scheme keeping constrainingcumulated coupons (yellow) and to a conditional analytic scheme (green).

52 / 105


Digital Caplet: Delta

Shiftin bp

0,0−0,10,1−0,50,5−1,01,0−2,02,0−3,03,0−4,04,0−5,05,0−6,06,0−8,0

8,0−10,010−1515−2020−2525−3030−4040−50

Direct Simulationmean ± std.dev.

3,35% ±13,38%2,04% ±1,83%1,91% ±1,10%1,98% ±0,78%2,02% ±0,58%1,95% ±0,52%1,94% ±0,46%1,95% ±0,37%1,98% ±0,31%1,99% ±0,31%1,98% ±0,25%1,97% ±0,21%1,94% ±0,20%1,98% ±0,17%1,98% ±0,16%1,99% ±0,13%

Partial Proxymean ± std.dev.

1,96% ±0,25%1,97% ±0,23%2,07% ±0,21%2,02% ±0,25%2,00% ±0,23%2,02% ±0,25%1,96% ±0,22%1,95% ±0,28%1,99% ±0,24%1,97% ±0,23%1,97% ±0,26%1,95% ±0,26%1,99% ±0,24%1,99% ±0,25%2,00% ±0,26%1,97% ±0,25%

Conditional Analyticmean ± std.dev.

1,98% ±0,05%1,97% ±0,04%1,98% ±0,04%1,98% ±0,05%1,98% ±0,04%1,98% ±0,04%1,99% ±0,05%1,97% ±0,04%1,97% ±0,04%1,98% ±0,04%1,98% ±0,04%1,97% ±0,05%1,97% ±0,05%1,98% ±0,04%1,98% ±0,04%1,97% ±0,04%

Table 2: Delta of a 5Y-digital caplet. Data corresponding to Figure 1.

53 / 105


Digital Caplet: Gamma

Gamma of Digital Caplet, exercise at t=5.0 (5000 paths)

0.0 2.5 5.0 7.5 10.0


-0.01

-0.00

-0.00

0.00

gam

ma

Gamma of Digital Caplet, exercise at t=5.0 (5000 paths)

0.0 10.0 20.0 30.0 40.0 50.0


-0.01

-0.00

-0.00

0.00

0.00

gam

ma

Figure 2: Gamma of a 5Y Digital Caplet: Finite difference is applied to a directsimulation (red), to a partial proxy simulation scheme keeping constrainingcumulated coupons (yellow) and to a conditional analytic scheme (green).

54 / 105


Digital Caplet: Gamma

Shiftin bp

0,0−0,10,1−0,50,5−1,01,0−2,02,0−3,03,0−4,04,0−5,05,0−6,06,0−8,08,0−10,0

10−1515−2020−2525−3030−4040−50


1,3E5% ±1,1E6%−174,3% ±1621,5%

35,14% ±265,9%1,30% ±112,2%5,43% ±43,43%−0,49% ±30,21%

4,08% ±18,84%−4,09% ±14,11%−0,57% ±9,23%−0,12% ±5,99%−0,27% ±4,26%−0,46% ±2,60%−0,28% ±1,68%−0,18% ±1,26%−0,37% ±0,91%−0,41% ±0,65%


−0,36% ±0,36%−0,29% ±0,31%−0,31% ±0,33%−0,31% ±0,30%−0,33% ±0,33%−0,30% ±0,31%−0,44% ±0,34%−0,31% ±0,37%−0,29% ±0,34%−0,34% ±0,35%−0,31% ±0,31%−0,34% ±0,31%−0,30% ±0,31%−0,32% ±0,34%−0,33% ±0,33%−0,30% ±0,33%


−0,34% ±0,03%−0,34% ±0,03%−0,34% ±0,03%−0,34% ±0,03%−0,34% ±0,03%−0,34% ±0,03%−0,34% ±0,03%−0,34% ±0,03%−0,34% ±0,03%−0,34% ±0,03%−0,34% ±0,03%−0,34% ±0,03%−0,34% ±0,03%−0,34% ±0,03%−0,34% ±0,03%−0,34% ±0,03%

Table 3: Gamma of a 5Y-digital caplet. Data corresponding to Figure 2.

55 / 105


Digital Caplet: Vega

Vega of Digital Caplet, exercise at t=5.0 (5000 paths)

0.0 2.5 5.0 7.5 10.0


-16.00%

-15.00%

-14.00%

-13.00%

-12.00%

-11.00%

-10.00%

-9.00%

veg

a

Vega of Digital Caplet, exercise at t=5.0 (5000 paths)

0.0 2.5 5.0 7.5 10.0


-16.00%

-15.00%

-14.00%

-13.00%

-12.00%

-11.00%

-10.00%

-9.00%

veg

aFigure 3: Vega of a 5Y Digital Caplet: Finite difference is applied to a directsimulation (red), to a partial proxy simulation scheme keeping constrainingcumulated coupons (yellow) and to a conditional analytic scheme (green).

56 / 105


Digital Caplet: Vega

Shiftin bp

0,0−0,00,0−0,10,1−0,50,5−1,01,0−2,02,0−3,03,0−4,04,0−5,05,0−6,06,0−8,08,0−10,0


−6,77% ±0,18%−6,70% ±0,20%−16,49% ±36,87%−10,86% ±13,42%−11,99% ±11,29%−12,47% ±7,51%−11,71% ±5,73%−11,82% ±5,88%−11,92% ±5,09%−12,54% ±5,27%−12,44% ±4,47%


−11,91% ±4,84%−11,77% ±5,43%−12,66% ±5,87%−12,99% ±6,68%−12,77% ±5,21%−12,60% ±6,17%−13,64% ±5,55%−12,37% ±5,57%−13,68% ±5,75%−13,56% ±5,91%−12,97% ±5,89%


−12,72% ±0,18%−12,65% ±0,21%−12,66% ±0,18%−12,66% ±0,22%−12,68% ±0,18%−12,71% ±0,16%−12,62% ±0,19%−12,60% ±0,16%−12,68% ±0,17%−12,67% ±0,16%−12,66% ±0,17%

Table 4: Vega of a 5Y-digital caplet. Data corresponding to Figure 3.

