Lipton, A. and Sepp, A. (2011). Credit value adjustment in the extended structural default model. In The Oxford Handbook of Credit Derivatives, pages 406-463. Oxford University.

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c h a p t e r 12
................................................................................................................
credit value
adjustment in the
extended structural
default model
................................................................................................................
alexander lipton and artur sepp
1 Introduction
................................................................................................................................................
1.1 Motivation
In view of the recent turbulence in the credit markets and given a huge outstanding
notional amount of credit derivatives, counterparty risk has become a critical issue for
the financial industry as a whole. According to the most recent survey conveyed by
the International Swap Dealers Association (see <www.isda.org>), the outstanding
notional amount of credit default swaps is $38.6 trillion as of 31 December 2008 (it
has 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, while
the outstanding notional amount of equity derivatives was $8.7 trillion. The biggest
bankruptcy in US history filed by one of the major derivatives dealers, Lehman
Brothers Holdings Inc., in September of 2008 makes counterparty risk estimation and
management vital to the financial system at large and all the participating financial
institutions.
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 reference
name (RN) in exchange for periodic payments to the protection seller (PS) whose
magnitude is determined by the so-called CDS spread. When a PB buys a CDS from
a risky PS they have to cope with two types of risk: (a) market risk which comes

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credit value adjustment 407
directly from changes in the mark-to-market (MTM) value of the CDS due to credit
spread and interest rate changes; (b) credit risk which comes from the fact that PS
may be unable to honour their obligation to cover losses stemming from the default
of the corresponding RN. During the life of a CDS contract, a realized loss due to the
counterparty exposure arises when PS defaults before RN and, provided that MTM
of the CDS is positive, the counterparty pays only a fraction of the MTM value of the
existing CDS contract (ifMTMof the CDS is negative to PB, this CDS can be unwound
at 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 PS
default is especially big if the credit quality of RN and PS deteriorate simultaneously
but PS defaults first. We define the credit value adjustment (CVA), or the counterparty
charge (CC), as the maximal expected loss on a short position (protection bought) in
a CDS contract.
In order to describe CVA in quantitative rather than qualitative terms, in this
chapter we build a multi-dimensional structural default model. Below we concentrate
on its two-dimensional (2D) version and show that the evaluation of CVA is equivalent
to pricing a 2D down-and-in digital option with the down barrier being triggered
when the value of the PS’s assets crosses their default barrier and the option rebate
being determined by the value of the RN’s assets at the barrier crossing time. We also
briefly 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 MTM
value of the contract is zero.1 Thus, the option underlying CVA is at-the-money, so
that its value is highly sensitive to the volatility of the RN’s CDS spread, while the
barrier triggering event is highly sensitive to the volatility of the PS’s asset value. In
addition 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 model
the correlation between default times of RN and PS as well as CDS spread volatilities
for both of them. It turns out that our structural model is very well suited to accomplish
this 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 diffusion
and 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 on
an up-front basis with fixed coupon.

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value N. Following this ioneering insight, many authors proposed various extensions
of the basic model (Black and Cox 1976; Kim and Ramaswamy, and Sundaresan
1993; Nielsen, and Saa-Requejo, and Santa-Clara 1993; Leland 1994; Longstaff and
Schwartz 1995; Leland and Toft 1996; Albanese and Chen 2005) among others. They
considered more complicated forms of debt and assumed that the default event may
be triggered continuously up to the debt maturity. More recent research has been
concentrated 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 latter
task 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 barriers
stochastic (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 several
researchers (Zhou 2001b; Hull and White 2001;Haworth 2006;Haworth Reisinger, and
Shaw 2006; Valu˘zis 2008), who considered bivariate correlated log-normal dynamics
for two firms and derived analytical formulas for their joint survival probability;
Li (2000), who introduced the Gaussian copula description of correlated default times
in multi-dimensional structural models; Kiesel and Scherer (2007), who studied a
multi-dimensional structural model and proposed a mixture of semi-analytical and
Monte Carlo (MC) methods for model calibration and pricing.
While we build a general multi-dimensional structural model, our specific efforts
are aimed at a quantitative estimation of the counterparty risk. Relevant work on the
counterparty risk includes, among others, Jarrow and Turnbull (1995), who developed
the so called reduced-form default model and analysed the counterparty risk in this
framework; Hull and White (2001), Blanchet-Scalliet and Patras (2008), who modelled
the 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 considered
correlated 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 Fast
Fourier 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 and
Andreasen (2000), Lipton (2003), d’Halluin, Forsyth, and Vetzal (2005), Cont and
Voltchkova (2005), Carr and Mayo (2007), Lipton, Song, and Lee (2007), Toiva-nen
(2008), and Clift and Forsyth (2008).

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credit value adjustment 409
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 assumption
that firms’ values are driven by correlated additive processes. (Recall that an additive
process is a jump-diffusion process with time-inhomogeneous increments.) In order
to calibrate the 1D version of our structural model to the CDS spread curve observed in
themarket, we introduce jumps with piecewise constant intensity. We correlate jumps
of different firms via aMarshall-Olkin inspiredmechanism (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 to
implement, they require uniform grids and a large number of discretization steps.
At the same time, FD-based methods, while more complex, tend to provide greater
flexibility and stability. As part of our FD scheme development, we obtain new explicit
recursion formulas for the evaluation of the 2D convolution term for discrete and
exponential jumps. In addition, we present a closed-form formula for the joint survival
probability of two firms driven by correlated lognormal bivariate diffusion processes
by using the method of images, thus complementing results obtained byHe, Keirstead,
and Rebholz, (1998), Lipton (2001), and Zhou (2001b) via the classical eigenfunction
expansionmethod. As always, themethod of images works well for shorter times, while
the method of eigenfunction expansion works well for longer times.
We use the above results to develop an innovative approach to the estimation of
CVA for CDSs. Our approach is dynamic in nature and takes into account both
the correlation between RN and PS (or PB) and the CDS spread volatilities. The
approaches proposed by Leung and Kwok (2005), Pykhtin and Zhu (2006), andMisir-pashaev
(2008) do not account for spread volatility and, as a result, may underesti-mate
CVA. Blanchet-Patras consider a conceptually similar approach; however, their
analytical implementation is restricted to lognormal bivariate dynamics with constant
volatilities, which makes it impossible to fit the term structure of the CDS spreads
and CDS option volatilities implied by the market (Blanchet-Scalliet and Patras 2008).
Accordingly, the corresponding CVA valuation is biased. In contrast, our model can
be fitted to an arbitrary term structure of CDS spreads and market prices of CDS and
equity options. The approach by Hull and White (2001) uses MC simulations of the
correlated lognormal bivariate diffusions. In contrast, our approach assumes jump-diffusion
dynamics, potentially more realistic for default modelling, and uses robust
semi-analytical and numerical methods for model calibration and CVA valuation.
This chapter is organized as follows. In section 2 we introduce the structural default
model in one, two, and multi-dimensions. In section 3 we formulate the generic
pricing problem in one, two and multi-dimensions. In section 4 we consider the
pricing problem for CDSs, CDS options (CDSOs), first-to-default swaps (FTDSs),
and the valuation problem for CVA. In section 5 we develop analytical, asymptotic,

