credit value adjustment in the extended structural default model

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credit valueadjustment in the

extended structuraldefault model

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alexander lipton and artur sepp

1 Introduction................................................................................................................................................

1.1 Motivation

In view of the recent turbulence in the credit markets and given a huge outstandingnotional amount of credit derivatives, counterparty risk has become a critical issue forthe financial industry as a whole. According to the most recent survey conveyed bythe International Swap Dealers Association (see <www.isda.org>), the outstandingnotional amount of credit default swaps is $38.6 trillion as of 31 December 2008 (ithas decreased from $62.2 trillion as of 31, December 2007). By way of comparison,the outstanding notional amount of interest rate derivatives was $403.1 trillion, whilethe outstanding notional amount of equity derivatives was $8.7 trillion. The biggestbankruptcy in US history filed by one of the major derivatives dealers, LehmanBrothers Holdings Inc., in September of 2008 makes counterparty risk estimation andmanagement vital to the financial system at large and all the participating financialinstitutions.

The key objective of this chapter is to develop a methodology for valuing the coun-terparty credit risk inherent in credit default swaps (CDSs). For the protection buyer(PB), a CDS contract provides protection against a possible default of the referencename (RN) in exchange for periodic payments to the protection seller (PS) whosemagnitude is determined by the so-called CDS spread. When a PB buys a CDS froma risky PS they have to cope with two types of risk: (a) market risk which comes



credit value adjustment 407

directly from changes in the mark-to-market (MTM) value of the CDS due to creditspread and interest rate changes; (b) credit risk which comes from the fact that PSmay be unable to honour their obligation to cover losses stemming from the defaultof the corresponding RN. During the life of a CDS contract, a realized loss due to thecounterparty exposure arises when PS defaults before RN and, provided that MTMof the CDS is positive, the counterparty pays only a fraction of the MTM value of theexisting CDS contract (if MTM of the CDS is negative to PB, this CDS can be unwoundat its market price).

Since PB realizes positive MTM gains when the credit quality of RN deteriorates(since the probability of receiving protection increases), their realized loss due to PSdefault is especially big if the credit quality of RN and PS deteriorate simultaneouslybut PS defaults first. We define the credit value adjustment (CVA), or the counterpartycharge (CC), as the maximal expected loss on a short position (protection bought) ina CDS contract.

In order to describe CVA in quantitative rather than qualitative terms, in thischapter we build a multi-dimensional structural default model. Below we concentrateon its two-dimensional (2D) version and show that the evaluation of CVA is equivalentto pricing a 2D down-and-in digital option with the down barrier being triggeredwhen the value of the PS’s assets crosses their default barrier and the option rebatebeing determined by the value of the RN’s assets at the barrier crossing time. We alsobriefly discuss the complementary problem of determining CVA for a long position(protection sold) in a CDS contract.

Traditionally, the par CDS spread at inception is set in such a way that the MTMvalue of the contract is zero.1 Thus, the option underlying CVA is at-the-money, sothat its value is highly sensitive to the volatility of the RN’s CDS spread, while thebarrier triggering event is highly sensitive to the volatility of the PS’s asset value. Inaddition to that, the option value is sensitive to the correlation between RN and PS.This observation indicates that for dealing with counterparty risk we need to modelthe correlation between default times of RN and PS as well as CDS spread volatilitiesfor both of them. It turns out that our structural model is very well suited to accomplishthis highly non-trivial task.

1.2 Literature overview

Merton developed the original version of the so-called structural default model (Mer-ton 1974). He postulated that the firm’s value V is driven by a lognormal diffusionand that the firm, which borrowed a zero-coupon bond with face value N and matu-rity T , defaults at time T if the value of the firm V is less than the bond’s face

1 Subsequent to the so-called ‘big bang’ which occurred in 2009, CDS contracts frequently trade onan up-front basis with fixed coupon.



408 a . lipton & a . sepp

value N. Following this ioneering insight, many authors proposed various extensionsof the basic model (Black and Cox 1976; Kim and Ramaswamy, and Sundaresan1993; Nielsen, and Saa-Requejo, and Santa-Clara 1993; Leland 1994; Longstaff andSchwartz 1995; Leland and Toft 1996; Albanese and Chen 2005) among others. Theyconsidered more complicated forms of debt and assumed that the default event maybe triggered continuously up to the debt maturity. More recent research has beenconcentrated on extending the model in order to be able to generate the high short-term CDS spreads typically observed in the market. It has been shown that the lattertask can be achieved either by making default barriers curvilinear (Hyer et al. 1998;Hull and White 2001; Avellaneda and Zhou 2001), or by making default barriersstochastic (Finger et al. 2002), or by incorporating jumps into the firm’s value dynamics(Zhou 2001a; Hilberink and Rogers 2002; Lipton 2002b; Lipton, Song, and Lee 2007;Sepp 2004, 2006; Cariboni and Schoutens 2007; Feng and Linetsky 2008).

Multi-dimensional extensions of the structural model have been studied by severalresearchers (Zhou 2001b; Hull and White 2001; Haworth 2006; Haworth Reisinger, andShaw 2006; Valuzis 2008), who considered bivariate correlated log-normal dynamicsfor two firms and derived analytical formulas for their joint survival probability;Li (2000), who introduced the Gaussian copula description of correlated default timesin multi-dimensional structural models; Kiesel and Scherer (2007), who studied amulti-dimensional structural model and proposed a mixture of semi-analytical andMonte Carlo (MC) methods for model calibration and pricing.

While we build a general multi-dimensional structural model, our specific effortsare aimed at a quantitative estimation of the counterparty risk. Relevant work on thecounterparty risk includes, among others, Jarrow and Turnbull (1995), who developedthe so called reduced-form default model and analysed the counterparty risk in thisframework; Hull and White (2001), Blanchet-Scalliet and Patras (2008), who modelledthe correlation between RN and the counterparty by considering their bivariate corre-lated lognormal dynamics; Turnbull (2005), Pugachevsky (2006), who derived model-free upper and lower bounds for the counterparty exposure; Jarrow and Yu (2001),Leung and Kwok (2005) who studied counterparty risk in the reduced-form setting;Pykhtin and Zhu (2006), Misirpashaev 2008), who applied the Gaussian copula for-malism to study counterparty effects; Brigo and Chourdakis (2008), who consideredcorrelated dynamics of the credit spreads, etc.

Our approach requires the solution of partial integro-differential equations (PIDE)with a non-local integral term. The analysis of solution methods based on the FastFourier Transform (FFT) can be found in Broadie-Broadie and Yamamoto (2003),Jackson and Jaimungal, and Surkov (2007), Boyarchenko and Levendorski (2008),Fang and Oosterlee (2008), Feng and Linetsky (2008), and Lord et al. (2008).The treatment via finite-difference (FD) methods can be found in Andersen andAndreasen (2000), Lipton (2003), d’Halluin, Forsyth, and Vetzal (2005), Cont andVoltchkova (2005), Carr and Mayo (2007), Lipton, Song, and Lee (2007), Toiva-nen (2008), and Clift and Forsyth (2008).




1.3 Contribution

In this chapter, we develop a novel variant of the one-dimensional (1D), two-dimensional (2D), and multi-dimensional structural default model the assumptionthat firms’ values are driven by correlated additive processes. (Recall that an additiveprocess is a jump-diffusion process with time-inhomogeneous increments.) In orderto calibrate the 1D version of our structural model to the CDS spread curve observed inthe market, we introduce jumps with piecewise constant intensity. We correlate jumpsof different firms via a Marshall-Olkin inspired mechanism (Marshall and Olkin 1967).This model was presented for the first time by Lipton and Sepp (2009).

In this chapter, we develop robust FFT- and FD-based methods for model cali-bration via forward induction and for credit derivatives pricing via backward induc-tion in one and two dimensions. While the FFT-based solution methods are easy toimplement, they require uniform grids and a large number of discretization steps.At the same time, FD-based methods, while more complex, tend to provide greaterflexibility and stability. As part of our FD scheme development, we obtain new explicitrecursion formulas for the evaluation of the 2D convolution term for discrete andexponential jumps. In addition, we present a closed-form formula for the joint survivalprobability of two firms driven by correlated lognormal bivariate diffusion processesby using the method of images, thus complementing results obtained by He, Keirstead,and Rebholz, (1998), Lipton (2001), and Zhou (2001b) via the classical eigenfunctionexpansion method. As always, the method of images works well for shorter times, whilethe method of eigenfunction expansion works well for longer times.

We use the above results to develop an innovative approach to the estimation ofCVA for CDSs. Our approach is dynamic in nature and takes into account boththe correlation between RN and PS (or PB) and the CDS spread volatilities. Theapproaches proposed by Leung and Kwok (2005), Pykhtin and Zhu (2006), and Misir-pashaev (2008) do not account for spread volatility and, as a result, may underesti-mate CVA. Blanchet-Patras consider a conceptually similar approach; however, theiranalytical implementation is restricted to lognormal bivariate dynamics with constantvolatilities, which makes it impossible to fit the term structure of the CDS spreadsand CDS option volatilities implied by the market (Blanchet-Scalliet and Patras 2008).Accordingly, the corresponding CVA valuation is biased. In contrast, our model canbe fitted to an arbitrary term structure of CDS spreads and market prices of CDS andequity options. The approach by Hull and White (2001) uses MC simulations of thecorrelated lognormal bivariate diffusions. In contrast, our approach assumes jump-diffusion dynamics, potentially more realistic for default modelling, and uses robustsemi-analytical and numerical methods for model calibration and CVA valuation.

This chapter is organized as follows. In section 2 we introduce the structural defaultmodel in one, two, and multi-dimensions. In section 3 we formulate the genericpricing problem in one, two and multi-dimensions. In section 4 we consider thepricing problem for CDSs, CDS options (CDSOs), first-to-default swaps (FTDSs),and the valuation problem for CVA. In section 5 we develop analytical, asymptotic,




and numerical methods for solving the 1D pricing problem. In particular, we describeMC, FFT, and FD methods for solving the calibration problem via forward inductionand the pricing problem via backward induction. In section 6 we present analyticaland numerical methods for solving the 2D pricing problem, including FFT and FDmethods. In section 7 we provide an illustration of our findings by showing how tocalculate CVA for a CDS on Morgan Stanley (MS) sold by JP Morgan (JPM) and aCDS on JPM sold by MS. We formulate brief conclusions in section 8.

2 Structural model and default event................................................................................................................................................

In this section we describe our structural default model in one, two, and multi-dimensions.

2.1 Notation

Throughout the chapter, we model uncertainty by constructing a probability space(Ÿ,F , F, Q) with the filtration F = {F(t), t ≥ 0} and a martingale measure Q. Weassume that Q is specified by market prices of liquid credit products. The operationof expectation under Q given information set F(t) at time t is denoted by E

Qt [·]. The

imaginary unit is denoted by i, i =√−1.

The instantaneous risk-free interest rate r (t) is assumed to be deterministic; thecorresponding discount factor, D(t, T) is given by:

D(t, T) = exp{−∫ T

tr (t ′)dt ′

}(1)

It is applied at valuation time t for cash flows generated at time T , 0 ≤ t ≤ T < ∞.The indicator function of an event ˆ is denoted by 1ˆ:

1ˆ ={

1 if ˆ is true0 if ˆ is false (2)

The Heaviside step function is denoted by H(x),

H(x) = 1{x≥0} (3)

the Dirac delta function is denoted by δ(x); the Kronecker delta function is denoted byδn,n0 . We also use the following notation

{x}+ = max {x, 0} (4)

We denote the normal probability density function (PDF) by n (x); and the cumu-lative normal probability function by N (x); besides, we frequently use the functionP (a, b) defined as follows:




P(a, b) = exp{

ab + b2/2}

N(a + b) (5)

2.2 One-dimensional case

2.2.1 Asset value dynamicsWe denote the firm’s asset value by a(t). We assume that a(t) is driven by a 1D jump-diffusion under Q:

da(t) = (r (t) − Ê(t) − Î(t)Í)a(t)dt + Û(t)a(t)dW(t) + (e j − 1)d N(t) (6)

where Ê(t) is the deterministic dividend rate on the firm’s assets, W(t) is a standardBrownian motion, Û(t) is the deterministic volatility, N(t) is a Poisson process inde-pendent of W(t), Î(t) is its intensity, j is the jump amplitude, which is a randomvariable with PDF � ( j ); and Í is the jump compensator:

Í =∫ 0

−∞e j � ( j )d j − 1 (7)

To reduce the number of free parameters, we concentrate on one-parametric PDFswith negative jumps which may result in random crossings of the default barrier. Weconsider either discrete negative jumps (DNJs) of size −Ì, Ì > 0, with

� ( j ) = δ( j + Ì), Í = e−Ì − 1 (8)

or exponential negative jumps (ENJs) with mean size 1Ì, Ì > 0, with:

� ( j ) = ÌeÌj , j < 0, Í =Ì

Ì + 1− 1 = − 1

Ì + 1(9)

In our experience, for 1D marginal dynamics the choice of the jump size distributionhas no impact on the model calibration to CDS spreads and CDS option volatilities,however for the joint correlated dynamics this choice becomes very important, as wewill demonstrate shortly.

2.2.2 Default boundaryThe cornerstone assumption of a structural default model is that the firm defaultswhen its value crosses a deterministic or, more generally, random default boundary.The default boundary can be specified either endogenously or exogenously.

The endogenous approach was originated by Black and Cox (1976) who used it tostudy the optimal capital structure of a firm. Under a fairly strict assumption that thefirm’s liabilities can only be financed by issuing new equity, the equity holders have theright to push the firm into default by stopping issuing new equity to cover the interestpayments to bondholders and, instead, turning the firm over to the bondholders. Blackand Cox (1976) found the critical level for the firm’s value, below which it is not optimalfor equity holders to sell any more equity. Equity holders should determine the criticalvalue or the default barrier by maximizing the value of the equity and, respectively,




minimizing the value of outstanding bonds. Thus, the optimal debt-to-equity ratio andthe endogenous default barrier are decision variables in this approach. A nice review ofthe Black-Cox approach and its extensions is given by Bielecki and Rutkowski (2002),and Uhrig-Homburg (2002). However, in our view, the endogenous approach is notrealistic given the complicated equity-liability structure of large firms and the actualrelationships between the firm’s management and its equity and debtholders. Forexample, in July 2009 the bail-out of a commercial lender CIT was carried out bydebtholders, who proposed debt restructuring, rather than by equity holders, who hadno negotiating power.

In the exogenous approach, the default boundary is one of the model parameters.The default barrier is typically specified as a fraction of the debt per share estimatedby the recovery ratio of firms with similar characteristics. While still not very realistic,this approach is more intuitive and practical (see, for instance, Kim and Ramaswamy,and Sundaresan 1993; Nielsen, and Saa-Requejo, and Santa-Clara 1993; Longstaff andSchwartz 1995; etc.).

In our approach, similarly to Lipton (2002b); and Stamicar and Finger (2005), weassume that the default barrier of the firm is a deterministic function of time given by

l(t) = E (t)l(0) (10)

where E (t) is the deterministic growth factor:

E (t) = exp{∫ t

0(r (t ′) − Ê(t ′))dt ′

}(11)

and l(0) is defined by l(0) = RL (0), where R is an average recovery of the firm’sliabilities and L (0) is its total debt per share. We find L (0) from the balance sheetas the ratio of the firm’s total liability to the total common shares outstanding; R isfound from CDS quotes, typically, it is assumed that R = 0.4.