57 / 105


Target Redemption Note: Delta

Delta of LIBOR TARN Swap (5000 paths)

0.0 2.5 5.0 7.5 10.0


-10.00%

-9.00%

-8.00%

-7.00%

-6.00%

-5.00%

-4.00%

del

ta


0.0 10.0 20.0 30.0 40.0 50.0


-10.00%

-9.00%

-8.00%

-7.00%

-6.00%

-5.00%

-4.00%

del

taFigure 4: Delta of a LIBOR TARN: Finite difference is applied to a directsimulation (red), to a partial proxy simulation scheme keeping constrainingcumulated coupons (yellow) and to a conditional analytic scheme (green).

58 / 105


Target Redemption Note: Delta

Shiftin bp

0,0−0,10,1−0,50,5−1,01,0−2,02,0−3,03,0−4,04,0−5,05,0−6,06,0−8,08,0−10,0

10−1515−2020−2525−3030−4040−50


−6,72% ±1,87%−6,98% ±0,94%−6,85% ±0,60%−6,94% ±0,46%−6,92% ±0,38%−6,86% ±0,32%−6,87% ±0,26%−6,89% ±0,23%−6,87% ±0,20%−6,93% ±0,20%−6,87% ±0,17%−6,84% ±0,15%−6,82% ±0,15%−6,75% ±0,14%−6,72% ±0,12%−6,61% ±0,12%


−6,89% ±0,38%−6,88% ±0,33%−6,84% ±0,29%−6,93% ±0,31%−6,93% ±0,37%−6,86% ±0,30%−6,87% ±0,35%−6,84% ±0,32%−6,92% ±0,33%−6,89% ±0,33%−6,87% ±0,33%−6,86% ±0,33%−6,79% ±0,35%−6,81% ±0,34%−6,66% ±0,42%−6,66% ±0,63%


−6,92% ±0,18%−6,88% ±0,17%−6,93% ±0,18%−6,92% ±0,16%−6,90% ±0,16%−6,87% ±0,17%−6,87% ±0,16%−6,90% ±0,17%−6,88% ±0,14%−6,90% ±0,15%−6,87% ±0,14%−6,84% ±0,13%−6,82% ±0,13%−6,75% ±0,13%−6,71% ±0,11%−6,60% ±0,11%

Table 5: Delta of a LIBOR TARN. Data corresponding to Figure 4.

59 / 105


Target Redemption Note: Gamma

Gamma of LIBOR TARN Swap (5000 paths)

0.0 2.5 5.0 7.5 10.0


-0.30

-0.20

-0.10

0.00

0.10

gam

ma


0.0 10.0 20.0 30.0 40.0 50.0


-0.30

-0.20

-0.10

0.00

0.10

gam

ma

Figure 5: Gamma of a LIBOR TARN: Finite difference is applied to a directsimulation (red), to a partial proxy simulation scheme keeping constrainingcumulated coupons (yellow) and to a conditional analytic scheme (green).

60 / 105


Target Redemption Note: Gamma

Shiftin bp

0,0−0,10,1−0,50,5−1,01,0−2,02,0−3,03,0−4,04,0−5,05,0−6,06,0−8,08,0−10,0

10−1515−2020−2525−3030−4040−50


311,8% ±8827,7%−140,8% ±1144,0%−7,21% ±149,4%−14,13% ±70,14%−0,22% ±26,60%−9,20% ±14,89%−7,29% ±10,41%−8,06% ±7,17%−7,75% ±5,57%−8,11% ±3,20%−7,53% ±2,56%−7,67% ±1,39%−7,73% ±0,92%−7,69% ±0,73%−7,75% ±0,55%−7,74% ±0,37%


−7,61% ±1,78%−7,75% ±1,57%−7,51% ±1,98%−7,62% ±1,81%−7,69% ±1,81%−8,10% ±1,93%−7,63% ±1,79%−7,96% ±1,76%−7,84% ±1,70%−7,73% ±1,94%−7,72% ±1,66%−7,65% ±1,60%−7,59% ±1,91%−7,47% ±1,86%−7,61% ±2,06%−7,85% ±2,64%


−7,57% ±1,69%−7,96% ±1,33%−7,69% ±2,32%−7,58% ±1,32%−7,68% ±1,40%−7,77% ±1,41%−7,71% ±1,33%−8,03% ±1,32%−7,85% ±1,14%−7,87% ±1,17%−7,72% ±0,96%−7,84% ±0,81%−7,65% ±0,62%−7,71% ±0,48%−7,73% ±0,37%−7,73% ±0,30%

Table 6: Gamma of a LIBOR TARN. Data corresponding to Figure 5.

61 / 105


Target Redemption Note: Vega

Vega of LIBOR TARN Swap (5000 paths)

0.0 2.5 5.0 7.5 10.0


-30.00%

-25.00%

-20.00%

-15.00%

-10.00%

-5.00%

veg

a


0.0 10.0 20.0 30.0 40.0 50.0


-25.00%

-20.00%

-15.00%

-10.00%

-5.00%

veg

aFigure 6: Vega of a LIBOR TARN: Finite difference is applied to a directsimulation (red), to a partial proxy simulation scheme keeping constrainingcumulated coupons (yellow) and to a conditional analytic scheme (green).