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and numerical methods for solving the 1D pricing problem. In particular, we describe
MC, FFT, and FD methods for solving the calibration problem via forward induction
and the pricing problem via backward induction. In section 6 we present analytical
and numerical methods for solving the 2D pricing problem, including FFT and FD
methods. In section 7 we provide an illustration of our findings by showing how to
calculate CVA for a CDS on Morgan Stanley (MS) sold by JP Morgan (JPM) and a
CDS 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.
Qt
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. We
assume that Q is specified by market prices of liquid credit products. The operation
of expectation under Q given information set F(t) at time t is denoted by E[·]. The
imaginary unit is denoted by i, i =
√
−1.
The instantaneous risk-free interest rate r (t) is assumed to be deterministic; the
corresponding discount factor, D(t, T) is given by:
D(t, T) = exp
−
T
t
r (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 ifˆ is true
0 ifˆ 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 by
δn,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 function
P (a, b) defined as follows:

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credit value adjustment 411
P(a, b) = exp
ab + b2/2
N(a + b) (5)
2.2 One-dimensional case
2.2.1 Asset value dynamics
We 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)dN(t) (6)
where Ê(t) is the deterministic dividend rate on the firm’s assets, W(t) is a standard
Brownian 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 random
variable 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 PDFs
with negative jumps which may result in random crossings of the default barrier. We
consider 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 1
Ì , Ì 0, with:
( j) = ÌeÌj , j 0, Í =
Ì
Ì + 1
− 1 = − 1
Ì + 1
(9)
In our experience, for 1Dmarginal dynamics the choice of the jump size distribution
has 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 we
will demonstrate shortly.
2.2.2 Default boundary
The cornerstone assumption of a structural default model is that the firm defaults
when 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 to
study the optimal capital structure of a firm. Under a fairly strict assumption that the
firm’s liabilities can only be financed by issuing new equity, the equity holders have the
right to push the firm into default by stopping issuing new equity to cover the interest
payments to bondholders and, instead, turning the firm over to the bondholders. Black
and Cox (1976) found the critical level for the firm’s value, below which it is not optimal
for equity holders to sell any more equity. Equity holders should determine the critical
value or the default barrier by maximizing the value of the equity and, respectively,

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minimizing the value of outstanding bonds. Thus, the optimal debt-to-equity ratio and
the endogenous default barrier are decision variables in this approach. A nice review of
the 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 not
realistic given the complicated equity-liability structure of large firms and the actual
relationships between the firm’s management and its equity and debtholders. For
example, in July 2009 the bail-out of a commercial lender CIT was carried out by
debtholders, who proposed debt restructuring, rather than by equity holders, who had
no 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 estimated
by 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 and
Schwartz 1995; etc.).
In our approach, similarly to Lipton (2002b); and Stamicar and Finger (2005), we
assume 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’s
liabilities and L(0) is its total debt per share. We find L(0) from the balance sheet
as the ratio of the firm’s total liability to the total common shares outstanding; R is
found from CDS quotes, typically, it is assumed that R = 0.4.
2.2.3 Default triggering event
The key variable of the model is the random default time which we denote by Ù. We
assume that Ù is an F-adapted stopping time, Ù ∈ (0,∞]. In general, the default event
can be triggered in three ways.
First, when the firm’s value is monitored only at the debt’s maturity time T, then
the 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, {tdm
}m=1,. . .,M,
0 td
1 . . . tdM
≤ T, then the default event can only occur at some time tdm
. The
corresponding default time is specified by:

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credit value adjustment 413
Ù = min{tdm
: a(tdm
) ≤ 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. The
corresponding 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-diffusion
dynamics (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 rates
than those for the problem with CDM, because the solution is not smooth at default
monitoring times in the vicinity of the default barrier. However, MC-based methods
can be applied in the case of DDM in a robust way, because the firm’s asset values need
to be simulated only at default monitoring dates. Importantly, there is no conceptual
difficulty in applying MC simulations for the multi-dimensional model.
In the case of CDM closed-form solutions are available for the survival probability
in 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-diffusions
with 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 two
dimensions tend to have a better rate of convergence in space and time than in the
case of DDM. However, a serious disadvantage of the CDM assumption is that the
corresponding MC implementation is complex and slow because continuous barriers
are difficult to deal with, especially in the multi-dimensional case.
Accordingly, CDM is useful for small-scale problems which can be solved without
MC 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 marginal
dynamics of individual firms andMC techniques can be used to solve the pricing prob-lem
for several firms. In our experience, we have not observed noticeable differences
between 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 use
about 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 default
monitoring frequency in the first year and quarterly monitoring in the following years.
2.2.4 Asset value, equity, and equity options
We introduce the log coordinate x(t):
x(t) = ln
a(t)
l (t)
(15)

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414 a. lipton a. sepp
and represent the asset value as follows:
a(t) = E (t)l (0)e x(t) = l (t) e x(t) (16)
where x(t) is driven by the following dynamics under Q:
dx(t) = Ï(t)dt + Û(t)dW(t) + j dN(t) (17)
x(0) = ln
a(0)
l (0)
≡ Ó, Ó 0
Ï(t) = −1
2
Û2(t) − Î(t)Í
We observe that, under this formulation of the firm value process, the default time
is 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 respect
to the filtration F which is characterized by the following conditions: y(t) is adapted
to 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 additive
process and a Levy process is that the distribution of increments in the former process
is time dependent.
Without loss of generality, we assume that volatility Û(t) and jump intensity Î(t) are
piecewise constant functions of time changing at times {tc
k
}, k = 1, . . . , k:
Û(t) =
k
k=1
Û(k)1{tc
k−1t≤tc
k
} + Û(k)1{ttc
k
} (19)
Î(t) =
k
k=1
Î(k)1{tc
k−1t≤tc
k
} + Î(k)1{ttc
k
}
where Û(k) defines the volatility and Î(k) defines the intensity at time periods (tc
k−1, tc
k ]
0 = 0, k = 1, . . . , k. In the case of DDM we assume that {tc
with tc
k
} is a subset of {tdm
}, so
that parameters do not jump between observation dates.
We consider the firm’s equity share price, which is denoted by s (t), and, following
Stamicar and Finger (2005), assume that the value of s (t) is given by:
s (t) =
a(t) − l (t) = E (t)l (0)
e x(t) − 1
= l (t)
e x(t) − 1
, {t Ù}
0, {t ≥ Ù} (20)
At time t = 0, s (0) is specified by themarket price of the equity share. Accordingly, the
initial value of the firm’s assets is given by:
a(0) = s (0) + l (0) (21)

