COUNTERPARTY RISK FOR CREDIT DEFAULT SWAPS

Updated version forthcoming in the
International Journal of Theoretical and Applied Finance
COUNTERPARTY RISK FOR CREDIT DEFAULT SWAPS
impact of spread volatility and default correlation
Damiano Brigo
Fitch Solutions and Dept. of Mathematics, Imperial College
101 Finsbury Pavement, EC2A 1RS London.
E-mail: damiano.brigo@fitchsolutions.com
Kyriakos Chourdakis
Fitch Solutions and CCFEA, University of Essex
101 Finsbury Pavement, EC2A 1RS London.
E-mail: kyriakos.chourdakis@fitchsolutions.com
Abstract
We consider counterparty risk for Credit Default Swaps (CDS) in presence of correlation between default
of the counterparty and default of the CDS reference credit. Our approach is innovative in that, besides
default correlation, which was taken into account in earlier approaches, we also model credit spread volatil-
ity. Stochastic intensity models are adopted for the default events, and defaults are connected through a
copula function. We find that both default correlation and credit spread volatility have a relevant impact
on the positive counterparty-risk credit valuation adjustment to be subtracted from the counterparty-risk
free price. We analyze the pattern of such impacts as correlation and volatility change through some fun-
damental numerical examples, analyzing wrong-way risk in particular. Given the theoretical equivalence of
the credit valuation adjustment with a contingent CDS, we are also proposing a methodology for valuation
of contingent CDS on CDS.
AMS Classification Codes: 60H10, 60J60, 60J75, 62H20, 91B70
JEL Classification Codes: C15, C63, C65, G12, G13
Keywords: Counterparty Risk, Credit Valuation adjustment, Credit Default Swaps, Con-
tingent Credit Default Swaps, Credit Spread Volatility, Default Correlation, Stochastic Intensity,
Copula Functions, Wrong Way Risk.
First version: May 16, 2008. This version: October 3, 2008

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D. Brigo, K. Chourdakis, FitchSolutions, Counterparty risk valuation adjustment for CDS
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1
Introduction
We consider counterparty risk for Credit Default Swaps (CDS) in presence of correlation between
default of the counterparty and default of the CDS reference credit. We assume the party that
is computing the counterparty risk adjustment to be default free, as a possible approximation to
situations where this party has a much higher credit quality than the counterparty. Our approach is
innovative in that, besides default correlation, which was taken into account in earlier approaches,
we also model explicitly credit spread volatility. This is particularly important when the underlying
reference contract itself is a CDS, as the counterparty credit valuation adjustment involves CDS
options, and modeling options without volatility in the underlying asset is quite undesirable. We
investigate the impact of the reference volatility on the counterparty adjustment as a fundamental
feature that is ignored or not studied explicitly in other approaches.
Hull and White (2000) address the counterparty risk problem for CDS by resorting to default
barrier correlated models, without considering explicitly credit spread volatility in the reference
CDS. Leung and Kwok (2005), building on Collin-Dufresne et al. (2002), model default intensities
as deterministic constants with default indicators of other names as feeds. The exponential triggers
of the default times are taken to be independent and default correlation results from the cross feeds,
although again there is no explicit modeling of credit spread volatility. Furthermore, most models in
the industry, especially when applied to Collateralized Debt Obligations or k-th to default baskets,
model default correlation but ignore credit spread volatility. Credit spreads are typically assumed
to be deterministic and a copula is postulated on the exponential triggers of the default times to
model default correlation. This is the opposite of what used to happen with counterparty risk for
interest rate underlyings, for example in Sorensen and Bollier (1994) or Brigo and Masetti (2006),
where correlation was ignored and volatility was modeled instead. Here we rectify this, with a
model that takes into account credit spread volatility besides the still very important correlation.
Ignoring correlation among underlying and counterparty can be dangerous, especially when the
underlying instrument is a CDS. Indeed, this credit underlying case involves default correlation,
that is perceived in the market as more relevant than the dubious interest-rate/ credit-spread
correlation of the interest rate underlying case. It is not so much that the latter is less relevant
because it would have no impact in counterparty risk credit valuation adjustments. We have seen in
Brigo and Pallavicini (2007, 2008) that changing this correlation parameter has a relevant impact
for interest rate underlyings. Still, the value of said correlation is difficult to estimate historically or
imply from market quotes, and the historical estimation often produces a very low or even slightly
negative correlation parameter. So even if this parameter has an impact, it is difficult to assign
a value to it and often this value would be practically null. On the contrary, default correlation
is more clearly perceived, as measured also by implied correlation in the quoted indices tranches
markets (i-Traxx and CDX).
