Statistics and Probability Letters 85 (2014) 25–35

Contents lists available at ScienceDirect

Statistics and Probability Letters
journal homepage: www.elsevier.com/locate/stapro

Unilateral counterparty risk valuation of CDS using a
regime-switching intensity model
Yinghui Dong a,b,∗ , Kam C. Yuen c , Chongfeng Wu a
a

Financial Engineering Research Center, Shanghai Jiao Tong University, Shanghai 200052, PR China

b

Department of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, PR China

c

Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong

article

info

Article history:
Received 7 July 2013
Received in revised form 12 October 2013
Accepted 7 November 2013
Available online 12 November 2013

Keywords:
Credit default swaps
Counterparty risk
Credit valuation adjustment
Interacting intensities
Regime-switching

abstract
We consider the unilateral credit valuation adjustment (CVA) of a credit default swap (CDS)
under a contagion model with regime-switching interacting intensities. The model assumes that the interest rate, the recovery, and the default intensities of the protection seller
and the reference entity are all influenced by macro-economy described by a homogeneous
Markov chain. By using the idea of ‘‘change of measure’’ and some formulas for the Laplace
transforms of the integrated intensity processes, we derive the semi-analytical formulas
for the joint distribution of the default times and the unilateral CVA of a CDS.
© 2013 Elsevier B.V. All rights reserved.

1. Introduction
Since the 2007 Credit Crisis, credit risk has become a hot topic in connection with valuation and risk management of credit
derivatives. Whenever two parties enter a trade in the OTC market, they should take credit risk against each other. The risk
of financial loss due to default of trading counterparties is referred to as counterparty credit default risk. In most cases, this
risk is not considered in direct evaluation of the trades and, therefore, needs to be adjusted appropriately to reflect the risk
should either of the counterparties default on their commitments. The adjustment to the value of a default-free trading book
is what is usually referred to as counterparty valuation adjustment (CVA). More precisely, the difference between the price
of a contract with default-free counterparties and that with default-risky counterparties is the CVA.
Being one of the most popular classes of credit derivative contracts actively traded in credit markets around the world,
credit default swaps (CDSs) with counterparty risk have received a lot of attention in the literature. We also investigate
the valuation of the counterparty risk of a CDS. In this paper, we assume only one of the two counterparties is defaultable.
Therefore, from the point of view of the default-free counterparty, the positive risk of default before the maturity of the
defaultable counterparty leads to the default-free counterparty charging a risk premium, the unilateral CVA. This paper will
detail the definition and properties of a CDS contract with and without counterparty risk and provide a pricing model for

the unilateral CVA of a CDS contract.
To consider the pricing of the counterparty credit risk of a CDS in the reduced-form framework, the most important task
is to model the default intensities and the default correlation between the protection seller and the reference entity. The

∗

Corresponding author at: Financial Engineering Research Center, Shanghai Jiao Tong University, Shanghai 200052, PR China.
E-mail address: (Y. Dong).

0167-7152/$ – see front matter © 2013 Elsevier B.V. All rights reserved.
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26

Y. Dong et al. / Statistics and Probability Letters 85 (2014) 25–35

reduced-form contagion models are some of the most popular models for describing dependent defaults. Roughly speaking,
one assumes that there exists a certain explicit structure among the default intensities of a group of inter-dependent firms,
and the default of one firm could directly affect the default of its counterparties, and even trigger a cascade of defaults in the
group. See, for example, Shaked and Shanthikumar (1987), Jarrow and Yu (2001) and Yu (2007). Solving a contagion model
faces an obstacle of looping default problem. Collin-Dufresne et al. (2004) provide a ‘‘change of measure’’ method in dealing
with contagion models. We also aim at developing a flexible pricing model in the framework of the reduced-form contagion
models.
Reduced-form models treat default as the first jump time of a random jump process. Many authors model the jump intensities by some deterministic processes or diffusion processes—such as the Vasicek model and the Cox–Ingersoll–Ross (CIR)
model. But these models do not consider the effect of the changes of economic regimes on the default intensities. Intuitively,
default risk should be much influenced by the business cycles or macro-economy. Default risk typically declines during economic expansions because strong earnings keep overall default rates low. Default risk increases during economic recession
because earnings deteriorate, making it more difficult to repay loans or make bond payments. The recent subprime crisis
has had a significant impact on the global financial market, especially on credit risk. Therefore, there is a practical need to
develop some credit risk models, which can take into account the changes of market regimes due to the crisis.
Indeed, regime-switching models have gained immense popularity in the finance domain, see for example, Elliott and Siu