2.2.3 Default triggering eventThe key variable of the model is the random default time which we denote by Ù. Weassume that Ù is an F-adapted stopping time, Ù ∈ (0, ∞]. In general, the default eventcan be triggered in three ways.

First, when the firm’s value is monitored only at the debt’s maturity time T , thenthe default time is defined by:

Ù ={

T, a(T) ≤ l(T)∞, a(T) > l(T) (12)

This is the case of terminal default monitoring (TDM) which we do not use below.Second, if the firm’s value is monitored at fixed points in time, {td

m}m=1,. . .,M ,0 < td

1 < . . . < tdM ≤ T , then the default event can only occur at some time td

m. Thecorresponding default time is specified by:




Ù = min{tdm : a(td

m) ≤ l(tdm)}, min{�} = ∞ (13)

This is the case of discrete default monitoring (DDM).Third, if the firm’s value is monitored at all times 0 < t ≤ T , then the default

event can occur at any time between the current time t and the maturity time T . Thecorresponding default time is specified by:

Ù = inf{t, 0 ≤ t ≤ T : a(t) ≤ l(t)}, inf{�} = ∞ (14)

This is the case of continuous default monitoring (CDM).The TDM assumption is hard to justify and apply for realistic debt structures.The DDM assumption is reasonably realistic. Under this assumption, efficient

quasi-analytical methods can be applied in one and two dimensions under the log-normal dynamics (Hull and White 2001) and in one dimension under jump-diffusiondynamics (Lipton 2003; Lipton, Song, and Lee 2007; Feng and Linetsky 2008). Numer-ical PIDE methods for the problem with DDM tend to have slower convergence ratesthan those for the problem with CDM, because the solution is not smooth at defaultmonitoring times in the vicinity of the default barrier. However, MC-based methodscan be applied in the case of DDM in a robust way, because the firm’s asset values needto be simulated only at default monitoring dates. Importantly, there is no conceptualdifficulty in applying MC simulations for the multi-dimensional model.

In the case of CDM closed-form solutions are available for the survival probabilityin one dimension (see e.g. Leland 1994; Leland and Toft 1996) and two dimensions(Zhou 2001b) for lognormal diffusions; and in one dimension for jump-diffusionswith negative jumps (see e.g. Zhou 2001a; Hilberink and Rogers 2002; Lipton 2002b;Sepp 2004, 2006). In the case of CDM, numerical FD methods in one and twodimensions tend to have a better rate of convergence in space and time than in thecase of DDM. However, a serious disadvantage of the CDM assumption is that thecorresponding MC implementation is complex and slow because continuous barriersare difficult to deal with, especially in the multi-dimensional case.

Accordingly, CDM is useful for small-scale problems which can be solved withoutMC methods, while DDM is better suited for large-scale problems, such that semi-analytical FFT or PIDE-based methods can be used to calibrate the model to marginaldynamics of individual firms and MC techniques can be used to solve the pricing prob-lem for several firms. In our experience, we have not observed noticeable differencesbetween DDM and CDM settings, provided that the model is calibrated appropriately.We note in passing that, as reported by Davidson (2008), the industry practice is to useabout 100 time steps with at least 60 steps in the first year in MC simulations of deriv-atives positions to estimate the counterparty exposure. This implies weekly defaultmonitoring frequency in the first year and quarterly monitoring in the following years.

2.2.4 Asset value, equity, and equity optionsWe introduce the log coordinate x(t):

x(t) = ln(

a(t)l (t)

)(15)




and represent the asset value as follows:

a(t) = E (t)l(0)ex(t) = l (t) ex(t) (16)

where x(t) is driven by the following dynamics under Q:

dx(t) = Ï(t)dt + Û(t)dW(t) + j d N(t) (17)

x(0) = ln(

a(0)l(0)

)≡ Ó, Ó > 0

Ï(t) = −12Û2(t) − Î(t)Í

We observe that, under this formulation of the firm value process, the default timeis specified by:

Ù = min{t : x(t) ≤ 0} (18)

triggered either discretely or continuously. Accordingly, the default event is deter-mined only by the dynamics of the stochastic driver x(t).

We note that the shifted process y(t) = x(t) − Ó is an additive process with respectto the filtration F which is characterized by the following conditions: y(t) is adaptedto F(t), increments of y(t) are independent of F(t), y(t) is continuous in probability,and y(t) starts from the origin, Sato (1999). The main difference between an additiveprocess and a Levy process is that the distribution of increments in the former processis time dependent.

Without loss of generality, we assume that volatility Û(t) and jump intensity Î(t) arepiecewise constant functions of time changing at times {tc

k }, k = 1, . . . , k:

Û(t) =k∑

k=1

Û(k)1{tck−1<t≤tc

k } + Û(k)1{t>tck} (19)

Î(t) =k∑

k=1

Î(k)1{tck−1<t≤tc

k } + Î(k)1{t>tck}

where Û(k) defines the volatility and Î(k) defines the intensity at time periods (tck−1, tc

k ]with tc

0 = 0, k = 1, . . . , k. In the case of DDM we assume that {tck } is a subset of {td

m}, sothat parameters do not jump between observation dates.

We consider the firm’s equity share price, which is denoted by s (t), and, followingStamicar and Finger (2005), assume that the value of s (t) is given by:

s (t) ={

a(t) − l(t) = E (t)l(0)(ex(t) − 1

)= l(t)

(ex(t) − 1

), {t < Ù}

0, {t ≥ Ù} (20)

At time t = 0, s (0) is specified by the market price of the equity share. Accordingly, theinitial value of the firm’s assets is given by:

a(0) = s (0) + l(0) (21)




It is important to note that Û(t) is the volatility of the firm’s assets. The volatility ofthe equity, Ûeq(t), is approximately related to Û(t) by:

Ûeq(t) =(

1 +l(t)s (t)

)Û(t) (22)

As a result, for fixed Û(t) the equity volatility increases as the spot price s (t) decreasescreating the leverage effect typically observed in the equity market. The model withequity volatility of the type (22) is also know as the displaced diffusion model, whichwas introduced by Rubinstein (1983).

2.3 Two-dimensional case

To deal with the counterparty risk problem, we need to model the correlated dynamicsof two or more credit entities. We consider two firms and assume that their asset valuesare driven by the following stochascic differential equations(SDEs):

dai (t) = (r (t) − Êi (t) − ÍiÎi (t))ai (t) dt + Ûi (t) ai (t) dWi (t) +(e ji −1

)ai (t) d Ni (t)

(23)where

Íi =∫ 0

−∞e ji �i ( ji )d ji − 1 (24)

jump amplitudes ji has the same PDF �i ( ji ) as in the marginal dynamics, jumpintensities Îi (t) are equal to the marginal intensities calibrated to single-name CDSs,volatility parameters Ûi (t) are equal to those in the marginal dynamics, i = 1, 2. Thecorresponding default boundaries have the form:

li (t) = E i (t) li (0) (25)

where

E i (t) = exp{∫ t

0(r (t ′) − Êi (t ′))dt ′

}(26)

In log coordinates with

xi (t) = ln(

ai (t)li (t)

)(27)

we obtain:

dxi (t) = Ïi (t)dt + Ûi (t) dWi (t) + ji d Ni (t) (28)

xi (0) = ln(

ai (0)li (0)

)≡ Ói , Ói > 0

Ïi (t) = −12Û2

i (t) − ÍiÎi (t)




The default time of the i-th firm, Ùi , is defined by

Ùi = min{t : xi (t) ≤ 0} (29)

Correlation between the firms is introduced in two ways. First, standard Brownianmotions W1(t) and W2(t) are correlated with correlation Ò. Second, Poisson processesN1(t), N2(t) are represented as follows:

Ni (t) = N{i} (t) + N{1,2} (t) (30)

where N{1,2} (t) is the systemic process with the intensity:

Î{1,2}(t) = max{Ò, 0} min{Î1(t), Î2(t)} (31)

while N{i}(t) are idiosyncratic processes with the intensities Î{i}(t), specified by:

Î{1}(t) = Î1(t) − Î{1,2}(t), Î{2}(t) = Î2(t) − Î{1,2}(t) (32)

This choice, which is Marshall-Olkin (1967) inspired, guarantees that marginal distri-butions are preserved, while sufficiently strong correlations are introduced naturally.

Expressing the correlation structure in terms of one parameter Ò has an advantagefor model calibration. After the calibration to marginal dynamics is completed foreach firm, and the set of firm’s volatilities, jump sizes, and intensities is obtained, weestimate the parameter Ò by fitting the model spread of a FTDS to a given market quote.

It is clear that the default time correlations are closely connected to the instanta-neous correlations of the firms’ values. For the bivariate dynamics in question, wecalculate the instantaneous correlations between the drivers x1(t) and x2(t) as follows:

ÒDNJ12 =

ÒÛ1Û2 + Î{1,2}Ì1Ì2√Û2

1 + Î1Ì21

√Û2

2 + Î2Ì22

, ÒENJ12 =

ÒÛ1Û2 + Î{1,2}/(Ì1Ì2)√Û2

1 + 2Î1/Ì21

√Û2

2 + 2Î2/Ì22

(33)

where we suppress the time variable. Here ÒDNJ12 and Ò

ENJ12 are correlations for DNJs and

ENJs, respectively.For large systemic intensities Î{1,2}, we see that Ò

DNJ12 ∼ 1, while Ò

ENJ12 ∼ 1

2 . Thus, forENJs correlations tend to be smaller than for DNJs. In our experiments with differentfirms, we have computed implied Gaussian copula correlations from model spreadsof FTDS referencing different credit entities and found that, typically, the maximalimplied Gaussian correlation that can be achieved is about 90% for DNJs and about50% for ENJs (in both cases model parameters were calibrated to match the term struc-ture of CDS spreads and CDS option volatilities). Thus, the ENJs assumption is notappropriate for modelling the joint dynamics of strongly correlated firms belonging toone industry, such as, for example, financial companies.

2.4 Multi-dimensional case

Now we consider N firms and assume that their asset values are driven by the sameequations as before, but with the index i running from 1 to N, i = 1, . . . , N.




We correlate diffusions in the usual way and assume that:

dWi (t) dWj (t) = Òi j (t) dt (34)

We correlate jumps following the Marshall-Olkin (1967) idea. Let –(N) be the set ofall subsets of N names except for the empty subset {∅}, and be its typical member.With every we associate a Poisson process N (t) with intensity Î (t), and representNi (t) as follows:

Ni (t) =∑

∈–(N)

1{i∈}N (t) (35)

Îi (t) =∑

∈–(N)

1{i∈}Î (t)

Thus, we assume that there are both systemic and idiosyncratic jump sources. Byanalogy, we can introduce systemic and idiosyncratic factors for the Brownian motiondynamics.

3 General pricing problem................................................................................................................................................

In this section we formulate the generic pricing problem in 1D, 2D, and multi-dimensions.

3.1 One-dimensional problem

For DDM, the value function V (t, x) solves the following problem on the entire axisx ∈ R

1:

Vt(t, x) + L(x)V (t, x) − r (t) V (t, x) = 0 (36)

supplied with the natural far-field boundary conditions

V (t, x) →x→±∞ ı±∞(t, x) (37)

Here tdm−1 < t < td

m. At t = tdm, the value function undergoes a transformation

Vm−(x) = –{

Vm+(x)}

(38)

where – {.} is the transformation operator, which depends on the specifics of thecontract under consideration, and Vm± (x) = V (td

m±, x). Here tdm± = td

m ± ε. Finally,at t = T

V (T, x) = v (x) (39)




the terminal payoff function v(x) is contract specific. Here L(x) is the infinitesimaloperator of process x(t) under dynamics (17):

L(x) = D(x) + Î(t)J (x) (40)

D(x) is a differential operator:

D(x)V (x) =12Û2(t)Vxx (x) + Ï(t)Vx (x) − Î (t) V (x) (41)

and J (x) is a jump operator:

J (x)V (x) =∫ 0

−∞V (x + j )� ( j )d j (42)

For CDM, we assume that the value of the contract is determined by the terminalpayoff function v(x), the cash flow function c(t, x), the rebate function z(t, x) specify-ing the payoff following the default event (we note that the rebate function may dependon the residual value of the firm), and the far-field boundary condition. The backwardequation for the value function V (t, x) is formulated differently on the positive semi-axis x ∈ R

1+ and negative semi-axis R

1−:

Vt(t, x) + L(x)V (t, x) − r (t) V (t, x) = c(t, x), x ∈ R1+

V (t, x) = z(t, x), x ∈ R1−

(43)

This equation is supplied with the usual terminal condition on R1:

V (T, x) = v(x) (44)

where J (x) is a jump operator which is defined as follows:

J (x)V (x) =∫ 0

−∞V (x + j )� ( j )d j (45)

=∫ 0

−xV (x + j )� ( j )d j +

∫ −x

−∞z(x + j )� ( j )d j

In particular,

J (x)V (x) ={

V (x − Ì) 1{Ì≤x} + z (x − Ì) 1{Ì>x}, DNJsÌ∫ 0−x V (x + j ) eÌj d j + Ì

∫ −x−∞ z (x + j ) eÌj d j, ENJs

(46)

For ENJs J (x)V (x) also can be written as

J (x)V (x) = Ì

∫ x

0V (y) eÌ(y−x)dy + Ì

∫ 0

−∞z (y) eÌ(y−x)dy (47)

In principle, for both DDM and CDM, the computational domain for x is R1.

However, for CDM, we can restrict ourselves to the positive semi-axis R1+. We can

represent the integral term in problem eq. (46) as follows:

J (x)V (x) ≡ J (x)V (x) + Z(x)(x) (48)




where J (x), Z(x)(x) are defined by:

J (x)V (x) =∫ 0

−xV (x + j )� ( j )d j (49)

Z(x)(x) =∫ −x

−∞z(x + j )� ( j )d j (50)

so that Z(x)(x) is the deterministic function depending on the contract rebate functionz(x). As a result, by subtracting Z(x) from rhs of eq. (43), we can formulate the pricingequation on the positive semi-axis R

1+ as follows:

Vt(t, x) + L(x)V (t, x) − r (t) V (t, x) = c (t, x) (51)

It is supplied with the boundary conditions at x = 0, x → ∞:

V (t, 0) = z(t, 0), V (t, x) →x→∞ ı∞(t, x) (52)

and the terminal condition for x ∈ R1+:

V (T, x) = v(x) (53)

Here

L(x) = D(x) + Î(t)J (x) (54)

c(t, x) = c(t, x) − Î (t) Z(x) (t, x) (55)

We introduce the Green’s function denoted by G (t, x, T, X), representing the prob-ability density of x(T) = X given x(t) = x and conditional on no default between t andT . For DDM the valuation problem for G can be formulated as follows:

G T (t, x, T, X) − L(X)†G (t, x, T, X) = 0 (56)

G (t, x, T, X) →X→±∞

0 (57)

G (t, x, tm+, X) = G (t, x, tm−, X) 1{X>0} (58)

G (t, x, t, X) = δ(X − x) (59)

where L(x)† being the infinitesimal operator adjoint to L(x):

L(x)† = D(x)† + Î(t)J (x)† (60)

D(x)† is the differential operator:

D(x)†g (x) =12Û2(t)g xx (x) − Ï(t)g x (x) − Î (t) g (x) (61)




and J (x)† is the jump operator:

J (x)†g (x) =∫ 0

−∞g (x − j )� ( j )d j (62)

For CDM, the PIDE for G is defined on R1+ and the boundary conditions are applied

continuously:

G T (t, x, T, X) − L(X)†G (t, x, T, X) = 0 (63)

G (t, x, T, 0) = 0, G (t, x, T, X) →X→∞

0 (64)

G (t, x, t, X) = δ(X − x) (65)

3.2 Two-dimensional problem

We assume that the specifics of the contract are encapsulated by the terminal payofffunction v(x1, x2), the cash flow function c(t, x1, x2), the rebate functions z·(t, x1, x2),· = (−, +) , (−, −) , (+, −), the default-boundary functions v0,i (t, x3−i ), i = 1, 2, andthe far-field functions v±∞,i (t, x1, x2) specifying the conditions for large values of xi .We denote the value function of this contract by V (t, x1, x2).