To investigate the impact of both default correlation and credit spread volatility, tractable
stochastic intensity diffusive models with possible jumps are adopted for the default events and
defaults are connected through a copula function on the exponential triggers of the default times.
We find that both default correlation and credit spread volatility have a relevant impact on the
positive credit valuation adjustment one needs to subtract from the default free price to take into
account counterparty risk. We analyze the pattern of such impacts as volatility and correlation pa-
rameters vary through some fundamental numerical examples, and find that results under extreme
default correlation (wrong way risk) are very sensitive to credit spread volatility. This points out
that credit spread volatility should not be ignored in these cases. Given the theoretical equivalence

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D. Brigo, K. Chourdakis, FitchSolutions, Counterparty risk valuation adjustment for CDS
3
of the credit valuation adjustment with a contingent CDS, we are also proposing a methodology for
valuation of contingent CDS on CDS. This can be particularly relevant for a financial institution
that has bought protection or insurance on CDS from other institutions whose credit quality is
deteriorating. The case of mono-line insurers after the sub-prime crisis is just a possible example.
We finally describe the structure of the paper, and how to benefit most of it from the point of
view of readers with different backgrounds.
The essential results are described in the case study in Section 6, so the reader aiming at
getting the main message of the paper with minimal technical implications can go directly to this
section, that has been written to be as self-contained as possible. Otherwise, Section 2 describes
the counterparty risk valuation problem in quite general terms and, apart a few technicalities
on filtrations that can be overlooked at first reading, is quite intuitive. Section 3 describes the
reduced form model setup of the paper with stochastic intensities and a copula on the exponential
triggers. A detailed presentation of the shifted squared root (jump) diffusion (SSRJD) model
and of its calibration to CDS, previously analyzed in Brigo and Alfonsi (2005), Brigo and Cousot
(2006), and Brigo and El-Bachir (2008), is given. Section 4 details how the general formula for the
counterparty credit valuation adjustment given in Section 2 can be written under the specific CDS
payoff and modeling assumptions of the paper, although formulas derived here will not be used,
as we will proceed through a more direct numerical approach. These calculations can however
give a feeling for the complexity of the problem and for the kind of issues one has to face in these
situations, and for this reason are presented. Section 5 details the numerical techniques that are
used to compute the credit valuation adjustment in the case study. Finally, Section 6 briefly recaps
the modeling assumptions and illustrates the paper conclusions with the case study itself.
2
General valuation of counterparty risk
We denote by τ1 the default time of the credit underlying the CDS, and by τ2 the default time of
the counterparty. We assume the investor who is considering a transaction with the counterparty
to be default-free. We place ourselves in a probability space (Ω, G, Gt, Q). The filtration (Gt)t
models the flow of information of the whole market, including credit and defaults. Q is the risk
neutral measure. This space is endowed also with a right-continuous and complete sub-filtration Ft
representing all the observable market quantities but the default events (hence Ft ⊆ Gt := Ft ∨ Ht
where Ht = σ({τ1
u}, {τ2
u} : u
t) is the right-continuous filtration generated by the default
events).
We set Et(·) := E(·|Gt), the risk neutral expectation leading to prices.
Let us call T the final maturity of the payoff we need to evaluate. If τ2 > T there is no default
of the counterparty during the life of the product and the counterparty has no problems in repay-
ing the investors. On the contrary, if τ2
T the counterparty cannot fulfill its obligations and
the following happens. At τ2 the Net Present Value (NPV) of the residual payoff until maturity
is computed: If this NPV is negative (respectively positive) for the investor (defaulted counter-
party), it is completely paid (received) by the investor (counterparty) itself. If the NPV is positive
(negative) for the investor (counterparty), only a recovery fraction REC of the NPV is exchanged.