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credit value adjustment 415
It is important to note that Û(t) is the volatility of the firm’s assets. The volatility of
the 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) decreases
creating the leverage effect typically observed in the equity market. The model with
equity volatility of the type (22) is also know as the displaced diffusion model, which
was introduced by Rubinstein (1983).
2.3 Two-dimensional case
To deal with the counterparty risk problem, we need to model the correlated dynamics
of two ormore credit entities. We consider two firms and assume that their asset values
are driven by the following stochascic differential equations(SDEs):
dai (t) = (r (t) − Êi (t) − Íi Îi (t))ai (t) dt + Ûi (t) ai (t)dWi (t) +
e ji −1
ai (t)dNi (t)
(23)
where
Íi =
0
−∞
e jii ( ji )d ji −1 (24)
jump amplitudes ji has the same PDF i ( ji ) as in the marginal dynamics, jump
intensities 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. The
corresponding default boundaries have the form:
li (t) = Ei (t) li (0) (25)
where
Ei (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) + jidNi (t) (28)
xi (0) = ln
ai (0)
li (0)
≡ Ói , Ói 0
Ïi (t) = −1
2
Û2
i (t) − Íi Îi (t)

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416 a. lipton a. sepp
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 Brownian
motions W1(t) and W2(t) are correlated with correlation Ò. Second, Poisson processes
N1(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 advantage
for model calibration. After the calibration to marginal dynamics is completed for
each firm, and the set of firm’s volatilities, jump sizes, and intensities is obtained, we
estimate the parameter Ò by fitting themodel spread of a FTDS to a givenmarket 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, we
calculate the instantaneous correlations between the drivers x1(t) and x2(t) as follows:
ÒDNJ
12 =

12. ÒÛ1Û2 + Î{

13. 1,2}Ì1Ì2
Û+ Î1ÌÛ22
21
21
+ Î2Ì22
, ÒENJ
12 =

14. ÒÛ1Û2 + Î{

15. 1,2}/(Ì1Ì2)
Û+ 2Î1/ÌÛ22
21 21
+ 2Î2/Ì22
(33)
where we suppress the time variable. Here ÒDNJ
12 and ÒENJ
12 are correlations for DNJs and
ENJs, respectively.
For large systemic intensities Î{1,2}, we see that ÒDNJ
12
∼ 1, while ÒENJ
12
∼ 12
. Thus, for
ENJs correlations tend to be smaller than for DNJs. In our experiments with different
firms, we have computed implied Gaussian copula correlations from model spreads
of FTDS referencing different credit entities and found that, typically, the maximal
implied Gaussian correlation that can be achieved is about 90% for DNJs and about
50% for ENJs (in both casesmodel parameters were calibrated to match the term struc-ture
of CDS spreads and CDS option volatilities). Thus, the ENJs assumption is not
appropriate for modelling the joint dynamics of strongly correlated firms belonging to
one 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 same
equations as before, but with the index i running from 1 to N, i = 1, . . . , N.

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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 of
all 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 represent
Ni (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. By
analogy, we can introduce systemic and idiosyncratic factors for the Brownian motion
dynamics.
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 axis
x ∈ R1:
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 tdm
. At t = tdm, the value function undergoes a transformation
Vm−(x) = –
Vm+(x)
(38)
dm
dm
dm
where –{.} is the transformation operator, which depends on the specifics of the
contract under consideration, and Vm± (x) = V(t±, x). Here t± ± = tε. Finally,
at t = T
V (T, x) = v (x) (39)

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418 a. lipton a. sepp
the terminal payoff function v(x) is contract specific. Here L(x) is the infinitesimal
operator 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) =
1
2
Û2(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 terminal
payoff 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 depend
on the residual value of the firm), and the far-field boundary condition. The backward
equation for the value function V(t, x) is formulated differently on the positive semi-axis
x ∈ R1
+ and negative semi-axis R1
−:
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
−x
V(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{Ìx}, DNJs
Ì
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
0
V (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)

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where J (x), Z(x)(x) are defined by:
J (x)V(x) =
0
−x
V(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 function
z(x). As a result, by subtracting Z(x) from rhs of eq. (43), we can formulate the pricing
equation on the positive semi-axis R1
+ as follows:
Vt (t, x) + L(x)V(t, x) − r (t) V(t, x) = ˆc (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)
ˆc (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 and
T. For DDM the valuation problem for G can be formulated as follows:
GT (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{X0} (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) =
1
2
Û2(t)gxx (x) − Ï(t)gx (x) − Î (t) g (x) (61)

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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:
GT (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 payoff
function 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, and
the 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 bivariate
dynamics (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 by
eq. (41) with Î (t) = Î{i } (t); J (x1) and J (x2) are the 1D orthogonal integral operators in
x1 and x2 directions defined by eq. (45) with appropriate model parameters; C(x1,x2) is
the correlation operator:

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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) = ˆc (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:
ˆc (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

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For DDM, the corresponding Green’s function G(t, x1, x2, T, X1, X2), satisfies the
following problem in the whole plane R2:
GT (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{X10,X20} (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 the
positive quadrant R2
+,+:
GT (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 pricing
problem for the value function V (t,

22. x) in the positive cone RN
+ as follows:
Vt (t,

23. x) + L(

24. x)V (t,

25. x) − r (t) V (t,

26. x) = ˆc (t,

27. x) (89)
V
t,

28. x0,k
= v0,k (t,

29. y
k), V (t,

30. x) →
xk→∞
v∞,k (t,

31. x) (90)
V (T,

32. x) = v (

33. x) (91)
where

34. x,

35. x0,k ,

36. y
k are N and N − 1 dimensional vectors, respectively,

37. x = (x1, . . . , xk, . . . xN)

38. x0,k =
x1, . . . ,0k
, . . . xN

39. y
k = (x1, . . . xk−1, xk+1, . . . xN)
(92)

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Here ˆc (t,

41. x), v0,k (t,

42. y
), v∞,k (t,

43. x), v (

44. x) are known functions which are contract
specific. The function ˆc (t,

45. x) incorporates the terms arising from rebates. The cor-responding
operator L(

46. x) can be written in the form
L(

47. x) f (

48. x) =
1
2
i
Û2
i ∂2
i f (

49. x) +
i, j, ji
Òi j ÛiÛj ∂i ∂j f (

50. x) (93)
+
i
Ïi ∂i f (

51. x) +
∈–(N)
Î
i∈
J (xi ) f (

52. x) − f (

53. x)
where
J (xi ) f (

54. 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)

55. 3.4 Green’s X
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,

56. T, , such that
GT
t,

57. x, T,

58. X
− L(

59. X
)†G
t,

60. x, T,

61. X
= 0 (95)
G
t,

62. x, T,

63. X
0k
= 0, G
t,

64. x, T,

65. X
→
Xk→∞0 (96)
G
t,

66. x, t,

67. X
= δ

68. X
−

69. x
(97)
Here L(

70. x)† is the corresponding adjoint operator
L(

71. x)†g (

72. x) =
1
2
i
Û2
i ∂2
i g (

73. x) +
i, j, ji
Òi j ÛiÛj ∂i ∂j g (

74. x) (98)
−
i
Ïi ∂i g (

75. x) +
∈–(N)
Î
i∈
J (xi )†g (

76. x) − g (

77. x)
where
J (xi )†g (

78. x) =
g (x1, . . . , xi + Ìi , . . . xN), DNJs
Ìi
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
+
d