(2009) and Siu (2010). In a regime-switching model, the market is assumed to be in different states depending on the state
of the economy. Regime shift from one economic state to another may occur due to various financial factors like changes in
business conditions, management decisions and other macro-economic conditions. Recently, by using an empirical analysis
of the corporate bond market over the course of the last 150 years, Giesecke et al. (2011) point out there exist three regimes,
associated with high, middle and low default risk, in the credit market. Motivated by Jarrow and Yu (2001), Giesecke et al.
(2011) and others, in this paper we develop a regime-switching pricing model for valuing the unilateral CVA of a CDS, in
which the dynamics of the default intensities, the interest rate and the recovery are driven by a continuous-time, finite-state
Markov chain, describing the economic conditions. Under the proposed regime-switching pricing model, the semi-analytical
formula for the unilateral CVA can be derived.
The paper is organized as follows. Section 2 describes the cash flows of a payer CDS with and without counterparty
credit risk, and further gives a formula for the unilateral CVA of this CDS in a general set-up. Section 3 introduces the
default dependence structure, the dynamics of the interest and the recovery under the regime-switching framework. We also
present some preliminary results. Section 4 gives the joint distribution of the default times by using the ‘‘change of measure’’
method. The semi-analytical formula for the unilateral CVA is presented in Section 5. Section 6 gives some numerical results.
Section 7 concludes.
2. Unilateral credit valuation adjustment
Given a filtered complete probability space {Ω , ℑ, {ℑt }0≤t ≤T , P }, all random variables of this paper are assumed to be
defined on it. Let Eτ stand for the conditional expectation under P given ℑτ .
A single name CDS is an insurance contract on the default of a single reference credit between a protection buyer
(investor) and a protection seller (counterparty). We assume the CDS buyer is default-free, and denote by rt the stochastic
risk-free interest rate. Consider a CDS contract with notional value one, continuous spread rate payments κ and maturity
time T . Indices 1, 2 refer to quantities related to the reference entity and the counterparty. Denote by τ1 and τ2 the default
times of the reference entity and the counterparty, respectively, denote by R1 (τ1 ) and R2 (τ2 ) the stochastic recoveries of the
reference entity and the counterparty upon default. Assume all the cash flows and prices are considered from the perspective
of the investor and there are no simultaneous defaults. Denote by x+ = max{x, 0} and x− = − min{x, 0} the positive part and
the negative part of x, x ∈ R, respectively. We first define the discounted cash flows of CDS with and without counterparty
credit risk.
Definition 2.1. The model price process of a CDS without counterparty credit risk is given by Pt = Et [pT (t )], where pT (t )
corresponds to the cumulative discounted cash flows of the CDS without counterparty risk on the time interval (t , T ], so
pT (t ) = −κ




T ∧τ1

e−

s

t rv dv

ds + (1 − R1 (τ1 ))e−

 τ1
t

rv dv

1{t <τ1 ≤T } ,

(2.1)

t ∧τ1

with pT (t ) = 0 for t ≥ τ1 ∧ T .
Now let us turn to the model price process of a CDS with counterparty risk. Under a CDS with counterparty risk, the
investor pays to the counterparty a stream of premia with spread κ , from the inception date (time 0 henceforth) until the
occurrence of a credit event (default of the counterparty or the reference entity) or the maturity time T of the contract,
whichever comes first. If the counterparty defaults while the reference entity is still alive, a ‘‘fair value’’ Pτ2 of the CDS is
computed at time τ2 . If Pτ2 is negative for the investor, it is completely paid by the investor. If Pτ2 is positive for the investor,

the counterparty is assumed to pay only a recovery fraction R2 of Pτ2 to the investor. Therefore, from the above description,
we have the following definition.


Y. Dong et al. / Statistics and Probability Letters 85 (2014) 25–35

27

Definition 2.2. The model price process of a CDS with counterparty credit risk is given by Πt = Et [πT (t )], where πT (t )
corresponds to the cumulative discounted cash flows of the CDS with counterparty risk on the time interval (t , T ], so

πT (t ) = −κ



T ∧τ1 ∧τ2

τ1 ∧τ2 ∧t
 τ2
− t rv dv

+e

e−

s

t rv dv

ds + (1 − R1 (τ1 ))e−


 τ1
t

rv dv

1{t <τ1 ≤T ,τ1 <τ2 }

1{τ2 <τ1 ,t <τ2 ≤T } (R2 (τ2 )Pτ+2 − Pτ−2 ),

(2.2)

with πT (t ) = 0 for t ≥ τ1 ∧ τ2 ∧ T .
The credit valuation adjustment is typically defined as the difference between the value of a CDS, assuming the counterparty is default-risk-free, and the value reflecting default risk of the counterparty. More specially, CVAt = 1{τ2 >t } (Pt − Πt ).
Then the formula for the unilateral CVA is given as in the following proposition.
Proposition 2.1. At valuation time t, and conditional on the event {τ2 > t },
1{τ2 >t } CVAt = 1{τ2 >t } Et [(1 − R2 (τ2 ))e−

 τ2
t

rv dv

Pτ+2 1{t <τ2 ≤T ,τ2 <τ1 } ],

(2.3)

where Pt is defined in Definition 2.1.
Proof. Conditional on {τ2 > t }, we can rewrite πT (t ) as


πT (t ) = 1{τ2 >T } pT (t ) + 1{t <τ2 ≤T } (pτ2 (t ) + e−

 τ2
t

rv dv

(R2 (τ2 )Pτ+2 − Pτ−2 )).

Similarly,
1{τ2 >t } pT (t ) = 1{τ2 >T } pT (t ) + 1{t <τ2 ≤T } (pτ2 (t ) + e−

 τ2
t

rv dv

pT (τ2 )).

So, conditional on the event {τ2 > t },
CVAt = Et [pT (t ) − πT (t )] = Et [1{t <τ2 ≤T } e−

= Et [Eτ2 [1{t <τ2 ≤T } e−
= Et [1{t <τ2 ≤T } e−

 τ2
t

 τ2
t


rv dv

= Et [(1 − R2 (τ2 ))e−

 τ2
t

rv dv

 τ2
t

rv dv

(pT (τ2 ) − (R2 (τ2 )Pτ+2 − Pτ−2 ))]

(pT (τ2 ) − (R2 (τ2 )Pτ+2 − Pτ−2 ))]]

(Pτ2 − (R2 (τ2 )Pτ+2 − Pτ−2 ))]
rv dv

Pτ+2 1{t <τ2 ≤T } ],

where the fourth equality holds since e−
holds.