Let us denote by ΠD(t, T ) the sum of all payoff terms between t and T , all terms discounted
back at t, and subject to counterparty default risk. We denote by Π(t, T ) the analogous quantity
when counterparty risk is not considered. All these payoffs are seen from the point of view of the
safe “investor” (i.e. the company facing counterparty risk). Then we have the net present value at

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D. Brigo, K. Chourdakis, FitchSolutions, Counterparty risk valuation adjustment for CDS
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time τ2 as NPV(τ2, T ) = Eτ {Π(τ
2
2, T )} and
ΠD(t, T ) = 1{τ2>T}Π(t, T ) +
1{t<τ2 T} Π(t, τ2) + D(t, τ2) REC (NPV(τ2, T ))+ − (−NPV(τ2, T ))+
(2.1)
being D(u, v) the stochastic discount factor at time u for maturity v. This last expression is the
general price of the payoff under counterparty risk. Indeed, if there is no early counterparty default
this expression reduces to risk neutral valuation of the payoff (first term in the right hand side); in
case of early default, the payments due before default occurs are received (second term), and then
if the residual net present value is positive only a recovery of it is received (third term), whereas
if it is negative it is paid in full (fourth term).
We notice incidentally that our definition involves an expectation Eτ , i.e. conditional on G
2
τ2
where
Gτ := σ(G
:= σ(F
2
t ∩ {t ≤ τ2}, t ≥ 0),
Fτ2
t ∩ {t ≤ τ2}, t ≥ 0).
It is possible to prove the following
Proposition 2.1. (General counterparty risk pricing formula). At valuation time t, and
on {τ2 > t}, the price of our payoff under counterparty risk is
Et{ΠD(t, T )} = Et{Π(t, T )}−
LGD Et{1{t<τ2 T}D(t, τ2) (NPV(τ2))+}
(2.2)
Positive counterparty-risk adj. (CR-CVA)
where LGD = 1 − REC is the Loss Given Default and the recovery fraction REC is assumed to
be deterministic. It is clear that the value of a defaultable claim is the value of the corresponding
default-free claim minus an option part, in the specific a call option (with zero strike) on the residual
NPV giving nonzero contribution only in scenarios where τ2
T . This adjustment, including the
LGD factor, is called counterparty-risk credit valuation adjustment (CR-CVA). Counterparty risk
adds an optionality level to the original payoff.
For a proof see for example Brigo and Masetti (2006).
Notice finally that the previous formula can be approximated as follows. Take t = 0 for
simplicity and write, on a discretization time grid T0, T1, . . . , Tb = T ,
E[ΠD(0, T
b
b)] = E[Π(0, Tb)]−
LGD
E[1
Π(τ
j=1
{Tj 1<τ2≤T
2 , Tb))+]
−
j } D(0, τ2 )(Eτ2
b
≈ E[Π(0, Tb)]−
LGD
E[1{T
Π(T
j
1 <τ2 ≤T
j , Tb))+]
(2.3)
−
j } D(0, Tj )(ETj
j=1
approximated (positive) adjustment
where the approximation consists in postponing the default time to the first Ti following τ2. From
this last expression, under independence between Π and τ2, one can factor the outer expectation
inside the summation in products of default probabilities times option prices. This way we would
not need a default model for the counterparty but only survival probabilities and an option model
for the underling market of Π. This is only possible, in our case of a CDS as underlying contract,
if the default correlation between the CDS reference credit and the counterparty is zero. This is

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D. Brigo, K. Chourdakis, FitchSolutions, Counterparty risk valuation adjustment for CDS
5
what led to earlier results on swaps with counterparty risk in interest rate payoffs in Brigo and
Masetti (2006). In this paper we do not assume zero correlation, so that in general we need to
compute the counterparty risk without factoring the expectations. To do so we need a default
model for the counterparty, to be correlated with the default model for the underlying CDS.
2.1
Contingent CDS
A Contingent Credit Default Swap (CCDS) is a CDS that, upon the default of the reference credit,
pays the loss given default on the residual net present value of a given portfolio if this is positive.
It is immediate then that the default leg CCDS valuation, when the CCDS underlying portfolio
constituting the protection notional is Π, is simply the counterparty credit valuation adjustment
in Formula (2.2). When Π is an underlying CDS, our adjustments calculations above can then be
interpreted also as examples of pricing contingent CDS on CDS.
