We consider the unilateral credit valuation adjustment (CVA) of a credit default swap (CDS) under a contagion model with regimeswitching 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 macroeconomy 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 semianalytical formulas for the joint distribution of the default times and the unilateral CVA of a CDS.
Trang 1Contents lists available atScienceDirect Statistics and Probability Letters journal homepage:www.elsevier.com/locate/stapro
Unilateral counterparty risk valuation of CDS using a
regime-switching intensity model
aFinancial Engineering Research Center, Shanghai Jiao Tong University, Shanghai 200052, PR China
bDepartment of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, PR China
cDepartment of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong
a r t i c l e i n f o
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
a b s t r a c t
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 as-sumes 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:dongyinghui1030@163.com (Y Dong).
0167-7152/$ – see front matter © 2013 Elsevier B.V All rights reserved.
Trang 2reduced-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) andYu(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 inten-sities 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 eco-nomic expansions because strong earnings keep overall default rates low Default risk increases during ecoeco-nomic 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) andSiu(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 byJarrow 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 Section2describes 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 Section3introduces the default dependence structure, the dynamics of the interest and the recovery under the regime-switching framework We also present some preliminary results Section4gives 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 Section5 Section6gives some numerical results Section7concludes
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 Eτstand for the conditional expectation under P givenℑτ.
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 r tthe stochastic risk-free interest rate Consider a CDS contract with notional value one, continuous spread rate paymentsκand maturity
time T.Indices 1, 2 refer to quantities related to the reference entity and the counterparty Denote byτ1andτ2the 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 P t =E t[pT(t)], where p T(t)
corresponds to the cumulative discounted cash flows of the CDS without counterparty risk on the time interval(t,T], so
p T(t) = −κ
T∧ τ 1
t∧ τ 1
e−
s
rvdvds+ (1−R1(τ1))e−
τ 1
t rvdv1{
with p T(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 spreadκ, 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’’ Pτ 2 of the CDS is computed at timeτ2 If Pτ 2is negative for the investor, it is completely paid by the investor If Pτ 2is positive for the investor,
the counterparty is assumed to pay only a recovery fraction R2of Pτ 2to the investor Therefore, from the above description,
we have the following definition
Trang 3Definition 2.2 The model price process of a CDS with counterparty credit risk is given byΠt = E t[ πT(t)],whereπT(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
∧ τ 1 ∧ τ 2
τ 1 ∧ τ 2 ∧t
e−
s
rvdvds+ (1−R1(τ1))e−
τ 1
t rvdv1{
t<τ 1 ≤T,τ 1 <τ 2 }
+e−
τ 2
t rvdv1{
withπT(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
counter-party is default-risk-free, and the value reflecting default risk of the countercounter-party More specially, CVA t =1{τ 2 >t}(P t−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{τ 2 >t}CVA t =1{τ 2 >t}E t[ (1−R2(τ2))e−
τ 2
t rvdvP+
where P t is defined in Definition 2.1
Proof Conditional on{ τ2>t}, we can rewriteπT(t)as
πT(t) =1{τ 2 >T}p T(t) +1{t<τ 2 ≤T}(pτ 2(t) +e−
τ 2
t rvdv(R2(τ2)Pτ+2−Pτ−2)).
Similarly,
1{τ 2 >t}p T(t) =1{τ 2 >T}p T(t) +1{t<τ 2 ≤T}(pτ 2(t) +e−
τ 2
t rvdvp
T(τ2)).