79. x = 0 (100)
J (xi ) f (

80. x) g (

81. x) − f (

82. x)J (xi )†g (

83. x)

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Accordingly, integration by parts yields
V (t,

85. x) = −
T
t
RN
+
ˆc
t
,

86. x
D
t, t
G
t,

87. x, t
,

88. x
d

89. xdt (101)
+
k
T
t
RN−1
+
v0,k
t
k
,

90. y
D
t, t
gk
t,

91. x, t
k
,

92. y
d

93. y
kdt
+D (t, T)
RN
+
v

94. x
G
t,

95. x, T,

96. x
d

97. x
where
gk
t,

98. x, T,

99. Yk
=
1
2
Û2k
∂kG
t,

100. x, T,

101. X
Xk=0
(102)

102. Yk = (X1, . . . , Xk−1, Xk+1, . . . , XN)
represents the hitting time density for the corresponding point of the boundary. In
particular, the initial value of a claim has the form
V
0,

103. Ó
= −
T
0
RN
+
ˆc
t
,

104. x
D
0, t
G
0,

105. Ó, t
,

106. x
d

107. xdt (103)
+
k
T
0
RN−1
+
v0,k
t
k
,

108. y
D
0, t
gk
0,

109. Ó, t
k
,

110. y
d

111. y
kdt
+D (0, T)
RN
+
v

112. x
G
0,

113. Ó, T,

114. x
d

115. x
This extremely useful formula shows that instead of solving the backward pricing
problem with inhomogeneous right hand side and boundary conditions, we can solve
the forward propagation problem for the Green’s function with homogeneous rhs and
boundary conditions and perform the integration as needed.
4 Pricing problem for credit derivatives
................................................................................................................................................
In this section we formulate the computational problem for several important credit
products. 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{Ùt}EQt
[1{ÙT}] (104)

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Using the default event definition (13), one can show that for DDM, Q(x)(t, x, T) solves
the following backward problem on R1:
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{x0} (107)
Q(x)(T, x, T) = 1{x0} (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
+:
t (t, x, T) + L(x)Q(x)(t, x, T) = 0 (109)
Q(x)
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 x
survival becomes certain. Green’s formula (101) yields
Q(x) (t, x, T) =
∞
0
G (t, x, T, X)dX (112)
We define the joint survival probability, Q(x1,x2)(t, x1, x2, T), as follows:
Q(x1,x2)(t, x1, x2, T) ≡ 1{Ù1t,Ù2t}EQt[1{Ù1T,Ù2T}] (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{x10,x20} (116)
Q(x1,x2)(T, x1, x2, T) = 1{x10,x20} (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).

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For CDM, Q(x1,x2)(t, x1, x2) solves the following problem:
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) →
Q(x1,x2)
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
∞
0
G (t, x1, x2, T, X1, X2)dX1dX2 (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 the
nearest payment date, in exchange for receiving loss given RN’s default from PS. For
standardized 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 current
time t up to the maturity time T. Assuming that RN has not defaulted yet, Ù t, we
compute the expected present value of the annuity leg, A(t, T), as:
A(t, T) =
T
t
D
t, t
Q(t, t)dt (122)
where Q(t, t) is the corresponding survival probability, and the expected present value
of the protection leg, P (t, T), as:
P (t, T) = −(1 − R)
T
t
D
t, t
dQ(t, t) (123)
= (1 − R)
1 − D (t, T) Q (t, T) −
T
t
r
t
D
t, t
Q(t, t)dt
where R is the expected debt recovery rate which is assumed to be given for valuation
purposes (typically, R is fixed at 40%).
For PB the present value of the CDS contract with coupon (or spread) c , is given by:
VCDS(t, T) = P (t, T) − c A(t, T) (124)

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The par coupon c (0, T) is defined in such a way the time t = 0 the value of the CDS
contract 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 used
to parametrize the risk-neutral survival probability of RN. Hazard rates are inferred
from the term structure of CDS spreads via bootstrapping.
One of the drawbacks of the reduced-form approach is that it assumes that CDS
spreads 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 CDS
spreads caused by changes in the firm’s value. Thus, the structural model can be used
for valuing credit contracts depending on the volatility of credit spreads.
For DDM, the value function for PB of a CDS contract, VCDS(t, x, T), solves eq.
(36), supplied with the following conditions:
VCDS(t, x, T) →
x→−∞ 1 − Rex , VCDS(t, x, T) →
x→∞
−c
M
m=m+1
‰tmD(t, tm) (126)
m+ (x, T) − ‰tmc )1{x0} + (1 − Rex )1{x≤0} (127)
VCDS
m− (x, T) = (VCDS
VCDS (T, x, T) = −‰tMc1{x0} + (1 − Rex )1{x≤0} (128)
where ‰tm = tm − tm−1.
For CDM, VCDS(t, x, T) solves eq. (51) with
ˆc (t, x) = c − Î (t) Z(x) (x) (129)
Z(x)(x) =
H(Ì − x)
1 − Rex−Ì
, DNJs
1 − R Ì
1+Ì
e−Ìx , ENJs (130)
Here we assume that the floating recovery rate, Rex , represents the residual value of the
firm’s assets after the default. The corresponding boundary and terminal conditions
are
VCDS(t, 0, T) = (1 − R), VCDS(t, x, T) →
x→∞
−c
T
t
D(t, t)dt (131)
VCDS(T, x, T) = 0 (132)
The boundary condition for x →∞ expresses the fact that for large positive x the
present value of CDS becomes a risk-free annuity.

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4.3 Credit default swap option
A CDSO contract serves as a tool for locking in the realized volatility of CDS rate up
to the option’s maturity. By using CDSOs quotes we can calibrate the model to this
volatility. The payer CDSO with maturity Te and tenor Tt gives its holder the right to
enter 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, the
payout of the payer CDSO is given by:
VCDS(Te , Te + Tt ; K) = 1{ÙTe } {P (Te , Te + Tt ) − K A(Te , Te + Tt )}
+ (133)
For DDM, the value function for the buyer of CDSO, VCDSO(t, x, T), solves eq.
(36) with c = 0, and the following conditions:
VCDSO(t, x) →
x→±∞0 (134)
VCDSO
m− (x) = VCDSO
m+ (x) 1{x0} (135)
VCDSO(Te , x) =
VCDS(Te , x, Te + Tt ; K)
+ 1{x0} (136)
For CDM, VCDSO(t, x) is governed by eq. (51) with ˆc = 0, supplied with the follow-ing
conditions:
VCDSO(t,0) = 0, VCDSO(t, x) →
x→∞0 (137)
VCDSO(Te , x) =
VCDS(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 by
eq. (20). In the context of studying credit products, the value of the equity put option
is themost relevant one, since such options provide protection against the depreciation
of 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 with
the following conditions:
V P (t, x) →
x→−∞ D(t, T)K, V P (t, x) →
x→∞0 (139)
V P
m− (x) = V P
m+ (x) 1{x0} + D (tm, T) K1{x≤0} (140)
V P (T, x) =
K − l (T) (e x − 1)
+ 1{x0} + K1{x≤0} (141)
For CDM, V P (t, x) solves eq. (51) with
ˆc (t, x) = −Î (t) Z(x) (t, x) (142)