62 / 105


Target Redemption Note: Vega

Shiftin bp

0,0−0,10,1−0,50,5−1,01,0−2,02,0−3,03,0−4,04,0−5,05,0−6,06,0−8,08,0−10,0

10−1515−2020−2525−3030−4040−50


−16,59% ±52,71%−15,90% ±15,51%−15,21% ±9,19%−16,97% ±7,61%−16,32% ±5,96%−16,90% ±4,33%−16,98% ±4,32%−16,55% ±4,01%−16,88% ±3,40%−16,67% ±2,99%−16,60% ±2,49%−16,46% ±2,06%−16,82% ±1,85%−16,43% ±1,78%−16,59% ±1,62%−16,77% ±1,36%


−16,86% ±1,72%−16,79% ±1,78%−16,34% ±1,92%−16,69% ±1,62%−16,89% ±1,95%−16,78% ±1,65%−16,49% ±1,68%−16,65% ±1,87%−16,60% ±1,76%−16,52% ±1,80%−16,51% ±1,73%−16,66% ±1,56%−16,46% ±1,70%−16,75% ±1,67%−16,57% ±1,78%−16,55% ±1,83%


−16,73% ±0,66%−16,62% ±0,67%−16,70% ±0,64%−16,79% ±0,59%−16,70% ±0,62%−16,62% ±0,69%−16,53% ±0,64%−16,72% ±0,68%−16,64% ±0,56%−16,72% ±0,63%−16,66% ±0,61%−16,66% ±0,59%−16,70% ±0,62%−16,58% ±0,69%−16,63% ±0,63%−16,65% ±0,61%

Table 7: Vega of a LIBOR TARN. Data corresponding to Figure 6.

63 / 105


Sensitivities of Target Redemption Note Close to Trigger ResetThe following example present delta, gamma and vega a targetredemption note with a short period of 0.05 to its next reset. The targetcoupon is 0.0575, such that under the market date assumed there isapproximately a 50:50 chance of knock out in the next period.In other words, we are approaching the discontinuity in time and space.Such a situation may indeed happen during the life-cycle of a targetredemption note. In the case sensitivities will blow up.

64 / 105


Target Redemption Note Close to Trigger Reset: Delta


0.0 2.5 5.0 7.5 10.0


0.75%

1.00%

1.25%

1.50%

1.75%

del

ta


0.0 10.0 20.0 30.0 40.0 50.0


0.75%

1.00%

1.25%

1.50%

1.75%

del

taFigure 7: Delta of a LIBOR TARN with short period to next reset. Finitedifference is applied to a direct simulation (red), to a partial proxy simulationscheme keeping constraining cumulated coupons (yellow) and to a conditionalanalytic scheme (green).

65 / 105


Target Redemption Note Close to Trigger Reset: DeltaShiftin bp

0,0−0,10,1−0,50,5−1,01,0−2,02,0−3,03,0−4,04,0−5,05,0−6,06,0−8,08,0−10,0

10−1515−2020−2525−3030−4040−50


1,54% ±1,31%1,30% ±0,69%1,24% ±0,36%1,32% ±0,27%1,34% ±0,20%1,30% ±0,15%1,29% ±0,13%1,28% ±0,13%1,21% ±0,11%1,19% ±0,10%1,03% ±0,10%0,78% ±0,10%0,51% ±0,10%0,20% ±0,11%−0,22% ±0,16%−0,79% ±0,15%


1,27% ±0,41%1,30% ±0,42%1,37% ±0,40%1,37% ±0,39%1,24% ±0,49%1,19% ±0,45%1,34% ±0,40%1,27% ±0,47%1,22% ±0,44%1,14% ±0,44%1,03% ±0,61%0,91% ±1,11%0,37% ±2,92%0,87% ±4,22%0,53% ±8,74%−2,78% ±31,78%


1,32% ±0,03%1,32% ±0,03%1,32% ±0,04%1,31% ±0,03%1,31% ±0,03%1,29% ±0,03%1,28% ±0,03%1,26% ±0,04%1,22% ±0,04%1,17% ±0,03%1,02% ±0,07%0,78% ±0,08%0,51% ±0,09%0,20% ±0,10%−0,23% ±0,16%−0,79% ±0,15%

Table 8: Delta of a LIBOR TARN with short period to next reset. Datacorresponding to Figure 7.

66 / 105


Target Redemption Note Close to Trigger Reset: Gamma


0.0 2.5 5.0 7.5 10.0


-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

gam

ma


0.0 10.0 20.0 30.0 40.0 50.0


-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

gam

ma

Figure 8: Gamma of a LIBOR TARN with short period to next reset. Finitedifference is applied to a direct simulation (red), to a partial proxy simulationscheme keeping constraining cumulated coupons (yellow) and to a conditionalanalytic scheme (green).