 τ2
t


rv dv

1{t <τ2 ≤T } is ℑτ2 -measurable. Note that Pt 1{τ1 ≤t } = 0. Consequently, (2.3)

Remark 2.1. The value of the unilateral counterparty risk is always positive and is equal to the value of a long position in
a call option with zero strike. If we relax the assumption that the CDS buyer is default-free, then the formula for the bilateral CVA can be obtained similarly. But this adjustment may change signs depending on the relative riskiness of the two
counterparties.
3. Regime-switching interacting intensities
In this section, we aim at modeling the default dependence between τ1 and τ2 under Markovian environment.
Let {Xt }t ≥0 be a homogeneous continuous-time, finite-state, irreducible Markov chain with generator Q = (qij )i,j=1,2,...,N ,
generating a filtration ℑXt . As in Buffington and Elliott (2002), the state space of X can be taken to be, without loss of
generality, the set of unit vectors E = {e1 , e2 , . . . , eN }, ei = (0, . . . , 0, 1, 0, . . . , 0)∗ ∈ RN , where ∗ denotes the transpose of
a vector or a matrix. Elliott (1993) and Elliott et al. (1994) provide the following semi-martingale decomposition for {Xt }t ≥0 :



t

Xt = X0 +

Q ∗ Xs ds + Mt ,

(3.1)

0

where {Mt }t ≥0 is an RN -valued martingale with respect to the filtration ℑXt .
We now introduce the default dependence under the reduced-form framework. Denote the filtration by

ℑt = ℑXt ∨ ℑ1t ∨ ℑ2t ,

where ℑit = σ (Hui : 0 ≤ u ≤ t ), with Hui = 1{τi ≤u} , i = 1, 2. For each i = 1, 2, assume τi possesses a nonnegative ℑt
t
predictable intensity process λit satisfying E [ 0 λis ds] < ∞, for all t < ∞ and the compensated process
Mti = 1{τi ≤t } −

t ∧τi


0

λis ds


28

Y. Dong et al. / Statistics and Probability Letters 85 (2014) 25–35

is a (ℑ, P )-martingale. Assume the default intensities of the reference entity and the counterparty are expressed as


λ1t = a1t + a2t 1{τ2 ≤t } ,

(3.2)

λ2t = a3t + a4t 1{τ1 ≤t } ,
j

where at = ⟨aj (t ), Xt ⟩, with aj (t ) = (aj1 (t ), aj2 (t ), . . . , ajN (t ))∗ ∈ RN , j = 1, 2, 3, 4. Here for each j = 1, 2, 3, 4, k =
t
1, 2, . . . , N , the ajk (t ) is a deterministic function of t valued on (0, ∞) which satisfies 0 ajk (s)ds < ∞, ∀t < ∞, and ⟨., .⟩


denotes the Euclidean inner product in RN , that is, for any x, y ∈ RN , ⟨x, y⟩ =
i=1 xi yi . Furthermore, throughout this paper
we assume rv = ⟨r, Xv ⟩, Rj (τj ) = ⟨Rj , Xτj ⟩, where r = (r1 , . . . , rN )∗ ∈ RN , Rj = (Rj1 , . . . , RjN )∗ ∈ RN , with ri > 0, 0 ≤ Rji < 1
for each j = 1, 2, i = 1, . . . , N .
Note that the default dependence modeled by (3.2) stems from two sources. First, the intensities of the two firms are
both affected by macro-economy, so we have the inter-dependence between their defaults through a Markov chain. Second,
the inter-dependent default structure arises from default contagion. The main obstacle of solving a contagion model is the
looping structure (3.2) of the intensities. This paper will follow the idea of ‘‘change of measure’’ proposed in Collin-Dufresne
et al. (2004).
Define the following survival measures:

N


 t

.
i
 = 1{τ >t } exp
λ
ds
= ηti ,
s
i
dP ℑt
0

dP i 


t ≤ T , i = 1, 2,

(3.3)

where P i is a firm-specific (firm i) probability measure which is absolutely continuous with
 t respect to P on the stochastic
interval [0, τi ). From Lemma A.2 in Collin-Dufresne et al. (2004), we have that 1{τi >t } exp( 0 λis ds) is a uniformly integrable
P-martingale with respect to ℑt and is almost surely strictly positive on [0, τi ) and almost surely equal to zero on [τi , ∞).
i

i

To proceed with the calculations under the measure P i , we enlarge the filtration to ℑ = (ℑt )t ≥0 as the completion of ℑ =
(ℑt )t ≥0 by the null sets of the probability measure P i . Let E i [.] denote the expectation taken under the measure P i . For noi

tational convenience, we still use ℑ instead of ℑ without changing the results. Then, for any P i -integrable random variable
Y and t < s ≤ T , it follows from Collin-Dufresne et al. (2004) that
E [Y ηti ] = E i [Y ],

E [ηsi Y |ℑt ] = ηti E i [Y |ℑt ].