3
Modeling assumptions
In this section we consider a reduced form model that is stochastic in the default intensity both
for the counterparty and for the CDS reference credit. We will not correlate the spreads with each
other, as typically spread correlation has a much lower impact on dependence of default times than
default correlation. The latter is rigorously defined as a dependence structure on the exponential
random variables characterizing the default times of the two names. This dependence structure is
typically modeled with a copula function.
More in detail, we assume that the counterparty default intensity λ2, and the cumulated in-
tensity Λ2(t) =
t λ
0
2(s)ds, are independent of the default intensity for the reference CDS λ1,
whose cumulated intensity we denote by Λ1. We assume intensities to be strictly positive, so that
t → Λ(t) are invertible functions.
We assume deterministic default-free instantaneous interest rate r (and hence deterministic
discount factors D(s, t), ...), but all our conclusions hold also under stochastic rates that are inde-
pendent of default times.
We are in a Cox process setting, where
τ1 = Λ−1
1 (ξ1),
τ2 = Λ−1
2 (ξ2),
with ξ1 and ξ2 standard (unit-mean) exponential random variables whose associated uniforms
Uj = 1 − exp(−ξj), j = 1, 2, are correlated through a copula function. We assume
Uj = 1 − exp(−ξj), j = 1, 2,
Q(U1 < u1, U2 < u2) =: C(u1, u2).
In the case study below we assume the copula C to be Gaussian and with correlation parameter
ρ, although the choice can be easily changed, as the framework is general.

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D. Brigo, K. Chourdakis, FitchSolutions, Counterparty risk valuation adjustment for CDS
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3.1
CIR++ stochastic intensity models
For the stochastic intensity model we set
λj(t) = yj(t) + ψj(t; βj) , t
0, j = 1, 2
(3.1)
where ψ is a deterministic function, depending on the parameter vector β (which includes y0), that
is integrable on closed intervals. The initial condition y0 is one more parameter at our disposal:
We are free to select its value as long as
ψ(0; β) = λ0 − y0 .
We take each y to be a Cox Ingersoll Ross (CIR) process (see for example Brigo and Mercurio
(2001)):
dyj(t) = κ(µ − yj(t))dt + ν
yj(t) dZj(t), j = 1, 2
where the parameter vectors are βj = (κj, µj, νj, yj(0)), with κ, µ, ν, y0 positive deterministic
constants. As usual, the Z are standard Brownian motion processes under the risk neutral measure,
representing the stochastic shock in our dynamics.
Usually, for the CIR model one assumes a condition ensuring the origin to be inaccessible, the
condition being 2κµ > ν2. However, this limits the CDS implied volatility generated by the model
when imposing also positivity of the shift ψ, which is a condition we will always impose in the
following to avoid negative intensities. This is why we do not enforce the condition 2κµ > ν2 and
in our case study below it will be violated.
Correlation in the spreads is a minor driver with respect to default correlation, so we assume
that the two Brownian motions Z are independent. We will often use the integrated quantities
t
t
t
Λ(t) =
λsds, Y (t) =
ysds,
and Ψ(t, β) =
ψ(s, β)ds.
0
0
0
This kind of models and the related calibration to CDS has been investigated in detail in Brigo
and Alfonsi (2005), while Brigo and Cousot (2006) examine the CDS implied volatility patterns
associated with the model.
Notice that we can easily introduce jumps in the diffusion process. Brigo and El-Bachir (2008)
consider a formulation where
dyj(t) = κ(µ − yj(t))dt + ν
yj(t)dZj(t) + dJj(t), j = 1, 2,
with
Nj(t)
Jj(t) =
Y i
j
i=1
and N standard Poisson process with intensity α counting the jumps, and the Y ’s i.i.d. exponential
random variables with mean γ representing the jump sizes. Besides deriving log-affine survival
probability formulas re-shaped exactly in the same form as in the CIR model without jumps,
Brigo and El-Bachir (2008) derive a closed form solution for CDS options as well.
In the sequel we take α = 0 and assume no jumps. However, all calculations and also the
fractional Fourier transform method are exactly applicable to the extended model with jumps.