67 / 105


Target Redemption Note Close to Trigger Reset: GammaShiftin bp

0,0−0,10,1−0,50,5−1,01,0−2,02,0−3,03,0−4,04,0−5,05,0−6,06,0−8,08,0−10,0

10−1515−2020−2525−3030−4040−50


−75,14% ±6514,7%79,61% ±913,8%6,10% ±106,6%0,34% ±44,81%2,36% ±16,28%1,78% ±9,59%−0,13% ±6,41%

2,05% ±4,20%0,95% ±3,26%1,18% ±2,31%0,56% ±1,56%0,26% ±0,82%−0,12% ±0,60%−0,64% ±0,50%−1,31% ±0,40%−2,12% ±0,30%


0,69% ±6,81%0,98% ±7,40%0,89% ±6,88%1,68% ±6,68%0,97% ±6,71%0,59% ±7,48%0,35% ±6,08%1,90% ±5,98%0,80% ±7,61%0,12% ±8,28%0,87% ±8,30%0,64% ±11,21%3,16% ±24,11%−3,72% ±32,95%−5,90% ±47,20%

1,12% ±179,2%


1,19% ±0,28%1,17% ±0,24%1,14% ±0,23%1,14% ±0,23%1,19% ±0,24%1,16% ±0,28%1,11% ±0,22%1,05% ±0,24%0,96% ±0,25%0,92% ±0,24%0,66% ±0,26%0,25% ±0,22%−0,17% ±0,24%−0,66% ±0,26%−1,33% ±0,28%−2,12% ±0,23%

Table 9: Gamma of a LIBOR TARN with short period to next reset. Datacorresponding to Figure 8.

68 / 105


Target Redemption Note Close to Trigger Reset: Vega


0.0 2.5 5.0 7.5 10.0


-3.50%

-3.00%

-2.50%

-2.00%

-1.50%

-1.00%

-0.50%

0.00%

0.50%

veg

a


0.0 10.0 20.0 30.0 40.0 50.0


-3.50%

-3.00%

-2.50%

-2.00%

-1.50%

-1.00%

-0.50%

0.00%

0.50%

veg

aFigure 9: Vega of a LIBOR TARN with short period to next reset. Finitedifference is applied to a direct simulation (red), to a partial proxy simulationscheme keeping constraining cumulated coupons (yellow) and to a conditionalanalytic scheme (green).

69 / 105


Target Redemption Note Close to Trigger Reset: VegaShiftin bp

0,0−0,10,1−0,50,5−1,01,0−2,02,0−3,03,0−4,04,0−5,05,0−6,06,0−8,08,0−10,0

10−1515−2020−2525−3030−4040−50


−0,25% ±5,11%−2,34% ±13,02%−0,87% ±1,28%−1,66% ±4,68%−1,85% ±3,19%−1,80% ±2,65%−1,19% ±1,55%−1,46% ±1,61%−1,42% ±1,37%−1,47% ±1,39%−1,43% ±1,14%−1,53% ±0,95%−1,51% ±0,80%−1,52% ±0,75%−1,48% ±0,72%−1,44% ±0,60%


−1,48% ±0,60%−1,42% ±0,59%−1,46% ±0,59%−1,36% ±0,55%−1,38% ±0,47%−1,46% ±0,59%−1,37% ±0,53%−1,38% ±0,59%−1,52% ±0,59%−1,48% ±0,57%−1,45% ±0,56%−1,53% ±0,56%−1,37% ±0,52%−1,56% ±0,57%−1,43% ±0,57%−1,48% ±0,52%


−1,42% ±0,11%−1,44% ±0,12%−1,44% ±0,12%−1,43% ±0,10%−1,41% ±0,11%−1,42% ±0,10%−1,43% ±0,12%−1,43% ±0,12%−1,45% ±0,11%−1,42% ±0,11%−1,43% ±0,11%−1,46% ±0,10%−1,42% ±0,11%−1,44% ±0,12%−1,44% ±0,11%−1,44% ±0,12%

Table 10: Vega of a LIBOR TARN with short period to next reset. Datacorresponding to Figure 9.

70 / 105

Conditional Analytic PricingConclusion

ConclusionsI We presented a reformulation of the pricing of a family of

generalized auto-callable products.I The pricing and sensitivities calculated by finite featured a greatly

reduced Monte-Carlo variance.Basic requirements of the method are

I The auto-callable value upon trigger hit may be valued analytically.I The trigger criteria may be formulated such that the trigger index is

linear in the increment of the numerical scheme. If not, alinearization may still work.

I The cumulative distribution function of the increment of thenumerical scheme as well as its inverse is known (or a suitableapproximation).

We have seen that this method is effective across a large range ofcases where other methods fail; this means that a practitioner can usethis method and be confident that it will work consistently.

71 / 105

REFERENCES

Outline




References


73 / 105

Further Reading ISome Books and Original Articles

FRIES, CHRISTIAN P.: Mathematical Finance. Theory, Modeling,Implementation. John Wiley & Sons, 2007. ISBN 0-470-04722-4.http://www.christian-fries.de/finmath/book.

GLASSERMAN, PAUL: Monte Carlo Methods in FinancialEngineering. (Stochastic Modelling and Applied Probability).Springer, 2003. ISBN 0-387-00451-3.

JÄCKEL, PETER: Monte-Carlo Methods in Finance. 238 Seiten.Wiley, Chichester, 2002. ISBN 0-471-49741-X.

JOSHI, MARK S.: The Concepts and Practice of MathematicalFinance. Cambridge University Press, 2003. ISBN 0-521-82355-2.

74 / 105

http://www.amazon.com/dp/0470047224/finmath-20

http://www.christian-fries.de/finmath/book

http://www.amazon.com/dp/0387004513?tag=finmath-20

http://www.amazon.com/dp/047149741X/finmath-20

http://www.amazon.com/dp/0521823552?tag=finmath-20

Further Reading IISome Books and Original Articles

BOYLE, PHELIM; BOADIE, MARK; GLASSERMAN, PAUL: MonteCarlo methods for security pricing. Journal of Economic Dynamicsand Control, 21, 1267-1321 (1997).