For DDM, the pricing equation defined in the entire plane R2 can be written as

follows:

Vt(t, x1, x2) + L(x)V (t, x1, x2) − r (t) V (t, x1, x2) = 0 (66)

As before, it is supplied with the far-field conditions

V (t, x1, x2) →xi →±∞ ı±∞,i (t, x1, x2), i = 1, 2 (67)

At times tdm the value function is transformed according to the rule

Vm−(x1, x2) = –{

Vm+(x1, x2)}

(68)

The terminal condition is

V (T, x1, x2) = v(x1, x2) (69)

Here L(x1,x2) is the infinitesimal backward operator corresponding to the bivariatedynamics (28):

L(x1,x2) = D(x1) + D(x2) + C(x1,x2) (70)

+Î{1}(t)J (x1) + Î{2}(t)J (x2) + Î{1,2}(t)J (x1,x2)

Here, D(x1) and D(x2) are the differential operators in x1 and x2 directions defined byeq. (41) with Î (t) = Î{i} (t); J (x1) and J (x2) are the 1D orthogonal integral operators inx1 and x2 directions defined by eq. (45) with appropriate model parameters; C(x1,x2) isthe correlation operator:




C(x1,x2)V (x1, x2) ≡ ÒÛ1(t)Û2(t)Vx1x2 (x1, x2) − Î{1,2} (t) V (x1, x2) (71)

and J (x1,x2) is the cross integral operator defined as follows:

J (x1,x2)V (x1, x2) ≡∫ 0

−∞

∫ 0

−∞V (x1 + j1, x2 + j2)�1( j1)�2( j2)d j1d j2 (72)

For CDM, V (t, x1, x2) solves the following problem in the positive quadrant R2+,+:

Vt(t, x1, x2) + L(x1,x2)V (t, x1, x2) − r (t) V (t, x1, x2) = c(t, x1, x2) (73)

V (t, 0, x2) = v0,1(t, x2), V (t, x1, x2) →x1→∞ v∞,1(t, x1,x2) (74)

V (t, x1, 0) = v0,2(t, x1), V (t, x1, x2) →x2→∞ v∞,2(t, x1, x2)

V (T, x1, x2) = v(x1, x2) (75)

where L(x1,x2) is the infinitesimal backward operator defined by:

L(x1,x2) = D(x1) + D(x2) + C(x1,x2) (76)

+Î{1}(t)J (x1) + Î{2}(t)J (x2) + Î{1,2}(t)J (x1,x2)

with

J (x1,x2)V (x1, x2) ≡∫ 0

−x1

∫ 0

−x2

V (x1 + j1, x2 + j2)�1( j1)�2( j2)d j1d j2 (77)

The ‘equivalent’ cash flows can be represented as follows:

c(t, x1, x2) = c(t, x1, x2) − Î{1} (t) Z(x1) (t, x1, x2) − Î{2} (t) Z(x2) (t, x1, x2) (78)

−Î{1,2}(

Z(x1,x2)−,+ (t, x1, x2) + Z(x1,x2)

−,− (t, x1, x2) + Z(x1,x2)+,− (t, x1, x2)

)

where

Z(x1) (t, x1, x2) =∫ −x1

−∞z−,+(x1 + j1, x2)�1( j1)d j1 (79)

Z(x2) (t, x1, x2) =∫ −x2

−∞z+,−(x1, x2 + j2)�2( j2)d j2

Z(x1,x2)−,+ (x1, x2) =

∫ −x1

−∞

∫ 0

−x2

z−,+(x1 + j1, x2 + j2)�1( j1)�2( j2)d j1d j2

Z(x1,x2)−,− (x1, x2) =

∫ −x1

−∞

∫ −x2

−∞z−,−(x1 + j1, x2 + j2)�1( j1)�2( j2)d j1d j2

Z(x1,x2)+,− (x1, x2) =

∫ 0

−x1

∫ −x2

−∞z+,−(x1 + j1, x2 + j2)�1( j1)�2( j2)d j1d j2




For DDM, the corresponding Green’s function G (t, x1, x2, T, X1, X2), satisfies thefollowing problem in the whole plane R

2:

G T (t, x1, x2, T, X1, X2) − L(X1,X2)†G (t, x1, x2, T, X1, X2) = 0 (80)

G (t, x1, x2, T, X1, X2) →Xi →±∞

0 (81)

G (t, x1, x2, tm+, X1, X2) = G (t, x1, x2, tm−, X1, X2) 1{X1>0,X2>0} (82)

G (t, x1, x2, t, X1, X2) = δ(X1 − x1)δ(X2 − x2) (83)

where L(x1,x2)† is the operator adjoint to L(x1,x2):

L(x1,x2)† = D(x1)† + D(x2)† + C(x1,x2) (84)

+Î{1}(t)J (x1)† + Î{2}(t)J (x2)† + Î{1,2}(t)J (x1,x2)†

and

J (x1,x2)†g (x1, x2) =∫ 0

−∞

∫ 0

−∞g (x1 − j1, x2 − j2)�1( j1)�2( j2)d j1d j2 (85)

For CDM, the corresponding Green’s function satisfies the following problem in thepositive quadrant R

2+,+:

G T (t, x1, x2, T, X1, X2) − L(X1,X2)†G (t, x1, x2, T, X1, X2) = 0 (86)

G (t, x1, x2, T, 0, X2) = 0, G (t, x1, x2, T, X1, 0) = 0 (87)

G (t, x1, x2, T, X1, X2) →Xi →∞

0

G (t, x1, x2, t, X1, X2) = δ(X1 − x1)δ(X2 − x2) (88)

3.3 Multi-dimensional problem

For brevity, we restrict ourselves to CDM. As before, we can formulate a typical pricingproblem for the value function V (t, �x) in the positive cone R

N+ as follows:

Vt (t, �x) + L(�x)V (t, �x) − r (t) V (t, �x) = c (t, �x) (89)

V(t, �x0,k

)= v0,k (t, �yk), V (t, �x) →

xk→∞ v∞,k (t, �x) (90)

V (T, �x) = v (�x) (91)

where �x , �x0,k , �yk are N and N − 1 dimensional vectors, respectively,

�x = (x1, . . . , xk, . . . xN)

�x0,k =(

x1, . . . , 0k, . . . xN

)

�yk = (x1, . . . xk−1, xk+1, . . . xN)

(92)




Here c (t, �x), v0,k (t, �y), v∞,k (t, �x), v (�x) are known functions which are contractspecific. The function c (t, �x) incorporates the terms arising from rebates. The cor-responding operator L(�x) can be written in the form

L(�x) f (�x) =12

∑i

Û2i ∂

2i f (�x) +

∑i, j, j>i

Òi j ÛiÛ j ∂i∂ j f (�x) (93)

+∑

i

Ïi∂i f (�x) +∑

∈–(N)

Î

(∏i∈

J (xi ) f (�x) − f (�x)

)

where

J (xi ) f (�x) =

⎧⎨⎩

f (x1, . . . , xi − Ìi , . . . xN), xi > Ìi

0 xi ≤ Ìi, DNJs

Ìi

∫ 0−xi

f (x1, . . . , xi + ji , . . . xN) eÌi ji d ji , ENJs(94)

3.4 Green’s formula

Now we can formulate Green’s formula adapted to the problem under consideration.To this end we introduce the Green’s function G

(t, �x, T, �X

), such that

G T

(t, �x, T, �X

)− L( �X)†G

(t, �x, T, �X

)= 0 (95)

G(

t, �x, T, �X0k

)= 0, G

(t, �x, T, �X

)→

Xk→∞0 (96)

G(

t, �x, t, �X)

= δ(

�X − �x)

(97)

Here L(�x)† is the corresponding adjoint operator

L(�x)†g (�x) =12

∑i

Û2i ∂

2i g (�x) +

∑i, j, j>i

Òi j ÛiÛ j ∂i∂ j g (�x) (98)

−∑

i

Ïi∂i g (�x) +∑

∈–(N)

Î

(∏i∈

J (xi )†g (�x) − g (�x)

)

where

J (xi )†g (�x) ={

g (x1, . . . , xi + Ìi , . . . xN), DNJsÌi

∫ 0−∞ g (x1, . . . , xi − ji , . . . xN) eÌi ji d ji , ENJs

(99)

It is easy to check that for both DNJs and ENJs the following identity holds:∫

RN+

[J (xi ) f (�x) g (�x) − f (�x)J (xi )†g (�x)

]d�x = 0 (100)




Accordingly, integration by parts yields

V (t, �x) = −∫ T

t

∫RN

+

c(t ′, �x ′) D

(t, t ′)G

(t, �x, t ′, �x ′) d�x ′dt ′ (101)

+∑

k

∫ T

t

∫R

N−1+

v0,k

(t ′, �y ′

k

)D(t, t ′) gk

(t, �x, t ′, �y ′

k

)d�y ′

kdt ′

+D (t, T)∫

RN+

v(�x ′)G

(t, �x, T, �x ′) d�x ′

where

gk

(t, �x, T, �Yk

)=

12Û2

k∂k G(

t, �x, T, �X)∣∣∣∣

Xk =0(102)

�Yk = (X1, . . . , Xk−1, Xk+1, . . . , X N)

represents the hitting time density for the corresponding point of the boundary. Inparticular, the initial value of a claim has the form

V(

0, �Ó)

= −∫ T

0

∫RN

+

c(t ′, �x ′) D

(0, t ′)G

(0, �Ó, t ′, �x ′

)d�x ′dt ′ (103)

+∑

k

∫ T

0

∫R

N−1+

v0,k

(t ′, �y ′

k

)D(0, t ′) gk

(0, �Ó, t ′, �y ′

k

)d�y ′

kdt ′

+D (0, T)∫

RN+

v(�x ′)G

(0, �Ó, T, �x ′

)d�x ′

This extremely useful formula shows that instead of solving the backward pricingproblem with inhomogeneous right hand side and boundary conditions, we can solvethe forward propagation problem for the Green’s function with homogeneous rhs andboundary conditions and perform the integration as needed.

4 Pricing problem for credit derivatives................................................................................................................................................

In this section we formulate the computational problem for several important creditproducts. We also formulate the CVA problem for CDSs.

4.1 Survival probability

The single-name survival probability function, Q(x)(t, x, T), is defined as follows:

Q(x)(t, x, T) ≡ 1{Ù>t}EQt [1{Ù>T}] (104)




Using the default event definition (13), one can show that for DDM, Q(x)(t, x, T) solvesthe following backward problem on R

1:

Q(x)t (t, x, T) + L(x) Q(x)(t, x, T) = 0 (105)

Q(x)(t, x, T) →x→−∞ 0, Q(x)(t, x, T) →

x→∞ 1 (106)

Q(x)m−(x, T) = Q(x)

m+(x, T)1{x>0} (107)

Q(x)(T, x, T) = 1{x>0} (108)

with the infinitesimal operator L(x) defined by eq. (40).Likewise, using the default event definition (14), one can show that for CDM,

Q(x)(t, x, T) solves the following backward problem on the positive semi-axis R1+:

Q(x)t (t, x, T) + L(x) Q(x)(t, x, T) = 0 (109)

Q(x)(t, 0, T) = 0, Q(x)(t, x, T) →x→∞ 1 (110)

Q(x)(T, x, T) = 1 (111)

Here the far field condition for x → ∞ expresses the fact that for large values of xsurvival becomes certain. Green’s formula (101) yields

Q(x) (t, x, T) =∫ ∞

0G (t, x, T, X) d X (112)

We define the joint survival probability, Q(x1,x2)(t, x1, x2, T), as follows:

Q(x1,x2)(t, x1, x2, T) ≡ 1{Ù1>t,Ù2>t}EQt [1{Ù1>T,Ù2>T}] (113)

For DDM, the joint survival probability function Q(x1,x2)(t, x1, x2) solves the follow-ing problem:

Q(x1,x2)t + L(x1,x2) Q(x1,x2) = 0 (114)

Q(x1,x2)(t, x1, x2, T) →xi →−∞ 0 (115)

Q(x1,x2)(t, x1, x2, T) →xi →∞ Q(x3−i )(t, x3−i , T)

Q(x1,x2)m− (x1, x2, T) = Q(x1,x2)

m+ (x1, x2, T)1{x1>0,x2>0} (116)

Q(x1,x2)(T, x1, x2, T) = 1{x1>0,x2>0} (117)

where the infinitesimal operator L(x1,x2) is defined by eq. (70). Here Q(xi )(t, xi , T),i = 1, 2, are the marginal survival probabilities obtained by solving eq. (105).




For CDM, Q(x1,x2)(t, x1, x2) solves the following problem:

Q(x1,x2)t (t, x1, x2, T) + L(x1,x2) Q(x1,x2)(t, x1, x2, T) = 0 (118)

Q(x1,x2)(t, x1, 0, T) = 0, Q(x1,x2) (t, 0, x2, T) = 0 (119)

Q(x1,x2)(t, x1, x2, T) →xi →∞ Q(x3−i )(t, x3−i , T)

Q(x1,x2)(T, x1, x2, T) = 1 (120)

where the infinitesimal operator L(x1,x2) is defined by eq. (76). As before

Q(x1,x2) (t, x1, x2, T) =∫ ∞

0

∫ ∞

0G (t, x1, x2, T, X1, X2) d X1d X2 (121)

4.2 Credit default swap

A CDS is a contract designed to exchange the credit risk of RN between PB and PS.PB makes periodic coupon payments to PS, conditional on no default of RN up to thenearest payment date, in exchange for receiving loss given RN’s default from PS. Forstandardized CDS contracts, coupon payments occur quarterly on the 20th of March,June, September, and December. We denote the annualized payment schedule by {tm},m = 1, . . ., M. The most liquid CDSs have maturities of 5y, 7y, and 10y.