The next result shows that, under P i , the Markov chain Xt has the same distribution as that under P .
Proposition 3.1. The process
t



Q ∗ Xs ds

Mt = Xt − X0 −


(3.4)

0

is an RN -valued martingale under P i .
Proof. Since Mt is an RN -valued martingale under P , it suffices to prove that Mt ηti is a martingale under P. From Itô’s formula,
we have
Mt ηti = M0 +

t



ηsi − dMs +

t



0

Ms− dηsi +



0

∆Ms ∆ηsi .


s≤t

Since Mt and η
(ℑ, )
t
=τ
η ̸=
= τi , the summation in the last term
As exp( 0 λ )
η
above will be zero provided that ∆Mτi = 0, a.s. So, it remains to show that ∆Mτi = 0, a.s. In fact, P (∆Mt ̸= 0) = P (∆Xt ̸=
0) = 0 for any fixed time t > 0. Denote 0 < T1 < T2 < · · · the transition times of X . Then, conditioning on X, we have
∞
t
P (∆Xτi ̸= 0) = E [E [1{∆Xτi ̸=0} |Xs , s ≥ 0]] = E [E [ j=1 1{τi =Tj } |Xs , s ≥ 0]]. By using the definition τi = min{t > 0 : 0 λit dt ≥
i
P -martingales, it remains to show the last term vanishes.
t are both
i
i
i
0 at s
ds
is
continuous,
i . Note that, given ∆ s
t jumps only at t
s

Ei }, we obtain E [


∞

j =1

1{τi =Tj } |Xs , s ≥ 0] = E [

∞

j =1

1  Tj
{

0

λis ds=Ei }

|Xs , s ≥ 0] = 0, where the last equality holds because Ei and X

are independent and Ei is a continuous random variable. So, we can conclude ∆Mτi = ∆Xτi = 0, a.s. The proof is completed.
In order to derive the joint distribution of the default times and the expression for the unilateral CVA, we first give a
useful result. Define an RN valued process
V (t , T ) = E [e−

T

t fu du

XT |ℑXt ],


(3.5)

where ft = ⟨ft , Xt ⟩ with ft = (f1 (t ), . . . , fN (t )) . Here fi (u) is a deterministic function valued on (0, ∞) for each i = 1, . . . ,
N. For notational simplicity, we define diag(θ) as a diagonal matrix with the diagonal entries given by the vector θ =
(θ1 , . . . , θN )∗ ∈ RN . Note that X is a Markov chain with respect to ℑX . Consequently,
∗

V (t , T ) = E [e−

T

t fu du

XT |Xt ] =: F (t , T , Xt ).


Y. Dong et al. / Statistics and Probability Letters 85 (2014) 25–35

29

Lemma 3.1. Let V (t , T ) be an RN valued process defined by (3.5). Then we have
V (t , T ) = ⟨Φ (t , T ), Xt ⟩,

(3.6)

where the matrix Φ solves

∂Φ
+ (Q − diag(ft ))Φ (t , T ) = 0,

∂t

Φ (T , T ) = I,

(3.7)

with I = diag(1) and 1 = (1, . . . , 1)∗ ∈ RN . Furthermore,
E [e−

T

t fu du

|ℑXt ] = ⟨Φ (t , T )1, Xt ⟩.

(3.8)

Proof. Write Fi (t , T ) = F (t , T , ei ) for i = 1, 2, . . . , N and

Φ (t , T ) = (F1 (t , T ), F2 (t , T ), . . . , FN (t , T ))∗ ∈ RN .

N
− 0t fu du
Obviously, F (t , T , Xt ) =
F ( t , T , Xt )
i=1 Fi (t , T )⟨ei , Xt ⟩ = ⟨Φ (t , T ), Xt ⟩. Applying Itô’s differentiation rule to Zt = e
yields
dZt = −ft Zt dt + e−

t


0 fu du




t
∂F
+ ⟨Φ (t , T ), Q ∗ Xt ⟩ dt + e− 0 fu du ⟨Φ (t , T ), dMt ⟩.
∂t

Note that Zt is a bounded ℑXt martingale since ft > 0 for any t > 0. So, the bounded variation terms in the above equality
must sum to zero. That is to say, for any x ∈ E , the function (t , x) → F (t , T , x) solves

∂F
− f (t )F (t , T , x) + ⟨Q Φ (t , T ), x⟩ = 0, F (T , T , x) = x, x ∈ E .
∂t
Since x takes e1 , . . . , eN , we have


∂Φ
, ei − ⟨diag(ft )Φ (t , T ), ei ⟩ + ⟨Q Φ (t , T ), ei ⟩ = 0, i = 1, . . . , N .
∂t
That is to say, Φ is the fundamental matrix solution of the following equation:

∂Φ
+ (Q − diag(ft ))Φ (t , T ) = 0,
∂t
with Φ (T , T ) = I. So the proof of the first part is completed. Note that ⟨XT , 1⟩ = ⟨
1. Consequently,


N

E [e −

T

t fu du

T

i=1

1{XT =ei } ei , 1⟩ =

N

i=1

1{XT =ei } ⟨ei , 1⟩ =

T

|ℑXt ] = E [e− t fu du ⟨XT , 1⟩|ℑXt ] = ⟨E [e− t fu du XT |ℑXt ], 1⟩
= ⟨Φ ∗ (t , T )Xt , 1⟩ = 1∗ Φ ∗ (t , T )Xt = ⟨Φ (t , T )1, Xt ⟩.