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D. Brigo, K. Chourdakis, FitchSolutions, Counterparty risk valuation adjustment for CDS
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3.2
CIR++ model: CDS calibration
We focus on the calibration of the default model for the counterparty, the one for the reference
credit being completely analogous. Since we are assuming deterministic rates, the default time
τ2 and interest rate quantities r, D(s, t), ... are trivially independent. It follows that the (receiver)
CDS valuation, for a CDS selling protection at time 0 for defaults between times Ta and Tb in
exchange of a periodic premium rate S becomes
Tb
CDSa,b(0, S, LGD; Q(τ2 > ·)) = S −
P (0, t)(t − Tγ(t)−1)dt Q(τ2 ≥ t)
(3.2)
Ta
b
+
P (0, Ti)αi Q(τ2 ≥ Ti)
+
i=a+1
Tb
+LGD
P (0, t) dt Q(τ2 ≥ t)
,
Ta
where in general Tγ(t) is is the first Tj following t. This formula is model independent. This means
that if we strip survival probabilities from CDS in a model independent way at time 0, to calibrate
the market CDS quotes we just need to make sure that the survival probabilities we strip from
CDS are correctly reproduced by the CIR++ model. Since the survival probabilities in the CIR++
model are given by
Q(τ2 > t)model = E(e−Λ2(t)) = E exp (−Ψ2(t, β) − Y2(t))
(3.3)
we just need to make sure
E exp (−Ψ2(t, β2) − Y2(t)) = Q(τ2 > t)market
from which
E(e−Y2(t))
P CIR(0, t, y
Ψ
2(0); β2)
2(t, β2) = ln
= ln
(3.4)
Q(τ2 > t)market
Q(τ2 > t)market
where we choose the parameters β2 in order to have a positive function ψ2 (i.e. an increasing Ψ2)
and P CIR is the closed form expression for bond prices in the time homogeneous CIR model with
initial condition y2(0) and parameters β2 (see for example Brigo and Mercurio (2001)). Thus, if
ψ2 is selected according to this last formula, as we will assume from now on, the model is easily
and automatically calibrated to the market survival probabilities for the counterparty (possibly
stripped from CDS data).
A similar procedure goes for the reference credit default time τ1.
Once we have done this and calibrated CDS data through ψ(·, β), we are left with the parameters
β, which can be used to calibrate further products. However, this will be interesting when single
name option data on the credit derivatives market will become more liquid. Currently the bid-ask
spreads for single name CDS options are large and suggest to either consider these quotes with
caution, or to try and deduce volatility parameters from more liquid index options. At the moment
we content ourselves of calibrating only CDS’s. To help specifying β without further data we set
some values of the parameters implying possibly reasonable values for the implied volatility of
hypothetical CDS options on the counterparty and reference credit.

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D. Brigo, K. Chourdakis, FitchSolutions, Counterparty risk valuation adjustment for CDS
8
4
CDS options embedded in the counterparty risk adjustment
We now move to computing the counterparty risk adjustment, as in Equation (2.3).
The only non-trivial term to compute is
E[1{T
)+]
(4.1)
j
1 <τ2 ≤T
−
j } (E
Π(Tj, Tb)|GTj
Now let us assume we are dealing with a counterparty “2” from which we are buying protection
at a given spread S through a CDS on the relevant reference credit “1”. This is the position where
we would be in the most critical situation in case of counterparty default. We are thus holding a
payer CDS on the reference credit “1”. Therefore Π(Tj, Tb) is the residual NPV of a payer CDS
between Ta and Tb at time Tj, with Ta < Tj ≤ Tb. The NPV of a payer CDS at time Tj can be
written similarly to (3.2), except that now valuation occurs at Tj and has to be conditional on the
information available in the market at Tj, i.e. GT . We can write:
j
CDSa,b(Tj, S, LGD1) = 1{τ1>Tj}CDSa,b(Tj, S, LGD1)
(4.2)
Tb
= 1{τ
P (T
)
1 >Tj }
S −
j , t)(t − Tγ(t)−1)dt Q(τ1 ≥ t|GTj
max(Ta,Tj )
b
+
P (Tj, Ti)αi Q(τ1 ≥ Ti|GT ) 
j
i=max(a,j)+1
Tb
+
+ LGD1
P (Tj, t) dt Q(τ1 ≥ t|GT )
j
max(Ta,Tj )
The Tj-credit valuation adjustment for counterparty risk would read
E[1{T
)+] = E[1
j
1 <τ2≤T
{T
1 <τ2 ≤T
−
j } (E
Π(Tj, Tb)|GTj
j−
j } (CDSa,b (Tj , S, LGD1 ))+ ]
= E[1{Tj 1<τ2≤T
−
j } 1{τ1>Tj } (CDSa,b (Tj , S, LGD1 ))+ ]
= E[E{1{T
}]
j
1 <τ2 ≤T
−
j } 1{τ1>Tj } (CDSa,b (Tj , S, LGD1 ))+ |FTj
= E[(CDSa,b(Tj, S, LGD1))+E{1{T
}]
j
1 <τ2 ≤T
−
j } 1{τ1>Tj } |FTj
= E{(CDSa,b(Tj, S, LGD1))+ [exp(−Λ2(Tj−1)) − exp(−Λ2(Tj))
−C(1 − exp(−Λ1(Tj)), 1 − exp(−Λ2(Tj)))
+C(1 − exp(−Λ1(Tj)), 1 − exp(−Λ2(Tj−1)))]}
(4.3)
This can be easily computed through simulation of the processes λ up to Tj if we know the
formula for Q(τ1 ≥ u|GT ) for all u ≥ T
j
j in terms of λ1(Tj ).