We consider a CDS with the unit notional providing protection from the currenttime t up to the maturity time T . Assuming that RN has not defaulted yet, Ù > t , wecompute the expected present value of the annuity leg, A(t, T), as:

A(t, T) =∫ T

tD(t, t ′) Q(t, t ′)dt ′ (122)

where Q(t, t ′) is the corresponding survival probability, and the expected present valueof the protection leg, P (t, T), as:

P (t, T) = − (1 − R)∫ T

tD(t, t ′) d Q(t, t ′) (123)

= (1 − R)(

1 − D (t, T) Q (t, T) −∫ T

tr(t ′) D

(t, t ′) Q(t, t ′)dt ′

)

where R is the expected debt recovery rate which is assumed to be given for valuationpurposes (typically, R is fixed at 40%).

For PB the present value of the CDS contract with coupon (or spread) c , is given by:

V C DS(t, T) = P (t, T) − c A(t, T) (124)




The par coupon c(0, T) is defined in such a way the time t = 0 the value of the CDScontract is zero:

c(0, T) =P (0, T)A(0, T)

(125)

The market-standard computation of the value of a CDS relies on the reduced-form approach (see, for example, Jarrow and Turnbull 1995; Duffie and Singleton 1997;Lando 1998; Hull and White 2000). Typically, a piecewise constant hazard rate is usedto parametrize the risk-neutral survival probability of RN. Hazard rates are inferredfrom the term structure of CDS spreads via bootstrapping.

One of the drawbacks of the reduced-form approach is that it assumes that CDSspreads are static and evolve deterministically along with hazard rates. Importantly,this approach does not tell us how CDS spreads change when the RN’s value changes.In contrast, the structural approach does explain changes in the term structure of CDSspreads caused by changes in the firm’s value. Thus, the structural model can be usedfor valuing credit contracts depending on the volatility of credit spreads.

For DDM, the value function for PB of a CDS contract, V C DS(t, x, T), solves eq.(36), supplied with the following conditions:

V C DS(t, x, T) →x→−∞ 1 − Rex , V C DS(t, x, T) →

x→∞ −cM∑

m′=m+1

‰tm′ D(t, tm′) (126)

V C DSm− (x, T) = (V C DS

m+ (x, T) − ‰tmc)1{x>0} + (1 − Rex )1{x≤0} (127)

V C DS (T, x, T) = −‰tMc1{x>0} + (1 − Rex )1{x≤0} (128)

where ‰tm = tm − tm−1.For CDM, V C DS(t, x, T) solves eq. (51) with

c (t, x) = c − Î (t) Z(x) (x) (129)

Z(x)(x) ={

H(Ì − x)(1 − Re x−Ì

), DNJs(

1 − R Ì1+Ì

)e−Ìx , ENJs (130)

Here we assume that the floating recovery rate, Rex , represents the residual value of thefirm’s assets after the default. The corresponding boundary and terminal conditionsare

V C DS(t, 0, T) = (1 − R), V C DS(t, x, T) →x→∞ −c

∫ T

tD(t, t ′)dt ′ (131)

V C DS(T, x, T) = 0 (132)

The boundary condition for x → ∞ expresses the fact that for large positive x thepresent value of CDS becomes a risk-free annuity.




4.3 Credit default swap option

A CDSO contract serves as a tool for locking in the realized volatility of CDS rate upto the option’s maturity. By using CDSOs quotes we can calibrate the model to thisvolatility. The payer CDSO with maturity Te and tenor Tt gives its holder the right toenter in a CDS contract providing the protection starting at time Te up to time Te + Tt

with a given coupon K . The option knocks out if RN defaults before time Te . Thus, thepayout of the payer CDSO is given by:

V C DS(Te , Te + Tt ; K ) = 1{Ù>Te } {P (Te , Te + Tt) − K A (Te , Te + Tt)}+ (133)

For DDM, the value function for the buyer of CDSO, V C DS O (t, x, T), solves eq.(36) with c = 0, and the following conditions:

V C DS O (t, x) →x→±∞ 0 (134)

V C DS Om− (x) = V C DS O

m+ (x) 1{x>0} (135)

V C DS O (Te , x) ={

V C DS(Te , x, Te + Tt ; K )}

+ 1{x>0} (136)

For CDM, V C DS O (t, x) is governed by eq. (51) with c = 0, supplied with the follow-ing conditions:

V C DS O (t, 0) = 0, V C DS O (t, x) →x→∞ 0 (137)

V C DS O (Te , x) ={

V C DS(Te , x, Te + Tt ; K )}

+ (138)

4.4 Equity put option

In our model, we can value European style options on the firm’s equity defined byeq. (20). In the context of studying credit products, the value of the equity put optionis the most relevant one, since such options provide protection against the depreciationof the stock price and can be used for hedging against the default event.

For DDM, the value function of V P (t, x) solves eq. (36) with c = 0, supplied withthe following conditions:

V P (t, x) →x→−∞ D(t, T)K , V P (t, x) →

x→∞ 0 (139)

V Pm− (x) = V P

m+ (x) 1{x>0} + D (tm, T) K 1{x≤0} (140)

V P (T, x) ={

K − l(T) (ex − 1)}

+ 1{x>0} + K 1{x≤0} (141)

For CDM, V P (t, x) solves eq. (51) with

c (t, x) = −Î (t) Z(x) (t, x) (142)




Z(x) (t, x) ={

D (t, T) K H (Ì − x) , DNJsD (t, T) K e−Ìx ENJs (143)

and the following conditions:

V P (t, 0) = D(t, T)K , V P (t, x) →x→∞ 0 (144)

V P (T, x) ={

K − l(T) (ex − 1)}

+ (145)

We note that, in this model, put-call parity for European options should beexpressed in terms of defaultable forward contracts.

Since, in general, we have to solve the pricing problem numerically, American styleoptions can be handled along similar lines with little additional effort.

4.5 First-to-default swap

An FTDS references a basket of RNs. Similarly to a regular CDS, PB of an FTDS paysPS a periodic coupon up to the first default event of any of RNs, or the swap’s maturity,whichever comes first; in return, PS compensates PB for the loss caused by the firstdefault in the basket. The market of FTDSs is relatively liquid with a typical basket sizeof five underlying names.

In this chapter we consider FTDSs referencing just two underlying names. Thepremium leg and the default leg of a FTDS are structured by analogy to the single-name CDS. For brevity we consider only CDM. To compute the present valueV F T DS(t, x1, x2, T) for PB of a FTDS, we have to solve eq. (73) with c (t, x1, x2) ofthe form:

c (t, x1, x2) = c − Î{1} (t) Z(x1) (x1) − Î{2} (t) Z(x2) (x2) (146)

−Î{1,2} (t)(

Z(x1,x2)−,+ (x1, x2) + Z(x1,x2)

−,− (x1, x2) + Z(x1,x2)+,− (x1, x2)

)

Z(x1,x2)−,+ (x1, x2) = Z(x1) (x1) (147)

Z(x1,x2)−,− (x1, x2) =

12[

Z(x1) (x1) + Z(x2) (x2)]

Z(x1,x2)+,− (x1, x2) = Z(x2) (x2)

where Z(xi ) (xi ) are given by eq. (130). Here we assume that in case of simultaneousdefault of both RNs, PB receives the notional minus their average recovery. Thecorresponding boundary and terminal conditions are

V F T DS(t, 0, x2, T) = 1 − R1, V F T DS(t, x1, 0, T) = 1 − R2 (148)

V F T DS(t, x1, x2, T) →xi →∞ V C DS(t, x3−i , T) (149)

V F T DS(T, x1, x2, T) = 0 (150)




The market practice is to quote the spread on a FTDS by using the Gaussiancopula with specified pair-wise correlation, Ò, between default times of RNs (see, forexample, Li 2000 and Hull, Nelkin, and White 2004). Thus, we can calibrate the modelcorrelation parameter to FTDS spreads observed in the market.

4.6 Credit default swap with counterparty risk

4.6.1 Credit value adjustmentFirst, we consider a CDS contract sold by a risky PS to a non-risky PB. We denote byÙ1 the default time of RN and by Ù2 the default time of PS. We assume that this CDSprovides protection up to time T and its coupon is c . We also assume that the recoveryrate of RN is R1 and of PS is R2.

We denote by V C DS(t ′, T) the value of the CDS contract with coupon c maturingat time T conditional on PS defaulting at time t ′. We make the following assumptionsabout the recovery value of the CDS given PS default at time Ù2: if V C DS (t ′, T) < 0,PB pays the full amount of −V C DS (t ′, T) to PS; if V C DS (t ′, T) > 0, PB receives onlya fraction R2 of V C DS (t ′, T).

Thus, CVA for PB, V C V AP B (t, T), is defined as the expected maximal potential loss

due to the PS default:

V C V AP B (t, T) = E

Qt

[∫ T

tD(t, t ′)(1 − R2)

{V C DS

(t ′, T

)}+

dt ′]

(151)

Accordingly, to value CVA we need to know the survival probability Q (t ′, t ′′) forRN conditional on PS default at time t ′. In this context, Pykhtin and Zhu (2006) andMisirpashaev (2008) applied the Gaussian copula model, while (Blanchet-Scalliet andPatras (2008) applied a bivariate lognormal structural model to calculate the relevantquantity.

Similarly, for a CDS contract sold by a non-risky PS to a risky PB we have thefollowing expression for CVA for PS:

V C V AP S (t, T) = E

Qt

[∫ T

tD(t, t ′)(1 − R3)

{−V C DS

(t ′, T

)}+

dt ′]

(152)

where R3 is the recovery rate for PB.How to calculate CVA when both PS and PB are risky is not completely clear as of

this writing.

4.6.2 Credit value adjustment in the structural frameworkWe start with the risky PS case and denote by x1 the driver for the RN’s value and byx2 the driver for the PS’s value.

In the context of the structural default model, the 2D plane is divided into fourquadrants as follows: (A) R

2+,+, where both RN and PS survive; (B) R

2−,+, where RN




defaults and PS survives; (C) R2−,−, where both the reference name and the coun-

terparty default; (D) R2+,−, where the reference name survives while the counterparty

defaults. In R2−,+ CVA is set zero, because PS is able to pay the required amount to PB.

In R2−,− CVA is set to the fraction of the payout which is lost due to the counterparty

default, (1 − R1ex1 ) (1 − R2ex2 ). In R2+,− CVA is set to (1 − R2ex2 )

{V C DS(t, x1, T)

}+,

where V C DS(t, x1, T) is the value of the CDS on RN at time t and state x1, becausethe CDS protection is lost following PS default. The value of CVA is computed as thesolution to a 2D problem with given rebates in regions R

2−,+, R

2−,−, and R

2+,−.

For DDM, the value of CVA, V C V AP B (t, x1, x2, T), satisfies eq. (66) and the following

conditions

V C V AP B (t, x1, x2, T) →

x1→−∞ (1 − R1ex1 )(1 − R2ex2 )1{x2≤0} (153)

V C V AP B (t, x1, x2, T) →

x1→∞ 0

V C V AP B (t, x1, x2, T) →

x2→−∞ (1 − R2ex2 ){

V C DS(t, x1, T)}

+

V C V AP B (t, x1, x2, T) →

x2→∞ 0

V C V AP B,m− (x1, x2) = V C V A

P B,m+(x1, x2)1{x1>0,x2>0} (154)

+(1 − R1ex1 )(1 − R2ex2 )1{x1≤0,x2≤0}

+(1 − R2ex2 ){

V C DS(t, x1, T)}

+ 1{x1>0,x2≤0}

V C V AP B (T, x1, x2, T) = 0 (155)

where V C DS(t, x1, T) is the value of the non-risky CDS computed by solving thecorresponding 1D problem.

For CDM, we have to solve eq. (73) with c (t, x1, x2) of the form:

c (t, x1, x2) = −Î{2} (t) Z(x2) (t, x1, x2) (156)

−Î{1,2} (t)(

Z(x1,x2)−,− (t, x1, x2) + Z(x1,x2)

+,− (t, x1, x2))

where

Z(x2) (t, x1, x2) ={

V C DS(t, x1, T)}

+ Z(x2) (x2) (157)

Z(x1,x2)−,− (t, x1, x2) = Z(x1) (x1) Z(x2) (x2)

Z(x1,x2)+,− (t, x1, x2) = κ (t, x1) Z(x2) (x2)

Z(xi ) (xi ) are given by eq. (130), and

κ (t, x1) =

{H(x1 − Ì1)

{V C DS(t, x1 − Ì1, T)

}+ DNJs

Ì1∫ 0−x1

{V C DS(t, x1 + j1, T)

}+ eÌ1 j1 d j1 ENJs

(158)




The corresponding boundary and final conditions are

V C V AP B (t, 0, x2, T) = 0, V C V A

P B (t, x1, x2, T) →x1→∞ 0 (159)

V C V AP B (t, x1, 0, T) = (1 − R2)

{V C DS(t, x1, T)

}+

V C V AP B (t, x1, x2, T) →

x2→∞ 0

V C V AP B (T, x1, x2, T) = 0 (160)

For risky PB, the formulation is similar and we leave it to the reader.

5 One-dimensional problem................................................................................................................................................

5.1 Analytical solution

In this section we derive some analytic solutions for jump-diffusion dynamics withENJs. Unfortunately, similar solutions for DNJs are not readily available. Resultspresented in this section rely on certain exceptional features of the exponential dis-tribution and do not extend to other jump distributions. In this section, we assumeconstant model parameters, CDM, and restrict ourselves to ENJs. In more generalcases, we need to solve the corresponding problems directly. Analytical results canserve as a useful tool for testing the accuracy of numerical calculations needed for lessrestrictive cases.

5.1.1 Green’s functionDue to the timehomogeneity of the problem under consideration, the Green’s functionG (t, x, T, X) depends on Ù = T − t rather than on t, T separately, so that we canrepresent it as follows:

G (t, x, T, X) = √ (Ù, x, X) (161)

where √ (Ù, x, X) solves the following problem:

√Ù (Ù, x, X) − L(X)†√ (Ù, x, X) = 0 (162)

√(Ù, x, 0) = 0, √(Ù, x, X) →X→∞

0 (163)

√(0, x, X) = δ(X − x) (164)

The Laplace transform of √(Ù, x, X) with respect to Ù

√(Ù, x, X) → √ ( p, x, X) (165)




solves the following problem:

p√ ( p, x, X) − L(X)†√ ( p, x, X) = δ (X − x) (166)

√ ( p, x, 0) = 0, √ ( p, x, X) →X→∞

0 (167)

The corresponding forward characteristic equation is given by:

12Û2¯2 − Ï¯ − (Î + p) +

ÎÌ

−¯ + Ì= 0 (168)

This equation has three roots, which, to facilitate comparison with earlier work, wedenote by −¯ j , j = 1, 2, 3. It is easy to show that these roots can be ordered in such away that ¯1 < −Ì < ¯2 < 0 < ¯3. Hence, the overall solution has the form:

√ ( p, x, X) =

⎧⎨⎩

C3e−¯3(X−x), X ≥ x3∑

j =1D j e−¯ j (X−x), 0 < X ≤ x

(169)

where

Di = − 2Û2

(Ì + ¯i )(¯i − ¯3−i ) (¯i − ¯3)

, i = 1, 2 (170)

D3 = −e (¯1−¯3)x D1 − e (¯2−¯3)x D2, C3 = D1 + D2 + D3

The inverse Laplace transform of √ ( p, x, X) yields √ (Ù, x, X). A review of relevantalgorithms can be found in Abate, Choudhury, and Whitt (1999).