Remark 3.1. If all of the fi (t ) are constants, then Φ (t , T ) = e(Q −diag(f))(T −t ) . Consequently, V (t , T ) = ⟨e(Q −diag(f))(T −t ) , Xt ⟩,
which is consistent with Lemma A.1 in Buffington and Elliott (2002).
4. Joint distributions
In this section, we follow the idea of change of measure to derive the two-dimensional conditional and unconditional

joint distributions of the default times.
Proposition 4.1. For 0 < s ≤ T ,
s 1
3
P (τ1 > s, τ2 > s|ℑXT ) = e− 0 (au +au )du .



(4.1)

For 0 ≤ t < s ≤ T ,
P (τ1 > s, t < τ2 ≤ s|ℑXT ) =

s



a3v e−

t

v

s 1
s 2
3
0 au du− 0 au du− v au du

dv


(4.2)

dv.

(4.3)

and
P (τ2 > s, t < τ1 ≤ s|ℑ ) =
X
T


t

s

a1v e−

v

s 3
s 4
1
0 au du− 0 au du− v au du


30

Y. Dong et al. / Statistics and Probability Letters 85 (2014) 25–35


Proof. To prove (4.1), it suffices to prove that for any event A ∈ ℑXT , it holds that
s 1
3
E [1{A} E [1{τ1 >s,τ2 >s} |ℑXT ]] = E [1{A} e− 0 (au +au )du ].



To this end, using the ‘‘tower property’’ of conditional expectations yields
E [1{A} E [1{τ1 >s,τ2 >s} |ℑXT ]] = E [1{A} 1{τ1 >s,τ2 >s} ].
Recall that for an arbitrary but fixed time 0 < s ≤ T , the survival probability is defined by
dP 1
dP

s





λ

1
u du

|ℑs = 1{τ1 >s} exp

.
= ηs1 .

0


Furthermore, for any P -integrable random variable Y , the equality E [Y ηs1 ] = E 1 [Y ] holds. Therefore, changing measure P
to P 1 yields
1

E [1{A} 1{τ1 >s,τ2 >s} ] = E [1{A} 1{τ2 >s} e−
1

1

s
0

λ1u du 1

X
T

ηs ] = E 1 [1{τ2 >s} e−

= E [E [1{τ2 >s} |ℑ ]e
s

−

s

1
0 au du 1


{A} ]

1

−

= E [e

s

1
0 au du

1{A} ]

s

3
1
0 (au +au )du

1{A} ],

1

where the third equality holds since e− 0 au du 1{A} ∈ ℑXT , and the last equality holds because τ2 has the intensity a3u under P 1 .
Then the fact that the distribution of Xt under P 1 is the same as that under P concludes the proof.
The proof of (4.2) is similar. For any event A ∈ ℑXT , by changing measure P to P 1 we have
E [1{A} E [1{τ1 >s,t <τ2 ≤s} |ℑXT ]] = E [1{τ1 >s,t <τ2 ≤s} 1{A} ]


= E [1{t <τ2 ≤s} e−

s
0

λ1u du
−

1{A} ηs1 ] = E 1 [1{t <τ2 ≤s} e

−

s

s 2
1
0 au du− τ2 au du

1{A} ]

s

s 2
1
0 au du− τ2 au du

= E 1 [E 1 [1{t <τ2 ≤s} e
|ℑXT ]1{A} ]
 s


v 3
s 1
s 2
= E1
a3v e− 0 au du− 0 au du− v au du dv 1{A} ,
t

where the last equality holds because τ2 has the intensity a3u under measure P 1 . Then by using Proposition 3.1, we conclude
the proof. The proof of (4.3) is similar to the one of (4.2), so we omit it.
Proposition 4.2. Let Ψ 1 (t , s), Ψ 2 (t , s) and Ψ 3 (t , s) be determined by (3.7) with ft replaced by a1 (t ) + a3 (t ), a1 (t ) + a2 (t ) and
a3 (t ) + a4 (t ), respectively. Then for s > 0, we have
P (τ1 > s, τ2 > s) = ⟨Ψ 1 (0, s)1, X0 ⟩.

(4.4)

For s > t ≥ 0, we have
P (τ1 > s, t < τ2 ≤ s) =

s



⟨Ψ 1 (0, v)diag(a3 (v))Ψ 2 (v, s)1, X0 ⟩dv,

(4.5)

⟨Ψ 1 (0, v)diag(a1 (v))Ψ 3 (v, s)1, X0 ⟩dv.

(4.6)


t

and
P (τ2 > s, t < τ1 ≤ s) =

s


t

Proof. Since the proofs of Eqs. (4.4)–(4.6) are similar, we only prove (4.5). Using the ‘‘tower property’’ of conditional expectations and (4.2) yields
E [1{τ1 >s,t <τ2 ≤s} ] = E [E [1{τ1 >s,t <τ2 ≤s} |ℑXT ]]
s



a3v e−

=E
t
s


=E

v 3 1
a3 e− 0 (au +au )du E

s




s 1
s 2
3
0 au du− 0 au du− v au du

v

t

e−

=E

v

v

3
1
0 (au +au )du

dv



s 1 2
[e− v (au +au )du |ℑXv ]dv




⟨diag(a3 (v))Ψ 2 (v, s)1, Xv ⟩dv



t

where the second equality is obtained from (4.2), and the last second equality follows from Lemma 3.1. Then again using
Lemma 3.1 yields (4.5).
The following results are direct consequences of Proposition 4.2.