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Z(x) (t, x) =
D (t, T) KH(Ì − x) , DNJs
D (t, T) Ke−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) (e x − 1)
+ (145)
We note that, in this model, put-call parity for European options should be
expressed in terms of defaultable forward contracts.
Since, in general, we have to solve the pricing problem numerically, American style
options 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 pays
PS 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 first
default in the basket. The market of FTDSs is relatively liquid with a typical basket size
of five underlying names.
In this chapter we consider FTDSs referencing just two underlying names. The
premium 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 value
VF TDS(t, x1, x2, T) for PB of a FTDS, we have to solve eq. (73) with ˆc (t, x1, x2) of
the form:
ˆc (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) =
1
2
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 simultaneous
default of both RNs, PB receives the notional minus their average recovery. The
corresponding boundary and terminal conditions are
VF TDS(t, 0, x2, T) = 1 − R1, VF TDS(t, x1, 0, T) = 1 − R2 (148)
VF TDS(t, x1, x2, T) →
xi→∞ VCDS(t, x3−i , T) (149)
VF TDS(T, x1, x2, T) = 0 (150)

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The market practice is to quote the spread on a FTDS by using the Gaussian
copula with specified pair-wise correlation, Ò, between default times of RNs (see, for
example, Li 2000 and Hull, Nelkin, and White 2004). Thus, we can calibrate the model
correlation parameter to FTDS spreads observed in the market.
4.6 Credit default swap with counterparty risk
4.6.1 Credit value adjustment
First, we consider a CDS contract sold by a risky PS to a non-risky PB. We denote by
Ù1 the default time of RN and by Ù2 the default time of PS. We assume that this CDS
provides protection up to time T and its coupon is c .We also assume that the recovery
rate of RN is R1 and of PS is R2.
We denote by ˜V
CDS(t
, T) the value of the CDS contract with coupon c maturing
˜at time V
T conditional on PS defaulting at time t.We make the following assumptions
about the recovery value of the CDS given PS default at time Ù2: if CDS (t
, T) 0,
PB pays the full amount of −˜V
CDS (t
, T) to PS; if ˜V
CDS (t
, T) 0, PB receives only
a fraction R2 of ˜V
CDS (t
, T).
Thus, CVA for PB, VCV A
P B (t, T), is defined as the expected maximal potential loss
due to the PS default:
VCV A
P B (t, T) = EQt
T
t
D(t, t)(1 − R2)
˜V
CDS
t
, T
+
dt
(151)
Accordingly, to value CVA we need to know the survival probability Q (t
, t) for
RN conditional on PS default at time t. In this context, Pykhtin and Zhu (2006) and
Misirpashaev (2008) applied the Gaussian copula model, while (Blanchet-Scalliet and
Patras (2008) applied a bivariate lognormal structural model to calculate the relevant
quantity.
Similarly, for a CDS contract sold by a non-risky PS to a risky PB we have the
following expression for CVA for PS:
VCV A
P S (t, T) = EQt
T
t
D(t, t)(1 − R3)
−˜V
CDS
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 framework
We start with the risky PS case and denote by x1 the driver for the RN’s value and by
x2 the driver for the PS’s value.
In the context of the structural default model, the 2D plane is divided into four
quadrants as follows: (A) R2
+,+, where both RN and PS survive; (B) R2
−,+, where RN

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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 − R1e x1) (1 − R2e x2 ). In R2
+,− CVA is set to (1 − R2e x2 )
VCDS(t, x1, T)
+,
where VCDS(t, x1, T) is the value of the CDS on RN at time t and state x1, because
the CDS protection is lost following PS default. The value of CVA is computed as the
solution to a 2D problem with given rebates in regions R2
−,+, R2
−,−, and R2
+,−.
For DDM, the value of CVA, VCV A
P B (t, x1, x2, T), satisfies eq. (66) and the following
conditions
VCV A
P B (t, x1, x2, T) →
x1→−∞ (1 − R1e x1)(1 − R2e x2 )1{x2≤0} (153)
VCV A
P B (t, x1, x2, T) →
x1→∞ 0
VCV A
P B (t, x1, x2, T) →
x2→−∞ (1 − R2e x2 )
VCDS(t, x1, T)
+
VCV A
P B (t, x1, x2, T) →
x2→∞ 0
VCV A
P B,m− (x1, x2) = VCV A
P B,m+(x1, x2)1{x10,x20} (154)
+(1 − R1e x1)(1 − R2e x2 )1{x1≤0,x2≤0}
+(1 − R2e x2 )
VCDS(t, x1, T)
+ 1{x10,x2≤0}
VCV A
P B (T, x1, x2, T) = 0 (155)
where VCDS(t, x1, T) is the value of the non-risky CDS computed by solving the
corresponding 1D problem.
For CDM, we have to solve eq. (73) with ˆc (t, x1, x2) of the form:
ˆc (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) =
+ Z(x2) (x2) (157)
VCDS(t, x1, T)
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)
VCDS(t, x1 − Ì1, T)
+ DNJs
Ì1
0−
x1
+ eÌ1 j1d j1 ENJs
VCDS(t, x1 + j1, T)
(158)

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The corresponding boundary and final conditions are
VCV A
P B (t, 0, x2, T) = 0, VCV A
P B (t, x1, x2, T) →
x1→∞0 (159)
VCV A
P B (t, x1, 0, T) = (1 − R2)
VCDS(t, x1, T)
+
VCV A
P B (t, x1, x2, T) →
x2→∞ 0
VCV A
P 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 with
ENJs. Unfortunately, similar solutions for DNJs are not readily available. Results
presented 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 assume
constant model parameters, CDM, and restrict ourselves to ENJs. In more general
cases, we need to solve the corresponding problems directly. Analytical results can
serve as a useful tool for testing the accuracy of numerical calculations needed for less
restrictive cases.
5.1.1 Green’s function
Due to the timehomogeneity of the problem under consideration, the Green’s function
G(t, x, T, X) depends on Ù = T − t rather than on t, T separately, so that we can
represent 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)