So, conditional on the event{ τ2>t} ,
CVA t =E t[pT(t) − πT(t)] =E t[1{t<τ 2 ≤T}e−
τ 2
t rvdv(p T(τ2) − (R2(τ2)Pτ+2−Pτ−2))]
=E t[Eτ 2[1{t<τ 2 ≤T}e−
τ 2
t rvdv(p T(τ2) − (R2(τ2)Pτ+2−Pτ−2))]]
=E t[1{t<τ 2 ≤T}e−
τ 2
t rvdv(Pτ 2− (R2(τ2)Pτ+2−Pτ−2))]
=E t[ (1−R2(τ2))e−
τ 2
t rvdvP+
τ 21{t<τ 2 ≤T}] ,
where the fourth equality holds since e−
τ 2
t rvdv1{
t<τ 2 ≤T}isℑτ 2-measurable Note that P t1{τ 1 ≤t} = 0 Consequently,(2.3) holds
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 bilat-eral 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 betweenτ1andτ2under Markovian environment
Let{Xt}t≥0be a homogeneous continuous-time, finite-state, irreducible Markov chain with generator Q = (q ij)i,j= 1 , 2 , ,N, generating a filtrationℑX
t.As inBuffington and Elliott(2002), the state space of X can be taken to be, without loss of
generality, the set of unit vectorsE = {e1,e2, ,e N} ,e i= (0, ,0,1,0, ,0)∗∈R N, where∗denotes the transpose of
a vector or a matrix.Elliott(1993) andElliott et al.(1994) provide the following semi-martingale decomposition for{Xt}t≥0:
X t =X0+
t
0
where{Mt}t≥0is an R N-valued martingale with respect to the filtrationℑX
t.
We now introduce the default dependence under the reduced-form framework Denote the filtration by
ℑt= ℑX t ∨ ℑ1t ∨ ℑ2t,
whereℑi
t = σ(H i
u : 0 ≤ u ≤ t), with H i
u = 1{τi≤u},i = 1,2 For each i = 1,2, assumeτi possesses a nonnegativeℑt
predictable intensity processλi
t satisfying E[ 0tλi
s ds]< ∞, for all t< ∞and the compensated process
M t i=1{τi≤t}−
t∧ τi
λi
s ds
Trang 4is a(ℑ,P)-martingale Assume the default intensities of the reference entity and the counterparty are expressed as
λ1
t =a1t +a2t1{τ 2 ≤t},
λ2
where a j t = ⟨aj(t),X t⟩, with aj(t) = (a j1(t),a j2(t), ,a jN(t))∗ ∈ R N,j = 1,2,3,4 Here for each j = 1,2,3,4,k =
1,2, ,N,the a jk(t)is a deterministic function of t valued on(0, ∞)which satisfies0t a jk(s)ds< ∞, ∀t < ∞, and⟨ , ⟩
denotes the Euclidean inner product in R N, that is, for any x,y∈R N,⟨x ,y⟩ =N
i= 1x i y i.Furthermore, throughout this paper
we assume rv= ⟨r,Xv⟩, R j(τj) = ⟨Rj,Xτj⟩, where r= (r1, ,r N)∗∈R N , Rj= (R j1, ,R jN)∗∈R N, with r i>0,0≤R ji<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 inCollin-Dufresne
et al.(2004)
Define the following survival measures:
dP i
dP
ℑt
=1{τi>t}exp
t
0
λi
s ds
= ηi
where P i is a firm-specific (firm i) probability measure which is absolutely continuous with respect to P on the stochastic
interval[0, τi) From Lemma A.2 inCollin-Dufresne et al.(2004), we have that 1{τi>t}exp(t
0λi
s ds)is a uniformly integrable
P-martingale with respect toℑtand is almost surely strictly positive on[0, τi)and almost surely equal to zero on[ τi, ∞)
To proceed with the calculations under the measure P i, we enlarge the filtration toℑi = (ℑi
t)t≥ 0as the completion ofℑ = (ℑt)t≥ 0by the null sets of the probability measure P i Let E i[ ]denote the expectation taken under the measure P i For no-tational convenience, we still useℑinstead ofℑi without changing the results Then, for any P i-integrable random variable
Y and t<s≤T , it follows fromCollin-Dufresne et al.(2004) that
E[Yηi
t] =E i[Y ] , E[ηi
s Y|ℑ t] = ηi
t E i[Y|ℑt]
The next result shows that, under P i , the Markov chain X t has the same distribution as that under P.
Proposition 3.1 The process
M t =X t−X0−
t
0
is an RN-valued martingale under P i.
Proof Since M tis an R N-valued martingale under P,it suffices to prove that M tηi
t is a martingale under P From Itô’s formula,
we have
M tηi
t =M0+
t
0
ηi
s−dM s+
t
0
M s−dηi
s+
s≤t
∆M s∆ηi
s.