Without jumps, all the above formulas can be calculated explicitly. Specifically,method of images yields:

√ (Ù, x, X) =e−ˇ/8−(X−x)/2

√ˇ

(n(

X − x√ˇ

)− n

(X + x√

ˇ

))(171)

where ˇ = Û2Ù.

5.1.2 Survival probabilityBy using eqs. (112), (169) we compute the Laplace-transformed survival probability asfollows:

Q(x) (Ù, x) → Q(x) ( p, x) (172)

Q(x) ( p, x) =∫ ∞

0√ ( p, x, X) d X (173)

=∫ ∞

xC3e−¯3(X−x)d X +

3∑j =1

∫ x

0D j e

−¯ j (X−x)d X =2∑

j =0

E j e¯ j x




where

¯0 = 0, E 0 =1p, E 1 =

(¯1 + Ì) ¯2

(¯1 − ¯2) Ìp, E 2 =

(¯2 + Ì) ¯1

(¯2 − ¯1) Ìp(174)

This result was first obtained by Lipton (2002b).The default time density can be defined as follows:

q (Ù, x) = −∂ Q(x)(Ù, x)∂Ù

(175)

Using eq. (112) we obtain:

q (Ù, x) = −∫ ∞

0

∂√(Ù, x, X)∂Ù

d X = g (Ù, x) + f (Ù, x) (176)

where g (Ù, x) is the probability density of hitting the barrier:

g (Ù, x) =Û2

2∂√(Ù, x, X)

∂ X

∣∣∣∣X=0

(177)

and f (Ù, x) is the probability of the overshoot:

f (Ù, x) = Î

∫ ∞

0

(∫ −X

−∞� ( j )d j

)√(Ù, x, X)d X (178)

Formula (178) is generic and can be used for arbitrary jump size distributions. ForENJs, we obtain:

f (Ù, x) = Î

∫ ∞

0e−ÌX√(Ù, x, X)d X (179)

Using eq. (169), the Laplace-transformed default time density can be represented as:

q ( p, x) = g ( p, x) + f ( p, x) (180)

where

g ( p, x) =(Ì + ¯2)e¯2x − (Ì + ¯1)e¯1x

¯2 − ¯1(181)

and

f ( p, x) =2Î(e¯2x − e¯1x

)Û2(¯2 − ¯1)(Ì + ¯3)

(182)

Alternatively, by taking the Laplace transform of eq. (175) and using eq. (173), weobtain:

q ( p, x) =(¯1 + Ì) ¯2e¯1x

(¯2 − ¯1) Ì+

(¯2 + Ì) ¯1e¯2x

(¯1 − ¯2) Ì(183)

Straightforward but tedious algebra shows that expressions (180)–(182) and (183) are inagreement.




Without jumps, straightforward calculation yields

Q(x) (Ù, x) = N

(x√ˇ

−√

ˇ

2

)− ex N

(− x√

ˇ−

√ˇ

2

)(184)

= e−ˇ/8+x/2

(P

(x√ˇ, −

√ˇ

2

)− P

(− x√

ˇ, −

√ˇ

2

))

and,

q (Ù, x) = g (Ù, x) =x

Ù√

ˇn

(x√ˇ

−√

ˇ

2

), f (Ù, x) = 0 (185)

5.1.3 Credit default swapWe use the general formula (124) together with eq. (176), and express the present valueV C DS(Ù, x) of a CDS contract with coupon c as:

V C DS(Ù, x) = −c

∫ Ù

0e−r Ù′

Q(x)(Ù′, x)dÙ′ (186)

+(1 − R)∫ Ù

0e−r Ù′

g (Ù′, x)dÙ′ +(

1 − RÌ

1 + Ì

) ∫ Ù

0e−r Ù′

f (Ù′, x)dÙ′

By using eqs. (173), (181), (182), we can compute the value of the CDS via the inverseLaplace transform.

Without jumps V C DS(Ù, x) can be found explicitly. The annuity leg can be repre-sented in the form

A (Ù, x) =1r

(1 − e−r Ù

(Q(x) (Ù, x) + e−ˇ/8+x/2

(P

(− x√

ˇ,„√

ˇ

2

)+ P

(− x√

ˇ,−„

√ˇ

2

))))

(187)where „ =

√8r/Û2 + 1, while the protection leg can be represented as follows

P (Ù, x) = (1 − R)(1 − e−r Ù Q(x) (Ù, x) − r A (Ù, x)

)(188)

Accordingly,

V C DS(Ù, x) = (1 − R)(1 − e−r Ù Q(x) (Ù, x)

)− ((1 − R) r + c) A (Ù, x) (189)

5.1.4 Credit default swap optionIn the time-homogeneous setting of the present section, we can represent the price ofa CDSO as follows

V C DS O (Ùe , x) = e−r Ùe

∫ X∗

0√ (Ùe , x, X) V C DS(Ùt, X)d X (190)

where X∗ is chosen in such a way that V C DS(Ùt, X∗) = 0. We can use our previousresults to evaluate this expression via the Laplace transform.

As before, without jumps V C DS O can be evaluated explicitly.




5.1.5 Equity put optionWe use eq. (139) and represent the value of the put option with strike K and maturityT as follows:

V P (Ù, Ó, T) = e−r Ù[V P

0 (Ù, Ó, T) + K(1 − Q(x) (Ù, Ó)

)](191)

The Laplace transform of V P0 (Ù, Ó, T) is given by

V P0 ( p, Ó, T) =

∫ ∞

0

{K − l(T)

(e X − 1

)}+ √ ( p, Ó, X) d X (192)

Straightforward calculation yields:

V P0 ( p, Ó, T) = l (T)

3∑j =1

D j e¯ j Ó

(ek(T) − e(1−¯ j )k(T)

¯ j+

1 − e (1−¯ j )k(T)

1 − ¯ j

)(193)

for out-of-the-money puts with Ó ≥ k (T), and

V P0 ( p, Ó, T) = l(T)

⎧⎨⎩

3∑j =1

D j e¯ j Ó

(ek(T) − ek(T)−¯ j Ó

¯ j+

1 − e (1−¯ j )Ó

1 − ¯ j

)(194)

+C3e¯3Ó

(ek(T)−¯3Ó − e (1−¯3)k(T)

¯3+

e (1−¯3)Ó − e (1−¯3)k(T)

1 − ¯3

)}

for in-the-money puts with Ó < k (T). Here k (T) = ln ((K + l (T)) / l (T)).Without jumps, we can find V P

0 (Ù, Ó, T), and hence the price of a put option,explicitly:

V P0 (Ù, Ó, T) = l(T)e−ˇ/8+Ó/2

{ek(T)/2

[P

(k (T) − Ó√

ˇ,

√ˇ

2

)− P

(k (T) + Ó√

ˇ,

√ˇ

2

)

(195)

−P

(k (T) − Ó√

ˇ, −

√ˇ

2

)+ P

(k (T) + Ó√

ˇ, −

√ˇ

2

)]

−ek(T)

[P

(− Ó√

ˇ,

√ˇ

2

)− P

(Ó√ˇ,

√ˇ

2

)]

+P

(− Ó√

ˇ, −

√ˇ

2

)− P

(Ó√ˇ, −

√ˇ

2

)}

5.1.6 ExampleIn Figure 12.1 we illustrate our findings. We use the following set of parameters: a(0) =200, l(0) = 160, s (0) = 40, Ó = 0.22, r = Ê = 0, Û = 0.05, Î = 0.03, Ì = 1/Ó. We compareresults obtained for the jump-diffusion model with the ones obtained for the diffu-sion model the ‘equivalent’ diffusion volatility Ûdiff specified by Ûdiff =

√Û2 + 2Î/Ì2,




0.60

0.70

0.80

0.90

1.00

T

Surv

ival

Pro

babi

lity

s(T), Jump-diffusions(T), Diffusion

0

40

80

120

160

200

240

280

320

360

T

CDS

spre

ad, b

p s(T), Jump-diffusions(T), Diffusion

30%35%40%45%50%55%60%65%70%75%80%

K/S

CDS

Impl

ied

Vola

tilit

y

JD D

10%

50%

90%

130%

170%

210%

0 1 62 3 4 5 7 8 9 0 1 2 3 4 5 6 7 8 9

60% 80% 100% 120% 140% 160% 20% 40% 60% 80% 100% 120%

K/S

Equi

ty Im

plie

d Vo

lati

lity

JD, T = 1m D, T = 1mJD, T = 6m D, T = 6mJD, T = 12m D, T = 12m

figure 12.1 The model implied survival probabilities for 1y ≤ T ≤ 10y (top lhs); CDS spreadsfor 1y ≤ T ≤ 10y (top rhs); volatility skew for CDSO (bottom lhs); volatility skew for putoptions with T = 1m, 6m, 12m (bottom rhs).

Ûdiff = 0.074 for the chosen model parameters. First, we show the term structure ofthe implied spread generated by the jump-diffusion and diffusion models. We see that,unlike the diffusion model, the jump-diffusion model implies a non-zero probability ofdefault in the short term, so that its implied spread is consistent with the one observedin the market. If needed, we can produce different shapes of the CDS curve by usingthe term structure of the model intensity parameter Î. Second, we show the modelimplied volatility surface for put options with maturity of 0.5Y . We see that the jump-diffusion model generates the implied volatility skew that is steeper that the diffusionmodel, so that, in general, it can fit the market implied skew more easily. An interestingdiscussion of the equity volatility skew in the structural framework can be found inHull, Nelkin, and White, (2004/05).

5.2 Asymptotic solution

In this section, we derive an asymptotic expansion for the Green’s function-solvingproblem (63) assuming that the jump-intensity parameter Î is small. More details ofthe derivation (which is far from trivial) and its extensions will be given elsewhere.

We introduce a new function √(Ù, x, X) such that:

√(Ù, x, X) = exp{−(

Ï2

2Û2 + Î

)Ù +

Ï

Û2 (X − x)}

√(Ù, x, X) (196)




The modified Green’s function solves the following propagation problem on the posi-tive semi-axis:

√Ù(Ù, x, X) − L(X)†√(Ù, x, X) = 0 (197)

√(Ù, x, 0) = 0, √(Ù, x, X) →X→∞

0 (198)

√(0, x, X) = δ(X − x) (199)

where

L(x)†g (x) =12Û2g xx (x) + ÎÌ

∫ 0

−∞g (x − j )eÌ j d j (200)

and Ì = Ì − Ï/Û2.We assume that Î << 1 and represent √(Ù, x, X)as follows:

√(Ù, x, X) = √(0)

(Ù, x, X) + Î√(1)

(Ù, x, X) + . . . (201)

The zero-order term √(0)

(Ù, x, X) solves the following problem:

√(0)Ù (Ù, x, X) − 1

2Û2√

(0)X X (Ù, x, X) = 0 (202)

√(0)

(Ù, x, 0) = 0, √(0)

(Ù, x, X) →X→∞

0 (203)

√(0)

(0, x, X) = δ(X − x) (204)

It is wellknown that it can be written as follows:

√(0)

(Ù, x, X) =1√ˇ

(n(

X − x√ˇ

)− n

(X + x√

ˇ

))(205)

The first-order term √(1)

(Ù, x, X) solves the following problem:

√(1)Ù (Ù, x, X) − 1

2Û2√

(1)X X (Ù, x, X) = ƒ(Ù, x, X) (206)

√(1)

(Ù, x, 0) = 0, √(1)

(Ù, x, X) →X→∞

0 (207)

√(1)

(0, x, X) = 0 (208)

where

ƒ(Ù, x, X) = Ì

∫ 0

−∞√

(0)(Ù, x, X − j )eÌ j d j (209)

= ÌP(

− X − x√ˇ

, −Ì√

ˇ

)− ÌP

(− X + x√

ˇ, −Ì

√ˇ

)

and P (a, b) is defined by eq. (5). We use Duhamel’s principle and represent√(1)(Ù, x, X) as follows:




√(1)

(Ù, x, X) =∫ Ù

0

∫ ∞

0√

(0)(Ù − Ù′, x, X − X ′)ƒ(Ù′, x, X ′)d X ′dÙ′ (210)

Fairly involved algebra yields:

√(1)

(Ù, x, X) =Ì

ÌÛ2

(ÌˇP

(− X − x√

ˇ, −Ì

√ˇ

)(211)

+XP(

− X + x√ˇ

, −Ì√

ˇ

)− (X − Ìˇ)P

(− X + x√

ˇ, Ì

√ˇ

)

−(X + Ìˇ)e−Ìx P(

− X√ˇ, −Ì

√ˇ

)+ (X − Ìˇ)e−Ìx P

(− X√

ˇ, Ì

√ˇ

))

For DNJs we can derive a similar expression:

√(1)

(Ù, x, X) =eÏÌ/Û2

Û2

⎧⎪⎪⎪⎨⎪⎪⎪⎩

X(

N(− X−x+Ì√

ˇ

)− N

(− X+x+Ì√

ˇ

))x < Ì

X(

N(− X+x−Ì√

ˇ

)− N

(− X+x+Ì√

ˇ

))+√

ˇ(

n(

X−x+Ì√ˇ

)− n

(X+x−Ì√

ˇ

)) x ≥ Ì(212)

5.3 Numerical solution

In this section we describe several complementary numerical methods for solving thecalibration and pricing problems in 1D. Specifically, we present the MC, FFT, and FD-based methods. The MC method, due to its generic nature, is easily applicable to theproblem at hand, particularly for DDM. However, it comes with the usual drawbacksand is to be avoided whenever possible. The FFT method is well suited to solvingproblems with DDM, however it has several well-known disadvantages including theneed for uniform grids with large number of steps, and complicated treatment ofaliasing effects. In our opinion, the FD method is the most powerful of the three. Itcan be used for both DDM and CDM. The key difficulty in applying the FD methodfor jumpdiffusions is the treatment of the integral term. The direct integration method(see, for example, Cont and Tankov 2004; Cont and Voltchkova 2005) has a complexityof O(N2) operations per time step, where N is the spatial grid size. To obviate thisdifficulty we can use the FFT method to compute the convolution term with a com-plexity of O(N log N), see (Andersen and Andreasen 2000; d’Halluin, Forsyth, andVetzal 2005 among others), however, this approach shares the disadvantages of theconventional FFT method. It turns out that for DNJs and ENJs one can compute theintegral term explicitly with a complexity of O(N) (Lipton 2003; Carr and Mayo 2007;Lipton, Song, and Lee 2007; Toivanen 2008)

Let us briefly compare the FFT and FD-based methods. In 1D, the complexity of theFFT method is O(N log N) per each time step, so that the overall complexity with Mdefault monitoring is O(MN log N). The complexity of the FD method with explicittreatment of the integral term and L time steps is O(L N) (we note that if the set of




monitoring times is sparse, we would need to add extra time steps to improve accuracyof the FD method). Thus, the overall competitiveness of the methods depends on thenumber of time steps needed to achieve the desired accuracy. In the case of DDM,the value function is not continuous at the barrier, so that both the FFT and FD-based methods are expected to be only first-order accurate in space, which is indeedconfirmed by our numerical experiments. In the case of CDM, the FD-based method isexpected to have second-order accuracy in space; while the FFT-based method cannoteasily be applied in this case.