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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:
1
2
Û2¯2 − ϯ − (Î + p) +
ÎÌ
−¯ + Ì
= 0 (168)
This equation has three roots, which, to facilitate comparison with earlier work, we
denote by −¯j , j = 1, 2, 3. It is easy to show that these roots can be ordered in such a
way that ¯1 −Ì ¯2 0 ¯3. Hence, the overall solution has the form:
√
(p, x, X) =
⎧⎨
⎩
C3e−¯3(X−x), X ≥ x
3!
j=1
Dj e−¯j (X−x), 0 X ≤ x (169)
where
Di = − 2
Û2
(Ì + ¯i )
, i = 1,2 (170)
(¯i − ¯3−i) (¯i − ¯3)
D3 = −e(¯1−¯3)xD1 − e(¯2−¯3)xD2, C3 = D1 + D2 + D3
The inverse Laplace transform of √
(p, x, X) yields √(Ù, x, X). A review of relevant
algorithms 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 probability
By using eqs. (112), (169) we compute the Laplace-transformed survival probability as
follows:
Q(x) (Ù, x) → ˆQ
(x) (p, x) (172)
ˆQ
(x) (p, x) =
∞
0
√
(p, x, X)dX (173)
=
∞
x
C3e−¯3(X−x)dX +
3
j=1
x
0
Dj e−¯j (X−x)dX =
2
j=0
E j e¯j x

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where
¯0 = 0, E0 =
1
p
, E1 =
(¯1 + Ì) ¯2
(¯1 − ¯2) Ìp
, E2 =
(¯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
dX = 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)dX (178)
Formula (178) is generic and can be used for arbitrary jump size distributions. For
ENJs, we obtain:
f (Ù, x) = Î
∞
0
e−ÌX√(Ù, x, X)dX (179)
Using eq. (169), the Laplace-transformed default time density can be represented as:
ˆq
(p, x) = ˆg (p, x) + ˆ f (p, x) (180)
where
ˆg (p, x) =
(Ì + ¯2)e¯2x − (Ì + ¯1)e¯1x
¯2 − ¯1
(181)
and
ˆ 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), we
obtain:
ˆq
(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 in
agreement.

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Without jumps, straightforward calculation yields
Q(x) (Ù, x) = N
√x
ˇ
−
√
ˇ
2
− e xN
−√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
√
ˇ
n
√x
ˇ
−
√
ˇ
2
, f (Ù, x) = 0 (185)
5.1.3 Credit default swap
We use the general formula (124) together with eq. (176), and express the present value
VCDS(Ù, x) of a CDS contract with coupon c as:
VCDS(Ù, x) = −c
Ù
0
Q(x)(Ù
e−r Ù
, x)dÙ
(186)
+(1 − R)
Ù
0
g (Ù
e−r Ù
, x)dÙ
+
1 − R
Ì
1 + Ì
Ù
0
f (Ù
e−r Ù
, x)dÙ
By using eqs. (173), (181), (182), we can compute the value of the CDS via the inverse
Laplace transform.
Without jumps VCDS(Ù, x) can be found explicitly. The annuity leg can be repre-sented
in the form
A (Ù, x) =
1
r
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 „ =

127. 8r/Û2 + 1, while the protection leg can be represented as follows
P (Ù, x) = (1 − R)
1 − e−r ÙQ(x) (Ù, x) − r A(Ù, x)
(188)
Accordingly,
VCDS(Ù, x) = (1 − R)
1 − e−r ÙQ(x) (Ù, x)
− ((1 − R) r + c ) A (Ù, x) (189)
5.1.4 Credit default swap option
In the time-homogeneous setting of the present section, we can represent the price of
a CDSO as follows
VCDSO(Ùe , x) = e−r Ùe
X∗
0
√(Ùe , x, X) VCDS(Ùt , X)dX (190)
where X∗ is chosen in such a way that VCDS(Ùt , X∗) = 0. We can use our previous
results to evaluate this expression via the Laplace transform.
As before, without jumps VCDSO can be evaluated explicitly.

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5.1.5 Equity put option
We use eq. (139) and represent the value of the put option with strike K and maturity
T 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 P
0 (Ù, Ó, T) is given by
ˆV
P
0 (p, Ó, T) =
∞
0
K − l (T)
e X − 1
+
ˆ√
(p, Ó, X)dX (192)
Straightforward calculation yields:
ˆV
P
0 (p, Ó, T) = l (T)
3
j=1
Dj 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
ˆV
P
0 (p, Ó, T) = l (T)
⎧⎨
3
⎩
j=1
Dj 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 P
0 (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 Example
In 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 compare
results obtained for the jump-diffusion model with the ones obtained for the diffu-sion
model the ‘equivalent’ diffusion volatility Ûdiff specified by Ûdiff =

129. Û2 + 2Î/Ì2,

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1.00
0.90
0.80
0.70
0.60
T
Survival Probability
s(T), Jump-diffusion
s(T), Diffusion
360
320
280
240
200
160
120
80
40
0
T
CDS spread, bp
s(T), Jump-diffusion
s(T), Diffusion
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9
80%
75%
70%
65%
60%
55%
50%
45%
40%
35%
30%
K/S
CDS Implied Volatility
JD D
210%
170%
130%
90%
50%
10%
K/S
Equity Implied Volatility
JD, T = 1m D, T = 1m
JD, T = 6m D, T = 6m
JD, T = 12m D, T = 12m
60% 80% 100% 120% 140% 160% 20% 40% 60% 80% 100% 120%
figure 12.1 The model implied survival probabilities for 1y ≤ T ≤ 10y (top lhs); CDS spreads
for 1y ≤ T ≤ 10y (top rhs); volatility skew for CDSO (bottom lhs); volatility skew for put
options with T = 1m, 6m, 12m (bottom rhs).
Ûdiff = 0.074 for the chosen model parameters. First, we show the term structure of
the implied spread generated by the jump-diffusion and diffusionmodels. We see that,
unlike the diffusionmodel, the jump-diffusionmodel implies a non-zero probability of
default in the short term, so that its implied spread is consistent with the one observed
in the market. If needed, we can produce different shapes of the CDS curve by using
the term structure of the model intensity parameter Î. Second, we show the model
implied 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 diffusion
model, so that, in general, it can fit themarket implied skewmore easily. An interesting
discussion of the equity volatility skew in the structural framework can be found in
Hull, Nelkin, and White, (2004/05).
5.2 Asymptotic solution
In this section, we derive an asymptotic expansion for the Green’s function-solving
problem (63) assuming that the jump-intensity parameter Î is small. More details of
the 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)

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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) =
1
2
Û2gxx (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:
(0)(Ù, x, X) + Θ√
˜ √(Ù, x, X) = ˜√
(1)(Ù, x, X) + . . . (201)
(0)(Ù, x, X) solves the following problem:
The zero-order term ˜√
(0)
Ù (Ù, x, X) − 1
˜√
2
(0)
XX(Ù, x, X) = 0 (202)
Û2 ˜√
(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)
(1)(Ù, x, X) solves the following problem:
The first-order term ˜√
(1)
Ù (Ù, x, X) − 1
˜√
2
(1)
XX(Ù, x, X) = ƒ(Ù, x, X) (206)
Û2 ˜√
(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
√
ˇ
,−Ì
− ÌP
−X√+ x
ˇ
√
ˇ
,−Ì
and P (a, b) is defined by eq. (5). We use Duhamel’s principle and represent
√(1)(Ù, x, X) as follows:

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(1)(Ù, x, X) =
˜√
Ù
0
∞
0
(0)(Ù − Ù
˜√
, x, X − X)ƒ(Ù
, x, X)dXdÙ
(210)
Fairly involved algebra yields:
(1)(Ù, x, X) =
˜√
Ì
ÌÛ2
ÌˇP
−X − √ x
ˇ
√
ˇ
,−Ì
(211)
+XP
−X√+ x
ˇ
√
ˇ
,−Ì
− (X − Ìˇ)P
−X√+ x
ˇ
√
ˇ
, Ì
−(X + Ìˇ)e−ÌxP
−√X
ˇ
√
ˇ
,−Ì
+ (X − Ìˇ)e−ÌxP
−√X
ˇ
√
ˇ
, Ì
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 the
calibration and pricing problems in 1D. Specifically, we present theMC, FFT, and FD-based
methods. The MC method, due to its generic nature, is easily applicable to the
problem at hand, particularly for DDM. However, it comes with the usual drawbacks
and is to be avoided whenever possible. The FFT method is well suited to solving
problems with DDM, however it has several well-known disadvantages including the
need for uniform grids with large number of steps, and complicated treatment of
aliasing effects. In our opinion, the FD method is the most powerful of the three. It
can be used for both DDM and CDM. The key difficulty in applying the FD method
for jumpdiffusions is the treatment of the integral term. The direct integrationmethod
(see, for example, Cont and Tankov 2004; Cont and Voltchkova 2005) has a complexity
of O(N2) operations per time step, where N is the spatial grid size. To obviate this
difficulty 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, and
Vetzal 2005 among others), however, this approach shares the disadvantages of the
conventional FFT method. It turns out that for DNJs and ENJs one can compute the
integral 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 the
FFT method is O(N log N) per each time step, so that the overall complexity with M
default monitoring is O(MNlog N). The complexity of the FD method with explicit
treatment of the integral term and L time steps is O(LN) (we note that if the set of

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monitoring times is sparse, we would need to add extra time steps to improve accuracy
of the FD method). Thus, the overall competitiveness of the methods depends on the
number 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 indeed
confirmed by our numerical experiments. In the case of CDM, the FD-based method is
expected to have second-order accuracy in space; while the FFT-based method cannot
easily be applied in this case.
5.3.1 Monte Carlo method
We assume DDM and describe the corresponding MC simulations. There are two
methods for the simulation of the 1D dynamics. The firstmethod is based on the direct
integration 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, and
x(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 the
presence 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 Poisson
distribution with intensity Îm:
wk =
e−Îm(Îm)k
k!
, k = 0, 1, . . . (216)
and
÷0(Ë) = 1
Ûm n(Ë)
÷1(Ë) = 1
Ûm n
Ë + Ì
Ûm
DNJs
÷k(Ë) = 1
Ûm n
Ë + kÌ
Ûm

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÷0(Ë) = 1
Ûm 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 in
the standard fashion.
5.3.2 Fast Fourier Transform method
In 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 only
apply it in the case of DDM. The advantage of thismethod is that its implementation is
relatively 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 enough
in order to avoid aliasing effects becoming important.
The Green’s function
To 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 spatially
independent, so that the Green’s function depends on X − x rather than X and x
separately:
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:
ˆ ’(t, T, k) =
∞
−∞
eikY’(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) =
1
2
Û2(t)k2 − iÏ (t) k − Î(t)((k) − 1) (219)
with the function (k) given by:
(k) =
0
−∞
eik j( j )d j =
e−ikÌ, DNJs
Ì
Ì+ik , ENJs (220)
Given ˆ ’ we can compute the Green’s function ’ via the inverse Fourier transform as
follows:
’(t, T, Y) =
1
2
∞
−∞
e−ikY ˆ ’(t, T, k)dk (221)

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When parameters are time independent we have
ˆ ’(t, T, k) = e−Ù¯(k) (222)
¯ (k) =
1
2
Û2k2 − iÏk − Î((k) − 1)
where Ù = T − t. This formula can be trivially generalized to the case of piecewise
constant parameters but we don’t present the corresponding expression here since we
are only interested in time intervals between observation times where parameters are
constant by construction.
Backward problem
We 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 as
follows:
V(m−1)+(x) = D (tm−1, tm)
∞
−∞
Vm− (X) ’(tm−1, tm, X − x)dX (223)
For convenience, we introduce the inverse Fourier transform of Vm− (x):
»
Vm− (k) =
1
2
∞
−∞
e−ikxVm− (x) dx (224)
Here we assume that Vm− is regularized as appropriate, so that the above integral
converges. By applying the Fourier transformed density function (218) and exchanging
the integration order, we obtain:
V(t(m−1)+, x) (225)
= D (tm−1, tm)
∞
−∞
Vm− (X)
1
2
∞
−∞
e−ik(X−x)ˆ ’(tm−1, tm, k)dk
dX
= D (tm−1, tm)
∞
−∞
eikx
1
2
∞
−∞
e−ikXVm− (X)dX
ˆ ’(tm−1, tm, k)dk
= D (tm−1, tm)
∞
−∞
eikx »
Vm− (k) ˆ ’(tm−1, tm, k)dk
Variants of formula (225) for option pricing in 1D case were proposed and analysed by
Carr 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 algorithm
based on the recurrent application of eq. (225): (a) Set m = M and apply the terminal
condition by vM(x) = v(x); (b) Compute the auxiliary function V(m−1)+(x) by virtue of
eq. (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 is
stopped. For financial instruments of interest, explicit expressions for v,–{.} are given

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in section 4. To calculate the survival probability, we run the above scheme without
discounting.
To apply the above algorithm in practice, we restrict the value function V(t, x) and
the payoff function v(x) to the uniform spatial grid {x0, . . . , xN} with the uniform
step 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 spatial
grid is defined in such a way that xn0 = 0, and Ó = xnÓ, for some n0, nÓ, respectively.
The transformed density function ˆ ’(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)) ˆ ’(tm−1, tm, k)
(228)
where denotes element-wise multiplication
Forward problem
Similarly, 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 as
follows:
Gm−(X) =
∞
−∞
G(m−1)+(x)’(tm−1, tm, X − x)dx (229)
By analogy with eq. (225), we obtain:
Gm−(X) =
∞
−∞
G(m−1)+(x)
1
2
∞
−∞
e−ik(X−x) ’(tm−1, tm, k)dk
dx (230)
=
1
2
∞
−∞
e−ikX
∞
−∞
eikxG(m−1)+(x)dx
’(tm−1, tm, k)dk
=
1
2
∞
−∞
e−ikX ˆG(m−1)+(k) ˆ ’(tm−1, tm, k)dk
We note that the expression in the curly brackets can be viewed as the direct Fourier
transform of the initial value function G(m−1)+(x). Thus, the discrete version of eq.
(230) can be represented by analogy to eq. (228):
Gm−(X) = ifft
fft(G(m−1)+(X)) ’
(tm−1, tm, k)
(231)
For calibration purposes it is important to solve the problem for the survival
probability Q(x)(t, x, T) via forward induction. For this purpose we present forward
induction 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