Since M tandηi
tare both(ℑ,P)-martingales, it remains to show the last term vanishes
As exp(t
0λi
s ds)is continuous,ηi
t jumps only at t= τi Note that, given∆ηi
s̸=0 at s= τi, the summation in the last term above will be zero provided that∆Mτi =0, a.s So, it remains to show that∆Mτi =0, a.s In fact, P(∆M t ̸=0) =P(∆X t̸=
0) = 0 for any fixed time t > 0 Denote 0< T1 <T2 < · · ·the transition times of X Then, conditioning on X , we have
P(∆Xτi ̸=0) =E[E[1{∆Xτ i̸= 0 }|Xs,s≥0]] =E[E[∞
j= 11{τi=T j}|Xs,s≥0]] By using the definitionτi=min{t >0: 0tλi
t dt ≥
E i}, we obtain E[ ∞
j= 11{τi=T j}|Xs,s≥0] =E[∞
j= 11 { Tj
0 λi
s ds=E i}|Xs,s≥0] =0, where the last equality holds because E i and X are independent and E iis 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 R Nvalued process
V(t,T) =E[e−
T
where f t = ⟨ft,X t⟩with ft= (f1(t), ,f N(t))∗.Here f i(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 vectorθ = (θ1, , θN)∗∈R N.Note that X is a Markov chain with respect toℑX Consequently,
V(t,T) =E[e−
T
t f u du X |X ] =:F(t,T,X).
Trang 5Lemma 3.1 Let V(t,T)be an RNvalued process defined by(3.5) Then we have
where the matrixΦsolves
∂Φ
with I=diag(1)and 1= (1, ,1)∗∈R N Furthermore,
E[e−
T
Proof Write F i(t,T) =F(t,T,e i)for i=1,2, ,N and
Φ(t,T) = (F1(t,T),F2(t,T), ,F N(t,T))∗∈R N.
Obviously, F(t,T,X t) = N
i= 1F i(t,T)⟨e i,X t⟩ = ⟨Φ(t,T),X t⟩ Applying Itô’s differentiation rule to Z t =e− t
0f u du F(t,T,X t)
yields
dZ t = −ft Z t dt+e−
t
0f u du
∂F
∂t + ⟨Φ(t,T),Q∗X t⟩
dt+e−
t
0f u du⟨Φ(t,T),dM t⟩
Note that Z tis a boundedℑX
t martingale since f t >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
∂t −f(t)F(t,T,x) + ⟨QΦ(t,T),x⟩ =0, F(T,T,x) =x,x∈E.
Since x takes e1, ,e N, we have
∂Φ
∂t ,e i
− ⟨diag(ft)Φ(t,T),e i⟩ + ⟨QΦ(t,T),e i⟩ =0, i=1, ,N.
That is to say,Φis the fundamental matrix solution of the following equation:
∂Φ
∂t + (Q −diag(ft))Φ(t,T) =0,
withΦ(T,T) =I So the proof of the first part is completed Note that⟨XT,1⟩ = ⟨ N
i= 11{X T=e i}e i,1⟩ = N
i= 11{X T=e i}⟨ei,1⟩ =
1 Consequently,
E[e−
T
t f u du|ℑX t] =E[e−
T
t f u du⟨XT,1⟩|ℑX t] = ⟨E[e−
T
t f u du X T|ℑX t] ,1⟩
= ⟨Φ∗(t,T)X t,1⟩ =1∗Φ∗(t,T)X t= ⟨Φ(t,T)1,X t⟩
Remark 3.1 If all of the f i(t)are constants, thenΦ(t,T) =e(Q−diag(f))(T−t) Consequently, V(t,T) = ⟨e(Q−diag(f))(T−t),X t⟩, which is consistent with Lemma A.1 inBuffington 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,
P(τ1>s, τ2>s|ℑ X T) =e−
s
For 0≤t<s≤T,
P(τ1>s,t < τ2≤s|ℑ X T) =
s t
a3ve−
v
0a3du− s
0a1du− s
and
P(τ2>s,t < τ1≤s|ℑ X T) = s a1ve−
v
0a1du− s
0a3du− s
Trang 6Proof To prove(4.1), it suffices to prove that for any event A∈ ℑX T, it holds that
E[1{A}E[1{τ 1 >s,τ 2 >s}|ℑX T]] =E[1{A}e−
s
0 (a1+a3)du]
To this end, using the ‘‘tower property’’ of conditional expectations yields
E[1{A}E[1{τ 1 >s,τ 2 >s}|ℑX T]] =E[1{A}1{ τ 1 >s,τ 2 >s}]
Recall that for an arbitrary but fixed time 0<s≤T , the survival probability is defined by
dP1
dP|ℑs=1{τ 1 >s}exp
s
0
λ1
u du
= η1
s.