5.3.1 Monte Carlo methodWe assume DDM and describe the corresponding MC simulations. There are twomethods for the simulation of the 1D dynamics. The first method is based on the directintegration of dynamics (17):

ƒxm ≡ x(tm) − x(tm−1) = Ïm + ÛmÂm +nm∑k=1

jk (213)

where Âm are standard independent normals, jk are independent variables with PDF� ( j ), nm = N(tm) − N(tm−1) is the Poisson random variable with intensity Îm, andx(t0) = Ó. Here

Ïm =∫ tm

tm−1

Ï(t ′)dt ′, Ûm =

√∫ tm

tm−1

Û2(t ′)dt ′, Îm =∫ tm

tm−1

Î(t ′)dt ′ (214)

The second method is based on the simulation of the increment ƒxm by the inver-sion of the PDF corresponding to the exact Green’s function. We note that in thepresence of jumps the Green’s function can be represented as follows:

G (ƒtm, ƒxm) =∞∑

k=0

wk÷k(Ë) (215)

where ƒtm = tm − tm−1 and wk is the probability of exactly k jumps for the Poissondistribution with intensity Îm:

wk =e−Îm (Îm)k

k!, k = 0, 1, . . . (216)

and

÷0(Ë) = 1Ûm

n(Ë)

÷1(Ë) = 1Ûm

n(Ë + Ì

Ûm

)DNJs

÷k(Ë) = 1Ûm

n(Ë + kÌ

Ûm

)




÷0(Ë) = 1Ûm

n(Ë)÷1(Ë) = ÌP(−Ë, −Û−) ENJs

÷k(Ë) = Û−(−(Ë+Û−)÷k−1(Ë)+Û−÷k−2(Ë))k−1

where Ë = (ƒxm − Ïm)/Ûm, Û− = ÌÛm. Typically, we can restrict ourselves to the com-bination of the first two terms.

Once the evolution of the driver x is described, the valuation can be performed inthe standard fashion.

5.3.2 Fast Fourier Transform methodIn this section we show how to use the FFT method for valuing credit products in 1D.As we mentioned earlier, this method is not well suited to the case of CDM, so we onlyapply it in the case of DDM. The advantage of this method is that its implementation isrelatively easy and it can be applied for relatively wide class of jump-size distributions.Its disadvantages are the need for a dense uniform grid, which has to be wide enoughin order to avoid aliasing effects becoming important.

The Green’s functionTo start with, we consider the unbounded Green’s function governed by eqs. (56), (57),(59). We emphasize that the coefficients of the infinitessimal generator are spatiallyindependent, so that the Green’s function depends on X − x rather than X and xseparately:

G (t, x, T, X) ≡ ’(t, T, Y ) (217)

where Y = X − x . Due to this fact, the Fourier transform of ’ can be found explicitly:

’(t, T, k) =∫ ∞

−∞e ikY ’(t, T, Y )dY = e− ∫ T

t ¯(t ′,k)dt ′(218)

where k is the transform variable, and ¯(t, k) is the characteristic exponent:

¯(t, k) =12Û2(t)k2 − iÏ (t) k − Î(t)(� (k) − 1) (219)

with the function � (k) given by:

� (k) =∫ 0

−∞e ik j � ( j )d j =

{e−ikÌ, DNJs

ÌÌ+ik , ENJs (220)

Given ’ we can compute the Green’s function ’ via the inverse Fourier transform asfollows:

’(t, T, Y ) =1

2

∫ ∞

−∞e−ikY ’(t, T, k)dk (221)




When parameters are time independent we have

’(t, T, k) = e−Ù¯(k) (222)

¯ (k) =12Û2k2 − iÏk − Î(� (k) − 1)

where Ù = T − t . This formula can be trivially generalized to the case of piecewiseconstant parameters but we don’t present the corresponding expression here since weare only interested in time intervals between observation times where parameters areconstant by construction.

Backward problemWe start by considering the backward problem (36), (37) for the value function V (t, x)on the time interval (tm−1, tm). The value function V(m−1)+(x) can be represented asfollows:

V(m−1)+(x) = D (tm−1, tm)∫ ∞

−∞Vm− (X) ’(tm−1, tm, X − x)d X (223)

For convenience, we introduce the inverse Fourier transform of Vm− (x):»

Vm− (k) =1

2

∫ ∞

−∞e−ikx Vm− (x) dx (224)

Here we assume that Vm− is regularized as appropriate, so that the above integralconverges. By applying the Fourier transformed density function (218) and exchangingthe integration order, we obtain:

V (t(m−1)+, x) (225)

= D (tm−1, tm)∫ ∞

−∞Vm− (X)

{1

2

∫ ∞

−∞e−ik(X−x)’(tm−1, tm, k)dk

}d X

= D (tm−1, tm)∫ ∞

−∞e ikx

{1

2

∫ ∞

−∞e−ik X Vm− (X) d X

}’(tm−1, tm, k)dk

= D (tm−1, tm)∫ ∞

−∞e ikx »

Vm− (k) ’(tm−1, tm, k)dk

Variants of formula (225) for option pricing in 1D case were proposed and analysed byCarr and Madan (1999); Lewis (2001); Lipton (2001, 2002a), among others.

We can apply the above formula repeatedly for DDM at times {tdm}m=1,...,M . Let

today’s time be t0 = 0. We represent the following backward induction algorithmbased on the recurrent application of eq. (225): (a) Set m = M and apply the terminalcondition by vM(x) = v(x); (b) Compute the auxiliary function V(m−1)+(x) by virtue ofeq. (225); (c) Evaluate V(m−1)− by applying the projection operator – {V (x)} as needed

V(m−1)−(x) = –{

V(m−1)+(x)}

(226)

(d) Repeat steps (b), (c) until m = 1, when V (0, Ó) is calculated and the recursion isstopped. For financial instruments of interest, explicit expressions for v, – {.} are given




in section 4. To calculate the survival probability, we run the above scheme withoutdiscounting.

To apply the above algorithm in practice, we restrict the value function V (t, x) andthe payoff function v(x) to the uniform spatial grid {x0, . . . , xN} with the uniformstep ƒx , where x0 is a large negative number and xN is a large positive number,and N = 2n − 1, which is required for the standard FFT algorithm to be efficient(but is not necessary in general). For the sake of computational efficiency, the spatialgrid is defined in such a way that xn0 = 0, and Ó = xnÓ

, for some n0, nÓ, respectively.The transformed density function ’(t, T, k) is defined on the discrete Fourier grid{k0, . . . , kN} with uniform step ƒk, such that ƒxƒk = 2/2n, and

kn =2(n − N/2)

Nƒx(227)

The discretized version of eq. (225) can be written as

V(m−1)+(x) = D (tm−1, tm) fft(

ifft(Vm− (x)) � ’(tm−1, tm, k))

(228)

where � denotes element-wise multiplication

Forward problemSimilarly, we consider the forward problem for the Green’s function G (T, X) gov-erned by eqs. (56), (57) on the time interval (tm−1, tm). This function can be written asfollows:

G m−(X) =∫ ∞

−∞G (m−1)+(x)’(tm−1, tm, X − x)dx (229)

By analogy with eq. (225), we obtain:

G m−(X) =∫ ∞

−∞G (m−1)+(x)

{1

2

∫ ∞

−∞e−ik(X−x)’(tm−1, tm, k)dk

}dx (230)

=1

2

∫ ∞

−∞e−ik X

{∫ ∞

−∞e ikx G (m−1)+(x)dx

}’(tm−1, tm, k)dk

=1

2

∫ ∞

−∞e−ik X G (m−1)+(k)’(tm−1, tm, k)dk

We note that the expression in the curly brackets can be viewed as the direct Fouriertransform of the initial value function G (m−1)+(x). Thus, the discrete version of eq.(230) can be represented by analogy to eq. (228):

G m−(X) = ifft(fft(G (m−1)+(X)) � ’(tm−1, tm, k)

)(231)

For calibration purposes it is important to solve the problem for the survivalprobability Q(x)(t, x, T) via forward induction. For this purpose we present forwardinduction with time stepping based on the recursive application of eq. (231) as follows:(a) Set m = 1 and specify the initial condition for G as a Kronecker’s delta function




centred at n0:

G (t0, X) = g (Xn) = δn,nÓ(232)

(b) Given the value of G at time t(m−1)+, G (m−1)+(X), apply the forward convolution(231); (c) Set to zero the value function G m+(X) outside the barrier region:

G m+(X) = G m−(X)1{X>0} (233)

(d) Evaluate the current value of the survival probability Q(x)(0, Ó, tm) by computingthe sum over the discrete spatial grid:

Q(x)(0, Ó, tm) =N∑

n=n0

G m+(xn) (234)

(e) Repeat steps (b), (c) until m = M, then stop the recursion.

Implementation detailsWhen implementing the FFT method with time stepping we need to consider thefollowing important aspects: periodicity, finiteness, and convergence.

The applicability of FFT is based on the assumption that the relevant functions areperiodic. If it is not the case, so-called aliasing effects tend to spoil the solution nearthe end points of the computational domain. We first notice that, provided the spatialgrid is large enough, the periodicity is not an issue for the computation of the Green’sfunction since asymptotically it approaches zero. For the backward recursion, we candeal with aliasing effects by modifying the solution near the edges of the grid. Wenote that near the edges of the grid the second derivative of the value function shouldgradually approach zero (from above for convex payoffs, from below for concaveones). However aliasing effects destroy the convexity/concavity of the correspondingsolution. To rectify this fact, we alter the value function by detecting remote areaswhere its convexity/concavity is violated, and linearly extrapolating the function there,thus achieving zero convexity/concavity in these regions.

Formulas (225) and (230) are based on the requirement that the Fourier transformof the payoff function exists. This requirement can be satisfied by applying a dampingfactor as needed. For the FFT method, this requirement is not too restrictive since thecalculations are performed in a finite domain. In our experiments, we have found noadvantage in using damping.

The convergence of the integral formulas (225) and (230) is affected by the rate ofdecay of the transformed Green’s function (218) for large k. Asymptotically, using eq.(219) we obtain the following result:

lim|k|→±∞

� [’(tm−1, tm, k)]

= e− 12 ÙmÛ2k2

(235)

where Ùm = tm − tm−1.Typically, the volatility is small, Û ≈ 0.01, and Ùm ≈ 0.25y, sothat the upper bound for k, kN = /2ƒx needs to be large enough (kN ∼ 1000). In




our experiments, we have found that for typical model parameters it is safe to chooseN = 212 − 1, ƒx ∼ 10−4.

5.3.3 Finite difference methodBackward equationWe start with the backward problem for the value function V (t, x). For DDM, thisfunction is solving problem (43) on R

1, while for CDM this function is solving a similarproblem on R

1+. For brevity, we consider only the latter case.

We introduce a discrete spatial grid of size N + 1, {x0, x1, . . . , xN−1, xN}, wherex0 = 0; and xN is a large positive number, and a discrete time grid of size L ,{t0, t1, . . . , tL }, where t0 = 0 and tL = T , in such a way that the set of times whenparameters jump, and other special times, if any, belong to the grid. The valuesV (tl , xn) are denoted by Vl ,n, and similarly for other relevant quantities. For fixed lwe use the notation �Vl , and think of �Vl as an (N + 1)-component vector. We discretizethe evolution operator (54) at time tl :

L(x) (tl ) =⇒ L(x)l (236)

D(x)(tl ) + J (x)(tl ) =⇒ D(x)l + J

(x)l

where L(x)l , D

(x)l , J

(x)l are (N + 1) × (N + 1) matrices with elements L

(x)l ,n,n′ , D

(x)l ,n,n′ ,

J(x)l ,n,n′ .

As usual, D(x)l is a tridiagonal matrix. Typically the diffusion term is small compared

to the advection and jump terms, so that an appropriate discretization of the firstderivative is necessary for the stability of the numerical scheme, see, for example,d’Halluin, Forsyth, and Vetzal (2005).

In general, the matrix J(x)l is not tridiagonal. However, as we shall demonstrate

presently, for DNJs and ENJs, the product J(x)l

�Vl can be evaluated in a way whichrequires only O (N) operations. To this end, we introduce an auxiliary function

êl (x) ≡ J (x)V (tl , x) (237)

which we intend to treat fully explicitly.For DNJs, we can approximate êl on the grid via the linear interpolation to second

order accuracy:

ˆl ,n ={

˘nVl ,n−1 + (1 − ˘n)Vl ,n, xn ≥ Ì

0, xn < Ì(238)

where

n = min{ j : x j−1 ≤ xn − Ì < x j }, ˘n =xn − (xn − Ì)

xn − xn−1(239)




For ENJs, we choose a small step h, h > 0, and write:

êl (x + h) = Ì

∫ 0

−x−heÌj V (tl , x + h + j )d j (240)

= Ìe−Ìh

(∫ 0

−xeÌz V (tl , x + z)dz +

∫ h

0eÌz V (tl , x + z)dz

)

= e−Ìhêl (x) + w0(Ì, h)V (tl , x) + w1(Ì, h)V (tl , x + h) + O(h3)

where z = h + j , and

w0(Ì, h) =1 − (1 + Ìh) e−Ìh

Ìh, w1(Ì, h) =

−1 + Ìh + e−Ìh

Ìh(241)

Accordingly, we obtain a recursive scheme for computing ê(x) to second-orderaccuracy:

êl (x + h) = e−Ìhêl (x) + w0(Ì, h)V (tl , x) + w1(Ì, h)V (tl , x + h) (242)

On the grid, we choose hn = xn − xn−1, and represent eq. (242) as follows:

êl ,n+1 = e−Ìhn+1êl ,n + w0(Ì, hn+1)Vl ,n + w1(Ì, hn+1)Vl ,n+1 (243)

with the initial condition specified by êl ,0 = 0. Thus, by introducing an auxiliary vector�êl =

{êl ,n

}we can calculate the matrix product Kl �Vl with O (N) operations in both

cases.To compute the value �Vl−1 given the value �Vl , l = 1, . . . , L , we introduce auxiliary

vectors �êl , �V∗, �V∗∗,and apply the following scheme:

�V∗ = �Vl +12‰tl

(−→c l−1 +

−→c l

)+ ‰tlÎ(tl )�êl (244)

(I − ‰tlD

(x)l

)�V∗∗ = �V∗

�Vl−1 = D (tl−1, tl ) �V∗∗

where ‰tl = tl − tl−1, l = 1, . . . , L , and I is the identity matrix. Thus, we use anexplicit scheme to approximate the integral step and compute �V∗ given �Vl ; then weuse an implicit scheme to approximate the diffusive step and compute �V∗∗ given �V∗,finally, we apply the deterministic discounting. Boundary values V∗∗

0 and V∗∗N are

determined by the boundary conditions. The second implicit step leads to a systemof tridiagonal equations which can be solved via O(N) operations. The time-steppingnumerical scheme is straightforward. The terminal condition is given by:

�VL = �v (245)

The implicit diffusion step in scheme (244) is first-order accurate in time, but ittends to be more stable than the Crank-Nicolson scheme which is typically second-order accurate in time (without the jump part). The explicit treatment of the jumpoperator is also first-order accurate in time. With an appropriate spatial discretization,the scheme is second-order accurate in the spatial variable. As usual, see, for example,




d’d’Halluin, Forsyth, and Vetzal, (2005), predictor-corrector iteration improves thescheme convergence in both time and space. At each step in time, the scheme withfixed point iterations is applied as follows. We set

�U 0 = �Vl +12‰tl

(c l−1 + c l

)(246)

and, for p = 1, 2, . . . , p (typically, it is enough to use two iterations, p = 2) apply thefollowing scheme:

�U ∗ = �U 0 + ‰tlÎ(tl )�êp−1l (247)(

I − ‰tlD(x)l

)�U p = �U ∗

where �êp−1l = J

(x)l

�U p−1. Provided that the difference ||U p − U p−1|| is small in anappropriate norm, we stop and apply the discounting:

Vl−1 = D(tl−1, tl )U p (248)

Forward equationFor calibration purposes we have to calculate the survival probability forward in timeby finding the Green’s function, G (t, x, T, X), defined by eq. (63).