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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 Gm+(X) outside the barrier region:
Gm+(X) = Gm−(X)1{X0} (233)
(d) Evaluate the current value of the survival probability Q(x)(0, Ó, tm) by computing
the sum over the discrete spatial grid:
Q(x)(0, Ó, tm) =
N
n=n0
Gm+(xn) (234)
(e) Repeat steps (b), (c) until m = M, then stop the recursion.
Implementation details
When implementing the FFT method with time stepping we need to consider the
following important aspects: periodicity, finiteness, and convergence.
The applicability of FFT is based on the assumption that the relevant functions are
periodic. If it is not the case, so-called aliasing effects tend to spoil the solution near
the end points of the computational domain. We first notice that, provided the spatial
grid is large enough, the periodicity is not an issue for the computation of the Green’s
function since asymptotically it approaches zero. For the backward recursion, we can
deal with aliasing effects by modifying the solution near the edges of the grid. We
note that near the edges of the grid the second derivative of the value function should
gradually approach zero (from above for convex payoffs, from below for concave
ones). However aliasing effects destroy the convexity/concavity of the corresponding
solution. To rectify this fact, we alter the value function by detecting remote areas
where 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 transform
of the payoff function exists. This requirement can be satisfied by applying a damping
factor as needed. For the FFT method, this requirement is not too restrictive since the
calculations are performed in a finite domain. In our experiments, we have found no
advantage in using damping.
The convergence of the integral formulas (225) and (230) is affected by the rate of
decay of the transformed Green’s function (218) for large k. Asymptotically, using eq.
(219) we obtain the following result:
lim |k|→±∞
= e−12
’(tm−1, tm, k)
ÙmÛ2k2 (235)
where Ùm = tm − tm−1.Typically, the volatility is small, Û ≈ 0.01, and Ùm ≈ 0.25y, so
that the upper bound for k, kN = /2ƒx needs to be large enough (kN ∼ 1000). In

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our experiments, we have found that for typical model parameters it is safe to choose
N = 212 − 1, ƒx ∼ 10−4.
5.3.3 Finite difference method
Backward equation
We start with the backward problem for the value function V(t, x). For DDM, this
function is solving problem (43) onR1, while for CDMthis function is solving a similar
problem on R1
+. For brevity, we consider only the latter case.
We introduce a discrete spatial grid of size N + 1, {x0, x1, . . . , xN−1, xN}, where
x0 = 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 when
parameters jump, and other special times, if any, belong to the grid. The values
V(tl , xn) are denoted by Vl ,n, and similarly for other relevant quantities. For fixed l
we use the notation

139. V
l, and think of

140. V
l as an (N + 1)-component vector.We discretize
the evolution operator (54) at time tl :
L(x) (tl ) =⇒L
(x)
l (236)
l +J
D(x)(tl) + J (x)(tl) =⇒ D(x)
(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 tridiagonalmatrix. Typically the diffusion term is small compared
to the advection and jump terms, so that an appropriate discretization of the first
derivative 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

141. V
l can be evaluated in a way which
requires 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 =
˘¯n
Vl ,¯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 − Ì)
x¯n
x¯n
− x¯n
−1
(239)

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For ENJs, we choose a small step h, h 0, and write:
êl (x + h) = Ì
0
−x−h
eÌj V(tl , x + h + j )d j (240)
= Ìe−Ìh
0
−x
eÌzV(tl , x + z)dz +
h
0
eÌzV(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-order
accuracy:
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

143. ê
l =
êl ,n
we can calculate the matrix product Kl

144. V
l with O (N) operations in both
cases.
To compute the value

145. V
l−1 given the value

146. V
l , l = 1, . . . , L, we introduce auxiliary
l ,

147. V
vectors

148. ê
∗,

149. V
∗∗,and apply the following scheme:

150. V
∗ =

151. V
l +
1
2
‰tl
−→ˆ
c l−1 +
−→ˆ
c l
+ ‰tl Î(tl )

152. ê
l (244)
I − ‰tlD(x)
l

153. V
∗∗ =

154. V
∗
l−1 = D (tl−1, tl )

155. V

156. V
∗∗
where ‰tl = tl − tl−1, l = 1, . . . , L, and I is the identity matrix. Thus, we use an
explicit scheme to approximate the integral step and compute

157. V
∗ given

158. V
l; then we
use an implicit scheme to approximate the diffusive step and compute

159. V
∗∗ given

160. V
∗,
0 and V∗∗
finally, we apply the deterministic discounting. Boundary values V∗∗
N are
determined by the boundary conditions. The second implicit step leads to a system
of tridiagonal equations which can be solved via O(N) operations. The time-stepping
numerical scheme is straightforward. The terminal condition is given by:

161. V
L =

162. v (245)
The implicit diffusion step in scheme (244) is first-order accurate in time, but it
tends 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 jump
operator 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,

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

164. U
0 =

165. V
l +
1
2
‰tl
ˆcl−1 + ˆcl
(246)
and, for p = 1, 2, . . . , p (typically, it is enough to use two iterations, p = 2) apply the
following scheme:

166. U
∗ =

167. U
p−1
l (247)
0 + ‰tl Î(tl )

168. ê
I − ‰tlD(x)
l

169. U
p =

170. U
∗
p−1
l =J
where

171. ê
(x)
l

172. U
p−1. Provided that the difference ||U p − U p−1|| is small in an
appropriate norm, we stop and apply the discounting:
Vl−1 = D(tl−1, tl )U p (248)
Forward equation
For calibration purposes we have to calculate the survival probability forward in time
by 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

173. G
l =
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 =
˘¯n
Gl ,¯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 + Ì)
x¯n
x¯n
− x¯n
−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:
G0,n = 2δn,nÓ/
xnÓ+1 − xnÓ−1
(253)

174. where the spatial grid G

175. G
in chosen in such a way that x0 = 0 and xnÓ = Ó. To compute the
value l given the value l−1, l = 1, . . . , L, we introduce auxiliary vectors

176. ¯
,

177. G
∗, and
apply the following scheme:

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2.50
2.00
1.50
1.00
0.50
0.00
X
Expansion, lambda=0.03
PDE, lambda=0.03
Expansion, lambda=0.1
PDE, lambda=0.1
2.50
2.00
1.50
1.00
0.50
0.00
X
Analytic, lambda=0.03
Expansion, lambda=0.03
PDE, lambda=0.03
Analytic, lambda=0.1
Expansion, lambda=0.1
PDE, lambda=0.1
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
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 the
same as in Figure 1, T = 10y.

179. G
∗ =

180. G
l−1 + ‰tl Î(tl )

181. ¯
l−1 (254)
I − ‰tlD(x)†
l

182. G
l =