Furthermore, for any P1-integrable random variable Y , the equality E[Y η1
s] =E1[Y ]holds Therefore, changing measure P
to P1yields
E[1{A}1{ τ 1 >s,τ 2 >s}] = E[1{A}1{ τ 2 >s}e−0sλ 1duη1
s] =E1[1{τ 2 >s}e−0s a1du1{A}]
= E1[E1[1{τ 2 >s}|ℑX T]e−
s
0a1du1{A}] =E1[e−
s
0 (a1+a3)du1{A}] ,
where the third equality holds since e−
s
0a1du1{A}∈ ℑX T, and the last equality holds becauseτ2has the intensity a3under P1
Then the fact that the distribution of X t under P1is the same as that under P concludes the proof.
The proof of(4.2)is similar For any event A∈ ℑX , by changing measure P to P1we have
E[1{A}E[1{τ 1 >s,t<τ 2 ≤s}|ℑX T]] = E[1{τ 1 >s,t<τ 2 ≤s}1{A}]
= E[1{t<τ 2 ≤s}e−
s
0 λ 1du1{A}η1
s] =E1[1{t<τ 2 ≤s}e−
s
0a1du− s
τ 2a2du
1{A}]
= E1[E1[1{t<τ 2 ≤s}e−
s
0a1du− s
τ 2a2du
|ℑX T]1{A}]
= E1
s t
a3ve−
v
0a3du− s
0a1du− s
v a2du dv1{A}
,
where the last equality holds becauseτ2has the intensity a3under measure P1 Then by usingProposition 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 LetΨ1(t,s),Ψ2(t,s)andΨ3(t,s)be determined by(3.7)with f t replaced by a1(t) +a3(t), a1(t) +a2(t)and
a3(t) +a4(t), respectively Then for s>0, we have
For s>t ≥0, we have
P(τ1>s,t< τ2≤s) =
s 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 expec-tations and(4.2)yields
E[1{τ 1 >s,t<τ 2 ≤s}] = E[E[1{τ 1 >s,t<τ 2 ≤s}|ℑX T]]
= E
s t
a3ve−
v
0a3du− s
0a1du− s
v a2du dv
= E
s t
a3ve−
v
0 (a3+a1)du E[e−
s
v (a1+a2)du|ℑXv]d v
= E
s t
e−
v
0 (a3+a1)du⟨diag(a3(v))Ψ2(v,s)1,Xv⟩d v
where the second equality is obtained from(4.2), and the last second equality follows fromLemma 3.1 Then again using Lemma 3.1yields(4.5)
The following results are direct consequences ofProposition 4.2
Trang 7Corollary 4.1 For s>0, the marginal distributions of τ1andτ2are given by
P(τ1>s) = ⟨Ψ1(0,s)1,X0⟩ +
s
0
and
P(τ2>s) = ⟨Ψ1(0,s)1,X0⟩ +
s
0
respectively, whereΨi(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
calcu-late 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 ofτ1andτ2conditional 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 spreadκof a CDS without counterparty risk and the unilateral CVA of this CDS The spreadκof the CDS without counterparty risk on the reference entity can be obtained by setting the value of P0to
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 f t replaced by r+a1(t) +a3(t)and r+a1(t) +a2(t), respectively Then the spreadκis given by
κ =
T
0⟨A1(0,s)diag(L1)a1(s) +g1(s),X0⟩ds
T
where
g1(s) =
s
0
and
g2(s) =
s
0
with L1=1−R1∈R N.