By analogy with the backward equation, we use a time-stepping scheme with fixed-point iterations in order to calculate �Gl =

{Gl ,n

}= {G (tl , xn)}. We introduce

¯l (x) ≡ J (x)†G (tl , x) (249)

For DNJs, we approximate this function on the grid to second-order accuracy:

¯l ,n ={

˘nGl ,n−1 + (1 − ˘n)Gl ,n, xn + Ì ≤ xN

Gl ,N, xn + Ì > xN(250)

where

n = min{ j : x j−1 ≤ xn + Ì < x j }, ˘n =xn − (xn + Ì)

xn − xn−1(251)

For ENJs, by analogy with eq. (242), we use the following scheme which is second-order accurate:

¯l ,n−1 = e−Ìhn¯l ,n + w0(Ì, hn)Gl ,n + w1(Ì, hn)Gl ,n−1 (252)

where w0 and w1 are defined by eq. (241).The time-stepping numerical scheme is straightforward. The initial condition is

given by:

G 0,n = 2δn,nÓ/(xnÓ+1 − xnÓ−1

)(253)

where the spatial grid in chosen in such a way that x0 = 0 and xnÓ= Ó. To compute the

value �Gl given the value �Gl−1, l = 1, . . . , L , we introduce auxiliary vectors �, �G∗, andapply the following scheme:




0.00

0.50

1.00

1.50

2.00

2.50

X

Expansion, lambda=0.03

PDE, lambda=0.03

Expansion, lambda=0.1

PDE, lambda=0.1

0.00

0.50

1.00

1.50

2.00

2.50

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.000.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

X

Analytic, lambda=0.03Expansion, lambda=0.03PDE, lambda=0.03Analytic, lambda=0.1Expansion, lambda=0.1PDE, lambda=0.1

figure 12.2 Asymptotic and numerical Green’s functions for DNJs (lhs), and analytical,asymptotic and numerical Green’s functions for ENJs (rhs). The relevant parameters are thesame as in Figure 1, T = 10y.

�G∗ = �Gl−1 + ‰tlÎ(tl )�l−1 (254)(I − ‰tlD

(x)†l

)�Gl = �G∗

As before, we use predictor-corrector iterations to improve the rate of convergence. Itis extremely useful to construct schemes (244), (254) is such a way that they are exactly(rather than approximately) adjoint (without discounting).

Given the Green’s function at time tl , we can compute the survival probability via asimple summation rule:

Q(x)(t0, Ó, tl ) =N∑

n=1

(xn − xn−1) Gl ,n (255)

5.3.4 Useful comparisonsIn Figure 12.2 we compare analytical, asymptotic, and numerical solutions for ENJs,and asymptotic and numerical solutions for DNJs. This figure shows that analyticaland numerical solutions agree perfectly (when both are known), while asymptoticsolution is respectable if not perfect.

6 Two-dimensional problem................................................................................................................................................

6.1 Analytical solution

In this section we study the 2D CVA problem semi-analytically under bivariatedynamics without jumps assuming CDM.

We start by constructing the 2D Green’s function semi-analytically. We assumeconstant volatility parameters and consider the Green’s function for the correspondingdiffusion advection which solves the following problem in R

2+,+:




√Ù − 12(Û2

1(√X1 X1 + √X1

)+ 2ÒÛ1Û2√X1 X2 + Û2

2(√X2 X2 + √X2

))= 0 (256)

√ (Ù, X1, 0) = 0, √ (Ù, 0, X2) = 0 (257)

√ (0, X1, X2) = δ (X1 − Ó1) δ (X2 − Ó2) (258)

To simplify the problem, we introduce new independent and dependent variablesr, ˆ,G such that

r =

√(Û2 X1 − Û1ÒX2

ÒÛ1Û2

)2

+(

X2

Û2

)2

, ˆ = atan(

ÒÛ1 X2

Û2 X1 − Û1ÒX2

)(259)

G = ÒÛ1Û2 exp(

18(Û2

1 + Û22)Ù +

12

(X1 − Ó1) +12

(X2 − Ó2))

√

where Ò =√

1 − Ò2. In these variables the pricing problem becomes:

GÙ − 12

(Gr r +

1rGr +

1r 2Gˆˆ

)= 0 (260)

G (Ù, r, 0) = 0, G (Ù, r, ·) = 0 (261)

G (0, r, ˆ) =1r ′ δ

(r − r ′) δ (ˆ − ˆ′) (262)

where

r ′ =

√(Û2Ó1 − Û1ÒÓ2

ÒÛ1Û2

)2

+(

Ó2

Û2

)2

, ˆ′ = atan(

ÒÛ1Ó2

Û2Ó1 − Û1ÒÓ2

)(263)

Thus, we have reduced the original problem to the standard heat problem in an angle.This angle is formed by the horizontal axis ˆ = 0 and the sloping line ˆ = · where· = atan (−Ò/Ò); for convenience, we think of atan(.) as defined on the interval [0, ).This angle is acute when Ò < 0 and obtuse otherwise.

Problem (260)–(262) can be solved in two complementary ways: (a) via the eigen-function expansion method; (b) via the method of images. Solution of this problemvia the eigenfunction expansion method is classical; it has been introduced in mathe-matical finance independently by (He, Keirstead, and Rebholz 1998; Lipton 2001; andZhou 2001b). In this method the Green’s function is presented in the form:

G·

(Ù, r ′, ˆ′, r, ˆ

)=

2e−(r 2+r ′2)/2Ù

·Ù

∞∑n=1

I n·

(r r ′

Ù

)sin(

nˆ′

·

)sin(

nˆ

·

)(264)

where IÌ(x) is the modified Bessel function with index Ì.It turns out that the Green’s function in an angle also can be constructed via the

method of images by using the non-periodic Green’s function defined in the half-plane0 < r < ∞, −∞ < ˆ < ∞. The latter Green’s function is given by:

G(Ù, r ′, r, ˜

)= G1

(Ù, r ′, r, ˜

)− G2(Ù, r ′, r, ˜

)(265)




where

G1(Ù, r ′, r, ˜

)=

e−(r 2+r ′2)/2Ù

2Ù

(s+ + s−)2

e (r r ′/Ù) cos ˜ (266)

G2(Ù, r ′, r, ˜

)=

e−(r 2+r ′2)/2Ù

22Ù

∞∫0

s+e−(r r ′/Ù) cosh((+˜)Ê) + s−e−(r r ′/Ù) cosh((−˜)Ê)

Ê2 + 1dÊ

and ˜ = ˆ − ˆ′, s± =sign( ± ˜). This formula seems to be new; it was presented byLipton (2008), its detailed derivation will be given elsewhere. To solve the problemin an angle 0 ≤ ˆ ≤ ·, we use the method of images and represent the fundamentalsolution in the form

G·

(Ù, r ′, ˆ′, r, ˆ

)=

∞∑n=−∞

[G(Ù, r ′, r, ˆ − ˆ′ + 2n·

)− G(Ù, r ′, r, −ˆ − ˆ′ + 2n·

)](267)

This series converges since simple balancing of terms shows that

G2(Ù, r ′, r, ˜

)= O

(|˜|−2) (268)

As always, the eigenfunction expansion method works well when Ù is large, whilethe method of images works well when Ù is small.

For the survival probability, we obtain the following expression:

Q(x1,x2) (Ù, Ó1, Ó2) = e−(Û21+Û2

2)Ù/8+(Ó1+Ó2)/2 (269)

×∞∫

0

·∫0

e−Û12r cos(ˆ−‚)/2G·

(Ù, r ′, ˆ′, r, ˆ

)r dr dˆ

where Û12 =√

Û21 + 2ÒÛ1Û2 + Û2

2, ‚ =atan((Û1Ò + Û2) /Û1Ò). We combine formula (260)with the well-known identity

e−z cos Ë = I0 (z) + 2∞∑

k=1

(−1)k Ik (z) cos (kË) (270)

and obtain

Q(x1,x2) (Ù, Ó1, Ó2) =4e−(Û2

1+Û22)Ù/8+(Ó1+Ó2)/2−r ′2/2Ù

·Ù(271)

×∞∑

n=1

∞∑k=1

cn,k

∞∫0

e−r 2/2Ù I n·

(r r ′

Ù

)Ik

(Û12r

2

)r dr

where

cn,k =(−1)k (n/·) sin (nˆ′/·) (cos (k‚) − (−1)n cos (k (· − ‚)))

(n/·)2 − k2(272)




The corresponding integrals are easy to evaluate numerically. If necessary, one can usesome classical formulas for the product of modified Bessel function to simplify themfurther.

The probability of hitting the boundary Xi = 0, i = 1, 2, which is needed in orderto calculate the value of FTDS and CVA, can be computed by using definition (102).Tedious algebra yields:

gi (Ù, Ó1, Ó2, X3−i ) =Û2

i

2√Xi (Ù, Ó1, Ó2, X1, X2)

∣∣Xi =0 (273)

=e−(Û2

1+Û22)Ù/8+(Ó1+Ó2)/2−r ′2/2Ù−X3−i /2−X2

3−i /2Ò2Û23−i Ù

·2ÙX3−i

∞∑n=1

I n·

(X3−i r ′

ÒÛ3−iÙ

)sin(

nˆ′

·

)

By using formula (101), we can represent V F T DS(Ù, Ó1, Ó2), and V C V AP B (Ù, Ó1, Ó2) as

follows

V F T DS(Ù, Ó1, Ó2) = −c

∫ Ù

0e−r Ù′

Q(x1,x2)(Ù′, Ó1, Ó2)dÙ′ (274)

+∑i=1,2

(1 − Ri )∫ Ù

0

∫ ∞

0e−r Ù′

gi (Ù′, Ó1, Ó2, X3−i )d X3−i dÙ′

V C V AP B (Ù, Ó1, Ó2) = (1 − R2)

∫ Ù

0

∫ ∞

0e−r Ù′ {

V C DS(Ù − Ù′, X1)}

+ g2(Ù′, Ó1, Ó2, X1

)d X1dÙ′

(275)where V C DS(Ù, X1) is given by eq. (189). Here r is the interest rate (not to be confusedwith the radial independent variable).

6.2 Numerical solution

In this section we develop robust numerical methods for model calibration and pricingin 2D. As before, we describe the MC, FFT, and FD methods.

To our knowledge, only Clift and Forsyth (2008) develop an FD method for solvingthe 2D pricing problem with jumps by computing the 2D convolution term usingthe FFT method with a complexity of O(N1 N2 log(N1 N2)), where N1 and N2 are thenumber of points in each dimension. As we have noticed, the application of the FFTmethod is not straightforward due to its inherent features. In contrast, we apply anexplicit second-order accurate method for the computation of the jump term with acomplexity of O(N1 N2).

In two dimensions, the complexity of the FFT method is O(L N1 N2 log(N1 N2))and the complexity of the FD method with explicit treatment of the integral term isO(L N1 N2). Accordingly the FFT method is attractive provided that N2 is not too large,say N2 < 100, otherwise the FD method is more competitive.




6.2.1 Monte Carlo methodThe MC simulation of the multi-dimensional dynamics (28) is performed by analogywith eq. (213). As before, we restrict ourselves to the case of DDM. The simulation ofthe multi-dimensional trajectories is based on direct integration:

ƒxi,m = Ïi,m + Ûi,mÂi,m +ni,m∑k=1

ji,k (276)

where Âi,m and Â j,m are standard normals with correlation Òi, j , ji,k are independentvariables with PDF �i ( j ),

ni,m =∑

∈–(N)

1{i∈}n,m (277)

where n,m are Poisson random variables with intensities Î,m, and xi (t0) = Ói . Here

Ïi,m =∫ tm

tm−1

Ïi (t ′)dt ′, Ûi,m =

√∫ tm

tm−1

Û2i (t ′)dt ′, Î,m =

∫ tm

tm−1

Î(t ′)dt ′ (278)

As soon as sample trajectories of the drivers are determined, the relevant instrumentscan be priced in the standard fashion.

6.2.2 Fast Fourier Transform methodWe consider the 2D version of the FFT method for the case of DDM. As in the 1D case,we consider the Green’s function in R

2, which we denote by ’ (t, T, Y1, Y2), whereYi = Xi − xi . As before, without loss of efficiency, we can assume that the coefficientsare time independent. The Fourier transform of ’ (t, T, Y1, Y2) can be written asfollows:

’(t, T, k1, k2) =∫ ∞

−∞

∫ ∞

−∞e ik1Y1+ik2Y2’(t, T, Y1, Y2)dY1dY2 (279)

= e−Ù¯(k1,k2)

where (k1, k2) ∈ R2 are the transform variables, and ¯(k1, k2) is the characteristic

exponent:

¯(k1, k2) =12Û2

1k21 + ÒÛ1Û1k1k2 +

12Û2

2k22 − iÏ1k1 − iÏ2k2 (280)

−Î{1,2}(�1(k1)�2(k2) − 1) − Î{1}(�1(k1) − 1) − Î{2}(�2(k2) − 1)

with

�i (ki ) =∫ 0

−∞e iki ji �i ( ji )d ji , i = 1, 2 (281)




Given ’(t, T, k1, k2) we can compute the Green’s function via the inverse Fouriertransform as follows:

’(t, T, Y1, Y2) =1

(2)2

∫ ∞

−∞

∫ ∞

−∞e−ik1Y1−ik2Y2’(t, T, k1, k2)dk1dk2 (282)

Once ’(t, T, k1, k2) is obtained, we can generalize formulas (225), (230) in an obviousway. As before, we apply the backward recursion scheme with time stepping andcontrol functions specified in section 4. The forward recursion is performed by thesame token.

6.2.3 Finite Difference methodNow we develop an FD method for solving the 2D backward problem (73) for the valuefunction V (t, x1, x2). For brevity, we consider only the backward problem with CDM,a numerical scheme for the 2D forward equation can be developed by analogy.