Y. Dong et al. / Statistics and Probability Letters 85 (2014) 25–35

31

Corollary 4.1. For s > 0, the marginal distributions of τ1 and τ2 are given by
P (τ1 > s) = ⟨Ψ 1 (0, s)1, X0 ⟩ +

s



⟨Ψ 1 (0, v)diag(a3 (v))Ψ 2 (v, s)1, X0 ⟩dv

(4.7)

⟨Ψ 1 (0, v)diag(a1 (v))Ψ 3 (v, s)1, X0 ⟩dv,


(4.8)

0

and
P (τ2 > s) = ⟨Ψ 1 (0, s)1, X0 ⟩ +

s


0

respectively, where Ψ i (v, s), i = 1, 2, 3 are defined in Proposition 4.2.
Remark 4.1. Since the joint distribution and the marginal distributions of the default times have been derived, we can calculate various dependence measures which quantify the relation of pairwise default correlation, such as, the linear correlation
coefficient of the default events {τ1 ≤ t } and {τ2 ≤ t }, and the Spearman’s Rho coefficient of τ1 and τ2 conditional on
min{τ1 , τ2 } > t. Since this paper mainly focuses on the computation of the CVA, we do not discuss them in detail.
5. Arbitrage-free valuation of unilateral counterparty risk
In this section we aim at deriving the fair spread κ of a CDS without counterparty risk and the unilateral CVA of this CDS.
The spread κ of the CDS without counterparty risk on the reference entity can be obtained by setting the value of P0 to
be zero. Hence, we have the following proposition.
Proposition 5.1. Let A1 (t , s) and A2 (t , s) be determined by (3.7) with ft replaced by r + a1 (t ) + a3 (t ) and r + a1 (t ) + a2 (t ),
respectively. Then the spread κ is given by

T
κ=

0

⟨A1 (0, s)diag(L1 )a1 (s) + g1 (s), X0 ⟩ds
,

T
⟨A1 (0, s)1 + g2 (s), X0 ⟩ds
0

(5.1)

where
g1 (s) =

s



A1 (0, v)diag(a3 (v))A2 (v, s)diag(L1 )(a1 (s) + a2 (s))dv

(5.2)

A1 (0, v)diag(a3 (v))A2 (v, s)1dv,

(5.3)

0

and
g2 (s) =

s


0


with L1 = 1 − R1 ∈ RN .
Proof. The expected present value of the swap premium payment over [t , T ] is

κE

T



e−

s

0 rv dv


1{τ1 >s} ds

= κE

T



e−

s

0 rv dv




E [1{τ1 >s,τ2 >s} |ℑXT ] + E [1{τ1 >s,τ2 ≤s} |ℑXT ] ds





0

0

=κ

T







0

=κ

0
T






⟨A1 (0, s)1, X0 ⟩ +



⟨A1 (0, s)1, X0 ⟩ +



a3v e−

v

s
s 2
3
1
0 av du− 0 (ru +au )du− v au du

v

1
3
0 (au +au +ru )du


ds


s
1
2
E [e− v (ru +au +au )du |ℑXv ]]dv



s



0

=κ

E [a3v e−

0
T



s



0

=κ


s



s
1
3
E [e− 0 (rv +av +av )dv ] + E

⟨diag(a3 (v))A2 (v, s)1, E [e−

v

1
3
0 (au +au +ru )du



Xv ]⟩dv

ds



ds,

0
T


⟨A1 (0, s)1 + g2 (s), X0 ⟩ds,

0

where the second equality follows from Proposition 4.1, the last second equality holds because ⟨a3v A2 (v, s)1, Xv ⟩ = ⟨diag
(a3 (v))A2 (v, s)1, Xv ⟩, and the last equality follows from Lemma 3.1.
Similarly, the expected present value of the loss payment over [t , T ] is
E [(1 − R1 (τ1 ))e−

 τ1
0

rv dv

T



(1 − R1 (s))e−

1{τ1 ≤T } ] = E

s

0 rv dv

0
T




E [(1 − R1 (s))e−

=
0

s

0 rv dv

1{τ1 >s− } dHs1



(1{τ1 >s,τ2 >s} a1s + 1{τ1 >s,τ2 ≤s} (a1s + a2s ))ds]ds


32

Y. Dong et al. / Statistics and Probability Letters 85 (2014) 25–35
T



 



s

1
3
E e− 0 (rv +av +av )dv ⟨diag(L1 )a1 (s), Xs ⟩

=



0
T





E ⟨diag(L1 )(a1 (s) + a2 (s)), Xs ⟩

+
T



a3v e−

0

0

⟨A1 (0, s)diag(L1 )a1 (s), X0 ⟩ds +


=

s



T


0

0

s


0

v

s
s 2
3
1
0 au du− 0 (ru +au )du− v au du

s

⟨E [a3v e−


dv


ds

v

1
3
0 (ru +au +au )du

× E [e− v (ru +au +au )du Xs |ℑXv ]]dv, diag(L1 )(a1 (s) + a2 (s))⟩ds
 T
⟨A1 (0, s)diag(L1 )a1 (s) + g1 (s), X0 ⟩ds,
=
1

2

0

 t ∧τ

where the second equality holds because Hs1 − 0 1 λ1s ds is an ℑ-martingale, the third equality is obtained by using Proposition 4.1 and the equalities (1 − R1 (s))a1s = ⟨diag(L1 )a1 (s), Xs ⟩ and (1 − R1 (s))(a1s + a2s ) = ⟨diag(L1 )(a1 (s) + a2 (s)), Xs ⟩,
and the last two equalities follow from Lemma 3.1. Then equating the expected present value of the premium payment to
the expected present value of the loss payment yields the result. The proof is completed.
Now we turn to calculate the unilateral CVA. From Definition 2.1, we have 1{τ1 ≤t } Pt = 0, then
Pt = 1{τ1 >t } Pt = 1{τ1 >t ,τ2 >t } Pt + 1{τ1 >t ,τ2 ≤t } Pt .
In particular,
Pτ2 = 1{τ1 >τ2 ,τ2 ≤τ2 } Pτ2 .