Proof The expected present value of the swap premium payment over[t ,T]is
κE
T
0
e−
s
0rvdv1{
τ 1 >s}ds
= κE
T
0
e−
s
0rvdv
E[1{τ 1 >s,τ 2 >s}|ℑX] +E[1{τ 1 >s,τ 2 ≤s}|ℑX] ds
= κ T 0
E[e−
s
0 (rv+a1
v +a3
v )dv] +E s
0
a3ve−
v
0a3
vdu− s
0 (r u+a1 )du− s
v a2du
ds
= κ T 0
⟨A1(0,s)1,X0⟩ +
s
0
E[a3ve−
v
0 (a1 +a3 +r u)du E[e−vs(r u+a1 +a2 )du|ℑXv]]d v
ds
= κ T 0
⟨A1(0,s)1,X0⟩ +
s
0
⟨diag(a3(v))A2(v,s)1,E[e−
v
0 (a1 +a3 +r u)du Xv]⟩d v
ds,
= κ T 0
⟨A1(0,s)1+g2(s),X0⟩ds ,
where the second equality follows fromProposition 4.1, the last second equality holds because⟨a3
vA2(v,s)1,Xv⟩ = ⟨diag
(a3(v))A2(v,s)1,Xv⟩, and the last equality follows fromLemma 3.1
Similarly, the expected present value of the loss payment over[t,T]is
E[(1−R1(τ1))e−
τ 1rvdv1{
τ 1 ≤T}] =E
T
0 (1−R1(s))e−
s
0rvdv1 { τ 1 >s−}dH s1
=
T
E[(1−R1(s))e−
s
0rvdv(1{τ 1 >s,τ 2 >s}a1s+1{τ 1 >s,τ 2 ≤s}(a1s+a2s))ds]ds
Trang 8 T
0
E
e−
s
0 (rv+a1v +a3v )dv⟨diag(L1)a1(s),X s⟩
+
T
0
E
⟨diag(L1)(a1(s) +a2(s)),X s⟩
s
0
a3ve−
v
0a3du− s
0 (r u+a1)du− s
v a2du dv
ds
=
T
0
⟨A1(0,s)diag(L1)a1(s),X0⟩ds +
T
0
s
0
⟨E[a3ve−
v
0 (r u+a1+a3)du
×E[e−
s
v (r u+a1+a2)du X s|ℑXv]]d v,diag(L1)(a1(s) +a2(s))⟩ds
=
T
0
⟨A1(0,s)diag(L1)a1(s) +g1(s),X0⟩ds ,
where the second equality holds because H1
s− t∧τ 1
0 λ1
s ds is anℑ-martingale, the third equality is obtained by using Propo-sition 4.1and the equalities(1−R1(s))a1
s = ⟨diag(L1)a1(s),X s⟩and(1−R1(s))(a1
s+a2
s) = ⟨diag(L1)(a1(s) +a2(s)),X s⟩, and the last two equalities follow fromLemma 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 FromDefinition 2.1, we have 1{τ 1 ≤t}P t =0, then
P t=1{τ 1 >t}P t=1{τ 1 >t,τ 2 >t}P t+1{τ 1 >t,τ 2 ≤t}P t.
In particular,
Pτ 2 =1{τ 1 >τ 2 ,τ 2 ≤ τ 2 }Pτ 2.
Consequently, to derive the unilateral CVA, we need to calculate P t on the event{ τ1 > t, τ2 ≤ t} We first give a useful lemma
Lemma 5.1 For any 0<t ≤s≤T and anyℑX -measurable random variable Z , we have
1{τ 1 >t,τ 2 ≤t}E t[Z 1{ τ 1 >s}] =1{τ 1 >t,τ 2 ≤t}E[Ze−
s(a1v +a2v )dv|ℑX
Proof Changing measure from P to P1yields
1{τ 1 >t,τ 2 ≤t}E t[Z 1{ τ 1 >s}] =1{τ 1 >t,τ 2 ≤t}E P1[Ze−
s(a1
v +a2 v1{τ 2≤v} )dv|ℑ
t]
=1{τ 1 >t,τ 2 ≤t}E P1[Ze−
s(a1v +a2v1{τ 2 ≤v} )dv|ℑX
t, τ1>t, τ2≤t]
=1{τ 1 >t,τ 2 ≤t}
E P1[1{τ 1 >t,τ 2 ≤t}Ze− s(a1v +a2v )dv|ℑX
t]
E P1 [1{τ 1 >t,τ 2 ≤t}|ℑX t]
=1{τ 1 >t,τ 2 ≤t}
E P1[EP1[1{τ 2 ≤t}|ℑX]Ze− s(a1v +a2v )dv|ℑX
t]
E P1 [1{τ 2 ≤t}|ℑX t]
=1{τ 1 >t,τ 2 ≤t}E P1[Ze−
s(a1v +a2v )dv|ℑX
t] ,
where the fourth equality holds since 1{τ 1 >t} = 1 under P1and Ze−s(a1