BROADIE, MARK; GLASSERMAN, PAUL: Estimating Security PriceDerivatives using Simulation. Management Science, 1996, Vol. 42,No. 2, 269-285.

FRIES, CHRISTIAN P.; JOSHI, MARK S.: Partial Proxy SimulationSchemes for Generic and Robust Monte-Carlo Greeks. Journal ofComputational Finance, 12-1. (2008).http://www.christian-fries.de/finmath/proxyscheme.

75 / 105

http://www.christian-fries.de/finmath/proxyscheme

Further Reading IIISome Books and Original Articles

FRIES, CHRISTIAN P.; KAMPEN, JÖRG: Proxy Simulation Schemesfor generic robust Monte Carlo sensitivities, process orientedimportance sampling and high accuracy drift approximation. Journalof Computational Finance, 10-2. (2006).http://www.christian-fries.de/finmath/proxyscheme

FRIES, CHRISTIAN P.: Localized Proxy Simulation Schemes forGeneric and Robust Monte Carlo Greeks. (2007).http://www.christian-fries.de/finmath/proxyscheme

FRIES, CHRISTIAN P.; MARK, JOSHI S.: Conditional Analytic MonteCarlo Pricing Scheme for Auto-Callable Products. (2008). http://www.christian-fries.de/finmath/montecarlo4trigger

76 / 105



http://www.christian-fries.de/finmath/montecarlo4trigger

http://www.christian-fries.de/finmath/montecarlo4trigger

Further Reading IVSome Books and Original Articles

FOURNIÉ, ERIC; LASRY JEAN-MICHEL; LEBUCHOUX, JÉRÔME;LIONS, PIERRE-LOUIS; TOUZI, NIZAR: Applications of Malliavincalculus to Monte Carlo methods in finance. Finance Stochastics. 3,391-412 (1999). Springer- Verlag 1999.

GILES, MIKE B.: Multi-level Monte Carlo path simulation.Operations Research, 56(3):607-617, 2008.

GILES, MIKE B.; GLASSERMAN, PAUL: Smoking Adjoints: fastMonte Carlo Greeks.RISK, January 2006, 88-92.

GLASSERMAN, PAUL; STAUM, JEREMY: Conditioning on one-stepsurvival in barrier option simulations. Operations Research,49:923?937, 2001.

77 / 105

Further Reading VSome Books and Original Articles

JOSHI, MARK S.: Rapid computation of drifts in a reduced factorLIBOR Market Model. Wilmott Magazine, May 2003.

JOSHI, MARK S.; KAINTH, DHERMINDER S.: Rapid computation ofprices and deltas of nth to default swaps in the Li Model,Quantitative Finance, volume 4, issue 3, (June 04), pages 266 - 275

JOSHI, MARK S.; LEUNG, TERENCE: Using Monte Carlo simulationand importance sampling to rapidly obtain jump-diffusion prices ofcontinuous barrier options. 2005.http://ssrn.com/abstract=907386

JOSHI, MARK S.; LIESCH, LORENZO : Effective implementation ofgeneric market models, ASTIN Bulletin, Dec 2007. pp 453–473.

78 / 105

http://ssrn.com/abstract=907386

Further Reading VISome Books and Original Articles

KAMPEN, J.; KOLODKO, A.; SCHOENMAKERS, J.: Monte CarloGreeks for financial products via approximative transition densities.Siam J. Sc. Comp., vol. 31 , p. 1-22, 2008.

KIENITZ, JOERG: A Note on Monte Carlo Greeks for Jump Diffusionand Other Levy Processes. SSRN, 2008.

PIETERSZ, RAOUL; VAN REGENMORTEL, MARCEL Generic MarketModels, Finance and Stochastics, 10, 507–528, (2006)

PITERBARG, VLADIMIR V.: Computing deltas of callable LIBORexotics in forward LIBOR models. Journal of ComputationalFinance. 7 (2003).

PITERBARG, VLADIMIR V.: TARNs: Models, Valuation, RiskSensitivities. Wilmott Magazine, 2004.

79 / 105

Further Reading VIISome Books and Original Articles

ROTT, MARIUS G.; FRIES, CHRISTIAN P.: Fast and RobustMonte-Carlo CDO Sensitivities and their Efficient Object OrientedImplementation. 2005.http://www.christian-fries.de/finmath/cdogreeks

80 / 105

http://www.christian-fries.de/finmath/cdogreeks

BONUS/BACKUP: STABLE MONTE-CARLO

SENSITIVITIES FOR BERMUDAN CALLABLES

Outline




References


82 / 105

BONUS/BACKUP: STABLE MONTE-CARLOSENSITIVITIES FOR BERMUDAN CALLABLES

BERMUDAN PRICING BACKWARD ALGORITHM

Stable Bermudan SensitivitiesThe Backward Algorithm

The Backward Algorithm (e.g., within an pdf/lattice framework)The value of a Bermudan option can be defined recursively using thebackward algorithm where the time Ti value of future payoffs is given as

V (Ti) =

H(Ti) G(Ti) := H(Ti)−U(Ti) > 0U(Ti) else,

where H(Ti) := E(

V (Ti+1)N(Ti+1)N(Ti )

| FTi

)is the conditional expectation of

the discounted continuation value V (Ti+1), U(Ti) is the value receivedupon exercise, i.e., the value of the underlying and N is the numéraire.1

A shorter notation is

V (Ti) = max(H(Ti),U(Ti)), where H(Ti) := E(

V (Ti+1)N(Ti+1)

N(Ti)| FTi

).