We introduce a 2D discrete spatial grid{(

x1,n1, x2,n2

)}, and a time grid {t0, . . . , tL }

which includes all the special times. As before V(tl , x1,n1, x2,n2

)is denoted by Vl ,n1,n2 ,

and similarly for other relevant quantities.We denote the discretized 1D operators by D

(xi )l , J

(xi )l , i = 1, 2. The correlation

operator, C(x1,x2) (tl ) is discretized via the standard four-stencil scheme:

C(x1,x2) (tl ) = ⇒ C(x1,x2)l (283)

In order to discretize the cross-integral operator J (x1,x2) efficiently, we introduce anauxiliary function ˆ (x1, x2), such that

êl (x1, x2) ≡ J (x1,x2)V (tl , x1, x2) (284)

as well as an auxiliary function

˜l (x1, x2) ≡ J (x1)V (tl , x1, x2) (285)

It is clear that

êl (x1, x2) = J (x2)˜l (x1, x2) (286)

so that we can perform calculations by alternating the axis of integration.We consider DNJs and ENJs separately. For DNJs, we approximate êl (x1, x2) on the

grid in two steps via the linear interpolation:

˜l ,n1,n2 ={

˘n1 Vl ,n1−1,n2 + (1 − ˘n1 )Vl ,n1,n2, xn1 ≥ Ì10, xn1 < Ì1

(287)

êl ,n1,n2 ={

˘n2˜l ,n1,n2−1 +(1 − ˘n2

)˜l ,n1,n2, xn2 ≥ Ì2

0, xn2 < Ì2

where

ni = min{ j : xi, j−1 ≤ xi,ni − Ìi < xi, j }, ˘ni =xi,ni − (xi,ni − Ìi )

xi,n − xi,n−1, i = 1, 2 (288)




For ENJs jumps with jump sizes Ì1 and Ì2 we write:

˜l ,n1+1,n2 = e−Ì1h1,n1+1˜l ,n1,n2 + w0(Ì1, h1,n1+1)Vl ,n1,n2 + w1(Ì1, h1,n1+1)Vl ,n1+1,n2 (289)

êl ,n1,n2+1 = e−Ì2h2,n2+1êl ,n1,n2 + w0(Ì2, h2,n2+1)˜l ,n1,n2 + w1(Ì2, h2,n2+1)˜l ,n1,n2+1 (290)

where hi,ni = xi,ni − xi,ni −1. The initial conditions are set by

˜l ,0,n2 = 0, n2 = 0, . . . , N2, êl ,n1,0 = 0, n1 = 0, . . . , N1 (291)

Thus, in both cases, êl ,n1,n2 can be evaluated via O (N1 N2) operations to second-orderaccuracy.

For DDM, we can proceed along similar lines.We develop a modified Craig and Sneyd (1988) discretization scheme using explicit

time stepping for jump and correlation operators and implicit time stepping for dif-fusion operators to compute the solution at time tl−1, Vl−1,n1,n2 , given the solution attime tl , Vl ,n1,n2 , l = 1, . . . , L , as follows:

(I − ‰tlD(x1)l ) �V∗ =

(I + ‰tl

(D

(x2)l + C

(x1,x2)l

))�Vl + ‰tl

12(c l−1 + c l−1

)(292)

+‰tl

(Î{1} (tl ) �ê{1} + Î{2} (tl ) �ê{2} + Î{1,2} (tl ) �ê{1,2}

)(I − ‰tlD

(x2)l ) �V∗∗ = �V∗ − ‰tlD

(x2)l

�Vl

�Vl−1 = D (tl−1, tl ) �V∗∗

When solving the first equation, we keep the second index n2 fixed, apply the diffusionoperator in the x2 direction, the correlation operator, the coupon payments (if any),and the jump operators explicitly, and then solve the tridiagonal system of equationsto get the auxiliary solution V∗

·,n2. When solving the second equation, we keep n1

fixed, and solve the system of tridiagonal equations to obtain the solution V∗∗n1,.

. Theoverall complexity of this method is O(N1 N2) operations per time step. The terminalcondition is given by:

�VL = �v (293)

Similarly to the 1D case, we apply fixed point iterations at each step in time, in order toimprove convergence.

7 Evaluation of the credit valueadjustment in practice

................................................................................................................................................

In this section we discuss forward model calibration and backward credit value adjust-ment valuation.




We start with the calibration of the marginal dynamics to CDS spreads, CDSOprices, and equity option prices. First, we find the initial value for each firm and itsdefault barrier from the balance sheet as discussed in section 2.2. Second we calibratethe model parameters as follows. We specify the time horizon T and determine theterm structure of CDS spreads with maturities {tc

k }, tcK ≤ T , k = 1, . . . , K referencing

the i-th firm. We assume that the model parameters change at times {tck }, but otherwise

stay constant. Then, we make an initial guess regarding the term structure of the firm’svolatility {Ûk} and the magnitude of the parameter Ì. Specifically, we choose Û(k) in theform (

Û(k))2 = Û2∞ +

(Û2

0 − Û2∞)

e−� tck (294)

Starting with this guess, a fast bootstrapping is implemented for the jump intensity{Îk} by means of forward induction with time stepping, resulting in matching the termstructure of CDS spreads of the i-th name. Bootstrapping is done by first fitting Î1 tothe nearest CDS quote (via 1D numerical root searching) and inductive fitting of Îk′+1 ,given {Îk}, k = 1, . . . , k′, to the CDS quote with maturity time tc

k′+1 (via numerical rootsearching). Finally, given the parameter set thus obtained, we compute model pricesof a set of CDSO and equity options. If differences between model and market pricesare small, we stop; otherwise we adjust Û0, Û∞, � and Ì and recompute {Îk}. Typically,the market data for CDSO and equity options with maturities longer than one year arescarce, so that a parametric choice of Ûk is necessary.

For the purposes of illustration, we use market data for two financial institutions: JPMorgan (JPM) and Morgan Stanley (MS). In Table 12.1 we provide market data, as of2 November 2009, which we use to calculate the initial firm’s value a(0) and its defaultbarrier l(0) for both firms. In Table 12.2, we provide the term structure of spreads forCDSs on JPM and MS, their survival probabilities implied from these spreads usingthe hazard rate model, and the corresponding default and annuity legs. It is clear that,compared to MS, JPM is a riskier credit, since it has more debt per equity.

We use jump-diffusion models with DNJs and ENJs. We set the jump amplitude Ì

to the initial log-value of the firm, Ì = Ó in the model with DNJs, and Ì = 1/Ó in themodel with ENJs. The volatility of the firm’s value Û is chosen using eq. (22), so thatthe 1y at-the-money implied volatility of equity options is close to the one observed inthe market for both DNJs and ENJs.

The computations are done via the FD method with weekly DDM. For both choicesof jump distributions, the model is calibrated to the term structure of CDS spreadsgiven in Table 12.2 via forward induction.

Table 12.1 Market data, as of 2 October 2009, for JPM and MS.

s(0) L(0) R a(0) l(0) Ó ÌDNJ ÌENJ Û0 Û∞ �

JPM 41.77 508 40% 245 203 0.1869 0.1869 5.35 53% 2% 4MS 32.12 534 40% 246 214 0.1401 0.1401 7.14 42% 2% 4




Table 12.2 CDS spreads and other relevant outputs for JPM and MS.

CDS Sprd (%) Survival Prob Default Leg Annuity Leg

T DF JPM MS JPM MS JPM MS JPM MS

1y 0.9941 0.45 1.10 0.9925 0.9818 0.0045 0.0109 0.9924 0.98572y 0.9755 0.49 1.19 0.9836 0.9611 0.0097 0.0231 1.9632 1.93883y 0.9466 0.50 1.24 0.9753 0.9397 0.0145 0.0354 2.9006 2.84694y 0.9121 0.54 1.29 0.9644 0.9173 0.0206 0.0479 3.7971 3.70365y 0.8749 0.59 1.34 0.9515 0.8937 0.0274 0.0604 4.6474 4.50616y 0.8374 0.60 1.35 0.9414 0.8732 0.0326 0.0709 5.4519 5.25597y 0.8002 0.60 1.35 0.9320 0.8539 0.0372 0.0803 6.2135 5.95708y 0.7642 0.60 1.33 0.9228 0.8373 0.0415 0.0881 6.9338 6.61279y 0.7297 0.60 1.31 0.9137 0.8217 0.0455 0.0950 7.6148 7.227210y 0.6961 0.60 1.30 0.9046 0.8064 0.0494 0.1015 8.2583 7.8028

In Figure 12.3, we show the calibrated jump intensities. We see that for the specificchoice of the jump size, the model with ENJs implies higher intensity than the one withDNJs because, with small equity volatility, more than one jump is expected before thebarrier crossing. For JPM the term structure of the CDS spread is relatively flat, sothat the model intensity is also flat. For MS, the term structure of the CDS spread isinverted, so that the model intensity is decreasing with time.

In Figure 12.4, we show the implied density of the calibrated driver x(T) for matu-rities of 1y, 5y, and 10y for JPM and MS. We see that even for short-times the modelimplies a right tail close to the default barrier, which introduces the possibility ofdefault in the short term. We observe no noticeable differences between the modelwith DNJs and ENJs given that both are calibrated to the same set of market data.

JPM

0.001 2 3 4 5 6 7 8 9 10

0.02

0.04

T

1 2 3 4 5 6 7 8 9 10

T

DNJENJSpread

MS

0.00

0.02

0.04

0.06

0.08 DNJENJSpread

figure 12.3 Calibrated intensity rates for JPM (lhs) and MS (rhs) in the models with ENJs andDNJs, respectively.




JPM

0.000.00 0.10 0.20 0.30 0.40 0.50

0.01

0.02

0.03

0.04

0.05

0.06

X

0.00 0.10 0.20 0.30 0.40 0.50

X

0.00 0.10 0.20 0.30 0.40 0.50

X

0.00 0.10 0.20 0.30 0.40 0.50

X

1y5y10y

JPM

0.00

0.01

0.02

0.03

0.04

0.05

0.06

MS

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07 MS

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

1y5y10y

1y5y10y

1y5y10y

figure 12.4 PDF of the driver x(T) for JPM in the model with DNJs (top lhs) and ENJs (toprhs) and for MS in the model with DNJs (bottom lhs) and ENJs (bottom rhs). T = 1y, 5y,and 10 y.

JPM

80%75% 85% 95% 105% 115% 125%

100%

120%

140%

160%

K/S

75% 85% 95% 105% 115% 125%

K/S

CDS

Impl

ied

Vol DNJ ENJ

MS

40%

60%

80%

100%

CDS

Impl

ied

Vol

DNJ ENJ

figure 12.5 Lognormal credit default swaption volatility implied from model with Te = 1y,Tt = 5y as a function of the inverse moneyness K /c (Te , Te + Tt ).

In Figure 12.5, we show the lognormal CDSO volatility for one-year payer optionon five-year CDS contract (Te = 1y, Tt = 5y), which is calculated by using the model-implied option price. The price is converted into volatility via Black’s futures formula(Black 1976; Hull and White 2003; Schönbucher 2003), which assumes that the forwardspread c(Te , Te + Tt) has lognormal volatility Û. We plot the model implied volatility




as a function of inverse moneyness, K /c (Te , Te + Tt), where c(Te , Te + Tt) is the CDSspread at option expiry. We observe that the model implied lognormal volatility Û

exhibits a positive skew. This effect is in line with market observations because the CDSspread volatility is expected to increase when the CDS spread increases, so that optionmarket makers charge an extra premium for out-of-the-money CDSOs. Comparingmodels with different jump distributions, we see that the model with DNJs implieshigher spread volatility than the one with ENJs.

In Figure 12.6, we convert the model implied values for put options with maturity of6m into implied volatility skew via the Black-Scholes formula. We see that the modelimplies a remarkable skew in line with the skew observed in the market. The impliedequity skew appears to be less sensitive to the choice of the jump size distribution thanthe CDSO skew.

In Figure 12.7, we present the spread term structure for the CDS contract referencingMS, which should be paid by a risk-free PB to JPM as PS. This spread is determinedin such a way that the present value of CDS for PB is zero at the contract inception.Similarly, we present the spread term structure for the CDS contract referencing MS,which should be received by a risk-free PS from JPM as PB. We use the calibratedmodel with DNJs and five choices of the model correlation parameter: Ò = 0.90, 0.45,0.0, −0.45, −0.90. Spreads are expressed in basis points. We also present results for MSas counterparty and JPM as RN. We observe the same effects as before. Importantly,we see that CVA is not symmetric. As expected, riskier counterparty quality impliessmaller equilibrium spread for PB and higher equilibrium spread for PS.

Figure 12.7 illustrates the following effects. First, high positive correlation betweenRN and PS implies low equilibrium spread for a risk-free PB, because, given PS default,the loss of protection is expected be high. Second, high negative correlation betweenRN and PB implies high equilibrium spread for a risk-free PS, because, given PBdefault, the corresponding annuity loss is high. In general, for a risk-free PB, weobserve noticeable differences between the risk-free and equilibrium spreads, whilefor a risk-free PS, these differences are relatively small.

JPM

30%75% 85% 95% 105% 115% 125%

40%

50%

60%

K/S75% 85% 95% 105% 115% 125%

K/S

Equi

ty Im

plie

d Vo

l

DNJENJMarketVol

MS

30%

40%

50%

60%

Equi

ty Im

plie

d Vo

l

DNJENJMarketVol

figure 12.6 Lognormal equity volatility implied by the model as a function of inverse money-ness K /s for put options with T = 0.5y.




MS-JPM, Risky Seller

1001

110

120

130

140

T

Spre

ad, b

p

InputSpreadrho=0.90rho=0.45rho=0.0rho=-0.45rho=-0.90




MS-JPM, Risky Buyer

100

110

120

130

140

T

Spre

ad, b

p

JPM-MS, Risky Seller

35

45

55

65

Spre

ad, b

p

JPM-MS, Risky Buyer

35

45

55

65

Spre

ad, b

p

2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

1T T

2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

figure 12.7 Equilibrium spread for a PB (top lhs) and PS (top rhs) of a CDS on MS with JPMas the counterparty; same for a CDS on JPM with MS as the counterparty (bottom rhs) and(bottom lhs). In both cases input spread is the fair spread for non-risky counterparties.

8 Conclusion................................................................................................................................................

We have presented a multi-dimensional extension of Merton’s model (Merton 1974),where the joint dynamics of the firm’s values are driven by a multi-dimensional jump-diffusion process. Applying the FFT and FD methods, we have developed a forwardinduction procedure for calibrating the model, and a backward induction procedurefor valuing credit derivatives in 1D and 2D.

We have considered jump size distributions of two types, namely, DNJs and ENJs,and showed that for joint bivariate dynamics, the model with ENJs produces a notice-ably lower implied Gaussian correlation than the one produced by the model withDNJs, although for both jump specifications the corresponding marginal dynamics fitthe market data adequately. Based on these observations, and given the high level ofthe default correlation among financial institutions (above 50%), the model with DNJs,albeit simple, seems to provide a more realistic description of default correlations and,thus, the counterparty risk, than a more sophisticated model with ENJs.

We have proposed an innovative approach to CVA estimation in the structuralframework. We have found empirically that, to leading order, CVA is determinedby the correlation between RN and PS (or PB), the higher the correlation the higherthe charge; the second influential factor is their CDS spread volatilities. Thus, by




accounting for both the correlation and volatility, our model provides a robust esti-mation of the counterparty risk.

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