Consequently, to derive the unilateral CVA, we need to calculate Pt on the event {τ1 > t , τ2 ≤ t }. We first give a useful
lemma.
Lemma 5.1. For any 0 < t ≤ s ≤ T and any ℑXT -measurable random variable Z , we have
s 1
2
1{τ1 >t ,τ2 ≤t } Et [Z 1{τ1 >s} ] = 1{τ1 >t ,τ2 ≤t } E [Ze− t (av +av )dv |ℑXt ].



(5.4)

Proof. Changing measure from P to P 1 yields
−

s

= 1{τ1 >t ,τ2 ≤t } E P [Ze−

s

1

1{τ1 >t ,τ2 ≤t } Et [Z 1{τ1 >s} ] = 1{τ1 >t ,τ2 ≤t } E P [Ze
1

1
2
t (av +av 1{τ2 ≤v} )dv
1
2

t (av +av 1{τ2 ≤v} )dv

|ℑt ]
|ℑXt , τ1 > t , τ2 ≤ t ]

s 1
2
1
E P [1{τ1 >t ,τ2 ≤t } Ze− t (av +av )dv |ℑXt ]



= 1{τ1 >t ,τ2 ≤t }

1

E P [1{τ1 >t ,τ2 ≤t } |ℑXt ]
s 1
2
1
1
E P [E P [1{τ2 ≤t } |ℑXT ]Ze− t (av +av )dv |ℑXt ]



= 1{τ1 >t ,τ2 ≤t }

1

E P [1{τ2 ≤t } |ℑXt ]

1

= 1{τ1 >t ,τ2 ≤t } E P [Ze−

s

1
2
t (av +av )dv

|ℑXt ],

s 1
2
where the fourth equality holds since 1{τ1 >t } = 1 under P 1 and Ze− t (av +av )dv ∈ ℑXT , and the last equality holds because



1

1

E P [1{τ2 ≤t } |ℑXT ] = E P [1{τ2 ≤t } |ℑXt ]. Then the result follows from the fact the distribution of X under P 1 is the same as that
under P .
Proposition 5.2. On the event {τ1 > t , τ2 ≤ t }, the price of a CDS without counterparty risk with spread κ on the firm admits
the representation
1{τ1 >t ,τ2 ≤t } Pt = 1{τ1 >t ,τ2 ≤t } ⟨µt , Xt ⟩=
ˆ 1{τ1 >t ,τ2 ≤t } P¯t ,

(5.5)


where

µt =

T



A2 (t , s)(diag(L1 )(a1 (s) + a2 (s)) − κ 1)ds,

(5.6)

t

with A2 (t , s) and L1 defined in Proposition 5.1.
In particular,
Pτ2 = 1{τ1 >τ2 ,τ2 ≤τ2 } Pτ2 =
ˆ 1{τ1 >τ2 } P¯τ2 .

(5.7)


Y. Dong et al. / Statistics and Probability Letters 85 (2014) 25–35

33

Proof. By using Lemmas 3.1 and 5.1, we have
1{τ1 >t ,τ2 ≤t } κ Et


T



−

1{τ1 >s} e

s

t rv dv


ds

= 1{τ1 >t ,τ2 ≤t } κ



= 1{τ1 >t ,τ2 ≤t } κ



T

s
2
1
E [e− t (rv +av +av )dv |ℑXt ]ds




t

t

T

⟨A2 (t , s)1, Xt ⟩ds.

(5.8)

t

The expected present value of the loss payment over [t , T ] is
1{τ1 >t ,τ2 ≤t } Et [(1 − R1 (τ1 ))e−

 τ1
t

rv

T



(1 − R1 (s))e−

1{t <τ1 ≤T } ] = 1{τ1 >t ,τ2 ≤t } Et
t

T



Et [e−

= 1{τ1 >t ,τ2 ≤t }

s

t rv dv

t
T


= 1{τ1 >t ,τ2 ≤t }

s

t rv

1{τ1 >s− } dHs1



1{τ1 >s− } (1 − R1 (s))(a1s + a2s )]ds

s
1

2
E [e− t (rv +av +av )dv ⟨diag(L1 )(a1 (s)



t

+ a2 (s)), Xs ⟩|ℑXt ]ds
 T
= 1{τ1 >t ,τ2 ≤t }
⟨A2 (t , s)diag(L1 )(a1 (s) + a2 (s)), Xt ⟩ds,

(5.9)

t

 t ∧τ

where the second equality is obtained by using the fact that Hs1 − 0 1 λ1s ds is an ℑ-martingale, and the third equality
follows from Lemma 5.1. Then substituting (5.8) and (5.9) into (2.1) yields the result.
+ ∗
+
For notational convenience, for each a = (a1 , a2 , . . . , aN )∗ ∈ RN , denote by a+ = (a+
1 , . . . , aN ) , where ai = max{ai , 0}.
Proposition 5.3. At valuation time t and conditional on the event {τ2 > t },
T



⟨A1 (t , s)diag(a3 (s))diag(L2 )µ+

s , Xt ⟩ds,

1{τ2 >t } CVAt = 1{τ2 ∧τ1 >t }

(5.10)

t

where L2 = 1 − R2 ∈ RN , A1 (t , s) is defined in Proposition 5.1 and µs is defined in Proposition 5.2.
In particular, we obtain
T



⟨A1 (0, s)diag(a3 (s))diag(L2 )µ+
s , X0 ⟩ds.