1The value V and U are considered to be numéraire relative.84 / 105


Monte-Carlo Lower-Bound Version of the Backward AlgorithmIn a Monte-Carlo simulation modify the backward algorithms as follows:We define the (unconditioned (!)) pathwise value

V ∗(Ti ,ω) :=

V ∗(Ti+1,ω)

N(Ti+1)N(Ti )

G(Ti) > 0

U∗(Ti ,ω) else.

Here G(Ti) := H(Ti)−U(Ti) > 0 is the exercise criteria and N is thenuméraire. U∗(Ti) is the sum of the discounted, numéraire relative valueof the cashflows of the underlying.Reason: In a Monte-Carlo simulation it is naturally difficult andexpensive to calculate.

H(Ti) := E(

V (Ti+1)N(Ti+1)

N(Ti)| FTi

).

An accurate estimate is difficult.Note: We have

E(V ∗(Ti) | FTi

)= V (Ti).

85 / 105


Monte-Carlo Version of Backward AlgorithmNow, let Gest(Ti) be some estimate of G(Ti), e.g. obtained from theMonte-Carlo simulation through a regression. Consider now

V ∗,est(Ti ,ω) :=

V ∗,est(Ti ,ω)(Ti+1,ω)

N(Ti+1)N(Ti )

Gest(Ti) > 0

U∗(Ti ,ω) else.(5)

Properties:The advantage of formulation (5) is that for any estimate Hest we get at alower bound of V (T0):

E(V ∗,est(T0) | FT0

)≤ V (T0)

Note: The gap of the two becomes smaller as the estimation of theexercise criteria Hest(Ti)−U(Ti) becomes more accurate.The disadvantage of the formulation (3) and hence of (5) is that nowV ∗(T0,ω) and V ∗,est(T0,ω) is a discontinuous function of the modelparameters, which leads to the known noisy sensitivities.

86 / 105


LOCALLY SMOOTHED BACKWARD ALGORITHM

Stable Bermudan SensitivitiesLocally Smoothed Backward Algorithm

Locally Smoothed Backward AlgorithmWe modify the formulation (3) towards

V∼(Ti) =

α V∼(Ti+1)

N(Ti+1)N(Ti )

+ (1−α) H(Ti) H(Ti)−U(Ti) > 0

α U∗(Ti) + (1−α) U(Ti) else.(6)

where α is a random variable defined through

α := 1−g(|H(Ti)−U(Ti)|

ε

).

and g is smooth function with g(x) = 1 for x ≤ 0 and g(x) = 0 for x ≥ 1.

Note: This is a payoff smoothing, like, e.g., see “sausage method” in [9].

However, we have. . .

88 / 105


Locally Smoothed Backward AlgorithmObviously, the conditional expectation of V∼(Ti) again agrees with thetime Ti value of the Bermudan option, i.e., we have

EQN(

α V (Ti+1)N(Ti+1)

N(Ti)+ (1−α) H(Ti) | FTi

)= α EQN

(V (Ti+1)

N(Ti+1)

N(Ti)| FTi

)+ (1−α) EQN (

H(Ti) | FTi

)= α V (Ti) + (1−α) V (Ti) = V (Ti)

and

EQN (α U∗(Ti) + (1−α) U(Ti) | FTi

)= α U(Ti) + (1−α) U(Ti) = U(Ti)

and thusEQN

(V ∗(Ti)) = V (Ti).

In particular this allows us to obtain the unconditional price V (T0) fromthe formulation (6). 89 / 105


Monte-Carlo Version with Estimated Exercise CriteriaBeing equipped with an approximation for the exercise criteria, i.e., withan estimate Hest(Ti) of H(Ti) and Uest(Ti) of U(Ti) we finally arrive at:

V∼,est(Ti) =α V∼,est (Ti+1)

N(Ti+1)N(Ti )

+ (1−α) Hest(Ti) Hest(Ti)−Uest(Ti) > 0

α U∗(Ti) + (1−α) Uest(Ti) else.(7)

The disadvantage of the formulation (7) is that it is no longer known tobe an lower bound of the true Bermudan value. It can be biased highand biased low.The advantage of formulation (7) is that the pathwise valuesV∼,est (Ti ,ω) are now continuous, or to be precise, their are as smoothas g and U are.

90 / 105


NUMERICAL RESULTS

Stable Bermudan SensitivitiesNumerical Results

Benchmark Product and Benchmark Model

I To demonstrate the efficiency of the smoothed Monte-Carlobackward algorithm we consider a cancelable swap2.

I The cancelable swap we consider will be almost at the money inour simple benchmark model environment. This is a meaningfultest case since then most paths will fall into the smoothed region.Hence, for the atm option, the effect of the smoothing will be strong,and at the same time the method is most sensitive to an error in theestimate of the conditional expectation operator.

I As benchmark model we consider a Monte-Carlo implementation ofthe LIBOR market model. The model is simulated in spot measureusing a simple log-Euler scheme.

2The method can be formulated the same way for cancelable products.92 / 105


Simple LocalizersFor the smoothed backward algorithm we choose a simple piecewiselinear localization function g given through

g(x) =

1 for |x |< 0.21−|x | for 0.2≤ |x |< 1.20 for 1.2≤ |x | -1,5 -1 -0,5 0 0,5 1 1,5

0,5

1

and with the continuously differentiable interpolation function g giventhrough

g(x) =

1 for |x |< 0.212(1 + cos((x−0.2) π)) for 0.2≤ |x |< 1.20 for 1.2≤ |x |. -1,5 -1 -0,5 0 0,5 1 1,5

0,5

1

93 / 105


Benchmark Results

I We calculated interest rate sensitivities of a cancelable swap being(almost) at the money. The model used artificial (easy to reproduce)initial data. Forward rates are 10% flat with a log-volatility of 20%.