This valuation, leading to an easy formula for CDSa,b(Tj), would be simple if we were to
compute the above probabilities under the filtration G1 of the default time τ
T
1 alone, rather than
j
GT incorporating information on τ
j
2 as well. Indeed, in such a case we could write

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D. Brigo, K. Chourdakis, FitchSolutions, Counterparty risk valuation adjustment for CDS
9
u
Q(τ1 ≥ u|G1T ) = 1
λ1(s)ds |F1
(4.4)
j
{τ1>Tj }E
exp −
Tj
Tj
= 1{τ1>Tj}P CIR++(Tj, u; y1(Tj)) := 1{τ1>Tj} exp (−(Ψ(u) − Ψ(Tj))) P CIR(Tj, u; y1(Tj))
i.e. the bond price in the CIR++ model for λ1, P CIR(Tj, u; y1(Tj)) being the non-shifted time
homogeneous CIR bond price formula for y1. Substitution in (4.2) would give us the NPV at time
Tj, since CDS(Tj) would be computed using indeed (4.4) in (4.2). So finally, we would have all the
needed components to compute our counterparty risk adjustment (2.3) through mere simulation
of the λ’s up to Tj.
However, there is a fatal drawback in this approach. Indeed, the survival probabilities con-
tributing to the valuation of CDS(Tj) have to be calculated conditional also on the information on
the counterparty default τ2 available at time Tj.
We can write the correct formula for this survival probability as follows.
1{T
) = E 1
j
1 <τ2 ≤T
{T
1 <τ2 ≤T
−
j } Q(τ1 ≥ u|GTj
j−
j } 1{τ1>u} |GTj
= E 1{Tj 1<τ2≤T
−
j } 1{τ1>Tj } 1{τ1>u} |GTj
= 1{T
, τ
j
1 <τ2 ≤T
1 > Tj , Tj−1 < τ2 ≤ Tj
−
j } E
1{τ1>u}|GTj
= 1{T
, τ
j
1 <τ2 ≤T
1 > Tj , Tj−1 < τ2 ≤ Tj
−
j } E
1{τ1>u}|FTj
Q(τ1 > u, Tj−1 < τ2 ≤ Tj|FT )
= 1
j
{Tj 1<τ2≤T
−
j }
Q(τ1 > Tj, Tj−1 < τ2 ≤ Tj|FT )
j
Q(U
1 )
1 > 1 − e−Λ1(u), 1 − e−Λ2(Tj−
< U2 < 1 − e−Λ2(Tj)|FT )
= 1
j
{·} Q(U1 > 1 − e−Λ1(Tj), 1 − e−Λ2(Tj 1)
−
< U2 < 1 − e−Λ2(Tj)}|FT )
j
e−Λ2(Tj 1)
1 )
−
− e−Λ2(Tj) + E[C(1 − e−Λ1(u), 1 − e−Λ2(Tj− ) − C(1 − e−Λ1(u), 1 − e−Λ2(Tj))|FT ]
= 1
j
{·}
e−Λ2(Tj 1)
1 )
−
− e−Λ2(Tj) + C(1 − e−Λ1(Tj), 1 − e−Λ2(Tj− ) − C(1 − e−Λ1(Tj), 1 − e−Λ2(Tj))
The residual expectation in the numerator accounts for randomness of Λ1(u) − Λ1(Tj), that is not
accounted for in FT , and is thus incorporated by taking an expectation with respect to the density
j
of Λ1(u) − Λ1(Tj) (that, in case of the CIR model, can be obtained through Fourier methods).