CVA0 =

(5.11)

0

Proof. From Proposition 2.1, we have
T


1{τ2 >t } CVAt = 1{τ1 >t } Et

e


−

s

t rv dv

(1 − R2 (s))P¯s+ 1{τ1 ∧τ2 >s} dHs2



t


= 1{τ1 ∧τ2 >t } Et

T

e

−

t
T


= 1{τ1 ∧τ2 >t }

s


t rv dv


+
3
¯
(1 − R2 (s))Ps 1{τ1 ∧τ2 >s− } as ds

s
1
3
E [e− t (rv +av +av )dv ⟨diag(a3 (s))diag(L2 )µ+
s , Xs ⟩ds],



t

τ

∧t

where the second equality holds because Ht2 − 0 2 λ2s ds is an ℑ-martingale and the last equality follows from Propositions 4.1 and 5.2. Then using Lemma 3.1 yields the result.
6. Numerical results
In this section, we shall present some numerical calculations based on the results derived in Section 5. Since we focus
on investigating the impact of regime switching on the spread and the CVA, we just make some numerical analysis without
doing the calibration in this paper. One thing on our future research agenda is to use the credit market data to empirically
test our model.
For ease of illustration, we consider N = 2, that is X only switches between two states, where state e1 and state e2 represent a ‘‘good’’ economy and a ‘‘bad’’ economy, respectively. Let T = 10, r = (0.05, 0.02)∗ , R1 = R2 = (0.6, 0.2)∗ , a1 (t ) =
(0.01, 0.03)∗ , a2 (t ) = (0.002, 0.006)∗ , a3 (t ) = (0.005, 0.015)∗ . To investigate the regime switching effect, we compare

the regime switching contagion intensities model with the one that has no regime switching. So for each ft = ⟨f, Xt ⟩ with
T

f = (f1 , f2 )∗ , we choose a constant f in the model without regime switching, such that it satisfies e−fT = E [e− 0 ft dt |X0 = ei ].
Figs. 1 and 2 present the relationship between the spread and q12 . From them we can see, the spreads are higher when we
start at the state e2 at time t = 0. We can also see when X0 = e1 , the spread in the no regime-switching model is higher than


34

Y. Dong et al. / Statistics and Probability Letters 85 (2014) 25–35

Fig. 1. The relation between q12 and κ, q21 = 0.2.

Fig. 2. The relation between q12 and κ, q21 = 0.5.

that in the regime-switching model, and the reverse relationship holds when X0 = e2 . From them we can easily conclude that
the spreads increase with q12 increasing and decrease with q21 increasing, which is because a higher q12 leads to an increasing
probability of switching to the state e2 , while a higher q21 leads to an increasing probability of switching to the state e1 .
Figs. 3 and 4 present the impacts of some model parameters on the CVA0 . From them we can see that the CVA at time
t = 0 in the regime-switching model is higher than that in the no regime-switching model. We can also see that the CVA
at time t = 0 is higher when ξ0 = e2 . Figs. 3 and 4 also show that CVA0 increases with a3 increasing, in line with stylized
features and the financial intuition: the counterparty is more likely to default with a3 increasing, so the adjustment increases.
Therefore, numerical results reveal that if we do not incorporate the changes of market regimes into the pricing models, we
shall underestimate or overestimate the spreads and the CVA.
7. Conclusions
This paper considers a regime-switching interacting intensities model, in which one firm’s intensity will have a jump
when the other firm defaults, and the intensities of the protection seller and the reference entity are both affected by a
Markov chain. In particular, the interest rate and the recovery upon default are also stochastic. Under the Markov, regimeswitching pricing model, the joint distributions of the default times and the unilateral CVA of a CDS can be represented as
some fundamental matrix solutions of linear, matrix-valued, ordinary differential equations. So the model we propose is

very easy to implement. Numerical results reveal that the regime-switching effect in the valuation of the credit derivatives
is practically meaningful.


Y. Dong et al. / Statistics and Probability Letters 85 (2014) 25–35

35

Fig. 3. The relation between q12 and CVA0 , a3 (t ) = (0.005, 0.015)∗ .

Fig. 4. The relation between q12 and CVA0 , a3 (t ) = (0.01, 0.03)∗ .

Acknowledgments
The authors thank the editor and the anonymous referees for their helpful comments. The research of Yinghui Dong is
supported by the Natural Science Foundation of Jiangsu Province (Grant No. BK20130260), the National Natural Science
Foundation of China (Grant No. 11301369) and China Postdoctoral Science Foundation (Grant No. 2013M540371). The
research of Kam C. Yuen is supported by a grant from the Research Grants Council of the Hong Kong Special Administrative
Region, China (Project No. HKU 7057/13P). The research of Chongfeng Wu is supported by the National Natural Science
Foundation of China (Grant No. 71320107002).
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