I Our benchmark product is a 5Y semi-annual cancelable swap,which can be canceled at each period start, except for the firstperiod. Swap rate (strike) of the cancelable swap is at 10%.

I Delta and gamma denotes the first derivative, respectively secondorder derivative of V (T0) with respect to a parallel movement of theforward curve. They are calculated from centered finite differencewith various shift sizes ranging from 0.1 bp to 50 bp.

94 / 105


Delta using Piecewise Linear Interpolation Function

Delta of Cancelable Swap (5000 paths)

0.0 2.5 5.0 7.5 10.0


3.50%

4.00%

4.50%

5.00%

5.50%

6.00%d

elta

Figure 10: Delta of a Cancelable Swap. Standard algorithm (red) versusC0-Locally Smoothed (green). See also Table 11.

95 / 105


Delta using Piecewise Linear Interpolation FunctionShiftin bp

0,1−0,50,5−1,01,0−2,02,0−3,03,0−4,04,0−5,05,0−6,06,0−8,08,0−10,0

10−1515−2020−2525−3030−4040−50

Standard Algorithmmean ± std.dev.

4,7586% ±0,7453%4,8053% ±0,3824%4,8022% ±0,3024%4,7827% ±0,2161%4,8088% ±0,1659%4,7992% ±0,1483%4,7959% ±0,1441%4,8101% ±0,1480%4,8110% ±0,1150%4,7965% ±0,1051%4,7852% ±0,0996%4,7665% ±0,0908%4,7632% ±0,0805%4,7436% ±0,0760%4,7257% ±0,0693%

Locally Smoothedmean ± std.dev.

4,8454% ±0,0780%4,8430% ±0,0710%4,8484% ±0,0817%4,8442% ±0,0749%4,8431% ±0,0857%4,8316% ±0,0864%4,8358% ±0,0686%4,8337% ±0,0742%4,8280% ±0,0706%4,8138% ±0,0745%4,8105% ±0,0739%4,7959% ±0,0753%4,7944% ±0,0675%4,7757% ±0,0719%4,7585% ±0,0678%

Table 11: Delta of the benchmark cancelable swap. Data corresponding toFigure 10.

96 / 105


Gamma using Piecewise Linear Interpolation Function

Gamma of Cancelable Swap (5000 paths)

0.0 10.0 20.0 30.0 40.0 50.0


-0.75%

-0.50%

-0.25%

0.00%

0.25%

0.50%

0.75%

1.00%

1.25%

1.50%g

amm

a

Figure 11: Gamma of a Cancelable Swap. Standard algorithm (red) versusC0-Locally Smoothed (green). See also Table 12.

97 / 105


Gamma using Piecewise Linear Interpolation FunctionShiftin bp

0,1−0,50,5−1,01,0−2,02,0−3,03,0−4,04,0−5,05,0−6,06,0−8,0

8,0−10,010−1515−2020−2525−3030−4040−50


3,5826% ±948,5%2,5770% ±54,7926%0,7905% ±19,6356%0,6024% ±7,6364%0,6583% ±5,0807%1,3091% ±2,6422%0,5764% ±2,0666%0,3005% ±1,6574%0,4492% ±1,1600%0,4001% ±0,6742%0,3942% ±0,4192%0,4332% ±0,2898%0,4222% ±0,2044%0,4233% ±0,1414%0,4242% ±0,1044%

Locally Smoothedmean ± std.dev.

0,4326% ±1,2078%0,3602% ±0,4926%0,4158% ±0,3063%0,4103% ±0,2268%0,3809% ±0,2015%0,3981% ±0,1767%0,3881% ±0,1636%0,3854% ±0,1579%0,4025% ±0,1498%0,4067% ±0,1134%0,3858% ±0,0973%0,3822% ±0,0879%0,3829% ±0,0803%0,3823% ±0,0747%0,3831% ±0,0611%

Table 12: Gamma of the benchmark cancelable swap. Data corresponding toFigure 11.

98 / 105


Gamma using C1-Smooth Interpolation Function

Gamma of Cancelable Swap (5000 paths)

0.0 10.0 20.0 30.0 40.0 50.0


-1.00%

-0.50%

0.00%

0.50%

1.00%

1.50%g

amm

a

Figure 12: Gamma of a Cancelable Swap. Standard algorithm (red) versusC0-Locally Smoothed (green) versus C1-Locally Smoothed (yellow). See alsoTable 13.

99 / 105


Gamma using C1-Smooth Interpolation FunctionShiftin bp

0,1−0,50,5−1,01,0−2,02,0−3,03,0−4,04,0−5,05,0−6,06,0−8,0

8,0−10,010−1515−2020−2525−3030−4040−50


3,5826% ±948,5%2,5770% ±54,7926%0,7905% ±19,6356%0,6024% ±7,6364%0,6583% ±5,0807%1,3091% ±2,6422%0,5764% ±2,0666%0,3005% ±1,6574%0,4492% ±1,1600%0,4001% ±0,6742%0,3942% ±0,4192%0,4332% ±0,2898%0,4222% ±0,2044%0,4233% ±0,1414%0,4242% ±0,1044%

C0 Locally Smoothedmean ± std.dev.