It is clear that this last expression we obtained is much more complex than (4.4). One can check
that if the chosen copula is the independence copula, C(u1, u2) = u1u2, then our last expression
reduces indeed to (4.4).
The difference, in correctly taking into account the dependence of default time τ1 conditional
on the information on default time τ2, manifests itself in the copula terms. Indeed, with respect
to the earlier and incorrect formula taking into account only information of name 1, we made the
transition
− u λ
− u λ
E e
T
1 (u)du
1 (u)du
j
→ E C(1 − e
Tj
e−Λ1(Tj), 1 − e−Λ2(Tj or j 1)
−
)|Λ1(Tj), Λ2(Tj)
that clearly involves directly the copula.
By substituting our last formula for Q(τ1 ≥ u|GT ) in (4.2) and then the resulting expression
j
in (4.3), we conclude.

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D. Brigo, K. Chourdakis, FitchSolutions, Counterparty risk valuation adjustment for CDS
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This procedure is however quite demanding, and the idea of partitioning the default interval in
periods [Tj−1, Tj] is not as effective here as in other situations (such as Brigo and Masetti (2006))
and we approach the problem in a more direct numerical way in the next section.
5
Direct Numerical Methodology: Monte Carlo and Fourier Transform
In this section we abandon the choice of bucketing the counterparty default time τ2 in intervals and
move to implementing directly the original formula (2.2), whose relevant term in our case reads
+
Et{1{t<τ
}
2
Tb}D(t, τ2) (CDSa,b(τ2, S, Tb))+} = Et{1{t<τ2 Tb}D(t, τ2) 1{τ1>τ2}CDSa,b(τ2, S, Tb)
Recall the formula in (4.2) for CDS and keep in mind that this is to be computed at the random
time τ2. In the CDS formula, all we need to know is the survival probability 1{τ1≥τ2}Q(τ1 >
u|Gτ ) = 1
, τ
2
{τ1≥τ2}Q(τ1 > u|Gτ2
1 ≥ τ2).
Summarizing: To effectively compute counterparty risk, we aim at determining the value of
the CDS contract on the reference credit “1” at the point in time τ2 where the counterparty “2”
defaults. The reference name “1” has survived this point, and there is a copula C that connects the
two default times. The stochastic intensities λ1 and λ2 of names “1” and “2” are independent and
the default times are connected uniquely through the copula, that is however the most important
source of default dependence, correlation among the λ being in general only a secondary source of
dependence.
We need to compute the probability
Q(τ1 > T |Gτ , τ
, τ
2
1 > τ2) = Q ( U1 > 1 − exp {−Y1(T ) − Ψ1(T ; β1)}| Gτ2
1 > τ2)
for any T > τ2, where U1 is a uniform random variable, λ1 = y1 + ψ1 is the intensity process, Ψ1
is the integrated deterministic shift Ψ1(T ) = T ψ
0
1(t)dt and analogously Y1 is the integrated y1
process.
The information Gτ will determine uniquely τ
2
2 and hence the value U2, since the intensity λ2
is also measurable w.r.t. G. In addition, it includes the quantity Λ1(τ2), which is measurable as
well.
Now, by conditioning on the value U1, the above probability can be written as
E [ P (U1)| Gτ , τ
2
1 > τ2]
for
P (u1) = Q (u1 > 1 − exp {−Y1(T ) − Ψ1(T ; β1)}| Gτ )
2
The conditional probability can be expressed as the cumulative probability of the integrated
CIR process
P (u1) = Q (Y1(T ) − Y1(τ2) < − log(1 − u1) − Y1(τ2) − Ψ1(T ; β1)| Gτ )
2
The characteristic function of the integrated CIR process Y1(T )− Y1(τ2) is known in closed form at
time τ2, with a calculation much resembling the CIR bond price formula. The probabilities P (u1)
can therefore be retrieved for an array of u1 using fractional FFT methods.

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