0,4326% ±1,2078%0,3602% ±0,4926%0,4158% ±0,3063%0,4103% ±0,2268%0,3809% ±0,2015%0,3981% ±0,1767%0,3881% ±0,1636%0,3854% ±0,1579%0,4025% ±0,1498%0,4067% ±0,1134%0,3858% ±0,0973%0,3822% ±0,0879%0,3829% ±0,0803%0,3823% ±0,0747%0,3831% ±0,0611%


0,4093% ±0,2556%0,3840% ±0,1756%0,3608% ±0,1904%0,3682% ±0,1610%0,3449% ±0,1878%0,3643% ±0,1505%0,3583% ±0,1398%0,3900% ±0,1504%0,3864% ±0,1633%0,3864% ±0,1298%0,3799% ±0,1284%0,3555% ±0,1188%0,3626% ±0,1062%0,3651% ±0,0959%0,3700% ±0,0740%

Table 13: Gamma of the benchmark cancelable swap. Data corresponding toFigure 12.

100 / 105


Vega using C1-Smooth Interpolation Function

Vega of Cancelable Swap (5000 paths)

0.0 2.5 5.0 7.5 10.0


15.00%

20.00%

25.00%

30.00%

35.00%

40.00%

45.00%

50.00%ve

ga

Figure 13: Vega of a Cancelable Swap. Standard algorithm (red) versusC0-Locally Smoothed (green) versus C1-Locally Smoothed (yellow). See alsoTable 14.

101 / 105


Vega using C1-Smooth Interpolation Function

Shiftin bp

0,0−0,10,1−0,50,5−1,01,0−2,02,0−3,03,0−4,04,0−5,05,0−6,06,0−8,0

8,0−10,0


35,4446% ±9,4126%34,9479% ±3,5159%34,3114% ±4,1499%34,2750% ±2,7266%34,0695% ±2,5486%34,2161% ±1,5970%34,1300% ±2,4372%34,3347% ±1,4203%34,1753% ±1,6234%34,2296% ±1,1931%


34,6128% ±0,9058%34,6126% ±1,0045%34,7526% ±0,9383%34,7592% ±0,9690%34,6517% ±0,9940%34,5778% ±1,0143%34,5699% ±0,9926%34,7228% ±0,7685%34,6351% ±0,9448%34,6322% ±1,0096%


34,6670% ±0,8448%34,6522% ±0,9916%34,7794% ±0,9110%34,7810% ±0,9684%34,6658% ±0,9884%34,6806% ±1,0052%34,6066% ±1,0428%34,7381% ±0,7601%34,6592% ±0,9750%34,6456% ±0,9883%

Table 14: Vega of the benchmark cancelable swap. Data corresponding toFigure 13.

102 / 105


Volga using C1-Smooth Interpolation Function

Vega gamma of Cancelable Swap (5000 paths)

0.0 10.0 20.0 30.0 40.0 50.0


-0.00%

-0.00%

-0.00%

-0.00%

-0.00%

0.00%

0.00%

0.00%ve

ga

gam

ma

Figure 14: Volga (vega gamma) of a Cancelable Swap. Standard algorithm(red) versus C0-Locally Smoothed (green) versus C1-Locally Smoothed(yellow). See also Table 15.

103 / 105


Volga using C1-Smooth Interpolation FunctionShiftin bp

0,1−0,50,5−1,01,0−2,02,0−3,03,0−4,04,0−5,05,0−6,06,0−8,0

8,0−10,010−1515−2020−2525−3030−4040−50


6,7881% ±106,2%0,0511% ±4,0664%0,0567% ±1,8692%0,0716% ±0,7943%0,0069% ±0,4239%−0,0019% ±0,4328%−0,0386% ±0,1966%

0,0113% ±0,1328%−0,0098% ±0,1009%

0,0006% ±0,0542%−0,0002% ±0,0297%

0,0010% ±0,0222%−0,0007% ±0,0149%−0,0007% ±0,0132%−0,0012% ±0,0080%


−0,0013% ±0,0080%−0,0007% ±0,0020%−0,0009% ±0,0016%−0,0010% ±0,0015%−0,0008% ±0,0010%−0,0007% ±0,0009%−0,0009% ±0,0010%−0,0008% ±0,0007%−0,0009% ±0,0007%−0,0009% ±0,0006%−0,0009% ±0,0005%−0,0009% ±0,0004%−0,0009% ±0,0004%−0,0009% ±0,0003%−0,0009% ±0,0003%


−0,0009% ±0,0003%−0,0009% ±0,0001%−0,0009% ±0,0001%−0,0009% ±0,0001%−0,0009% ±0,0001%−0,0009% ±0,0001%−0,0009% ±0,0001%−0,0009% ±0,0001%−0,0009% ±0,0001%−0,0009% ±0,0001%−0,0009% ±0,0001%−0,0009% ±0,0001%−0,0009% ±0,0001%−0,0009% ±0,0001%−0,0009% ±0,0001%

Table 15: Volga (vega gamma) of the benchmark cancelable swap. Datacorresponding to Figure 14.

104 / 105

Stable Bermudan SensitivitiesConclusions

ConclusionsWe presented a modification of the backward algorithm for pricing earlyexercise rights (optimal exercise) within Monte-Carlo simulations. Themodification results in a smoothed payoff with the following properties:

I The smoothing result is greatly reduced Monte-Carlo variance ofsensitivities calculated from finite differences.

I The Monte-Carlo variance of the sensitivities is largely independentof the shift size used in for the approximating finite differences.

I If the estimator for conditional expectation of the continuation valueis exact, then the smoothing does not introduce a pricing error inthe Monte-Carlo limit.

105 / 105

monte-carlo pricing and sensitivities of auto-callable and ... · pdf filemonte-carlo pricing...

Documents