v +a2
v )dv ∈ ℑX
T, and the last equality holds because
E P1[1{τ 2 ≤t}|ℑX] =E P1[1{τ 2 ≤t}|ℑX
t] Then the result follows from the fact the distribution of X under P1is 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 κon the firm admits the representation
1{τ 1 >t,τ 2 ≤t}P t =1{τ 1 >t,τ 2 ≤t}⟨ µt,X t⟩ ˆ =1{τ 1 >t,τ 2 ≤t}P¯t, (5.5)
where
µt =
T
t
A2(t,s)(diag(L1)(a1(s) +a2(s)) − κ1)ds, (5.6)
with A2(t,s)and L1defined in Proposition 5.1
In particular,
Trang 9Proof By usingLemmas 3.1and5.1, we have
1{τ 1 >t,τ 2 ≤t}κE t
T t
1{τ 1 >s}e−
s
rvdvds = 1{
τ 1 >t,τ 2 ≤t}κ
T t
E[e−
s(rv+a1v +a2v )dv|ℑX
t]ds
= 1{τ 1 >t,τ 2 ≤t}κ
T t
The expected present value of the loss payment over[t ,T]is
1{τ 1 >t,τ 2 ≤t}E t[ (1−R1(τ1))e−
τ 1
t rv1{t<τ 1 ≤T}] =1{τ 1 >t,τ 2 ≤t}E t
T t
(1−R1(s))e−
s
rv1{τ 1 >s−}dH s1
=1{τ 1 >t,τ 2 ≤t}
T t
E t[e−
s
rvdv1{
τ 1 >s−}(1−R1(s))(a1s+a2s)]ds
=1{τ 1 >t,τ 2 ≤t}
T t
E[e−
s(rv+a1v +a2v )dv⟨diag(L1)(a1(s) +a2(s)),X s⟩|ℑX t]ds
=1{τ 1 >t,τ 2 ≤t}
T t
⟨A2(t,s)diag(L1)(a1(s) +a2(s)),X t⟩ds , (5.9)
where the second equality is obtained by using the fact that H s1− t∧ τ 1
0 λ1
s ds is anℑ-martingale, and the third equality follows fromLemma 5.1 Then substituting(5.8)and(5.9)into(2.1)yields the result
For notational convenience, for each a= (a1,a2, ,a N)∗∈R N , denote by a+= (a+1, ,a+N)∗
, where a+i =max{ai,0}
Proposition 5.3 At valuation time t and conditional on the event{ τ2>t},
1{τ 2 >t}CVA t =1{τ 2 ∧ τ 1 >t}
T t
⟨A1(t,s)diag(a3(s))diag(L2)µ+
where L2=1−R2∈R N, A1(t,s)is defined in Proposition 5.1 andµs is defined in Proposition 5.2
In particular, we obtain
CVA0 =
T
0
⟨A1(0,s)diag(a3(s))diag(L2)µ+
Proof FromProposition 2.1, we have
1{τ 2 >t}CVA t = 1{τ 1 >t}E t
T t
e−
s
rvdv(1−R2(s))¯P s+1{τ 1 ∧ τ 2 >s}dH s2
= 1{τ 1 ∧ τ 2 >t}E t
T t
e−
s
rvdv(1−R2(s))¯P s+1{τ 1 ∧ τ 2 >s−}a3s ds
= 1{τ 1 ∧ τ 2 >t}
T t
E[e−
s(rv+a1v +a3v )dv⟨diag(a3(s))diag(L2)µ+
s,X s⟩ds] ,
where the second equality holds because H t2− τ 2 ∧t
0 λ2
s ds is anℑ-martingale and the last equality follows from Proposi-tions 4.1and5.2 Then usingLemma 3.1yields the result
6 Numerical results
In this section, we shall present some numerical calculations based on the results derived in Section5 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 e1and state e2
repre-sent 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 f t = ⟨f,X t⟩with
f= (f1,f2)∗
, we choose a constant f in the model without regime switching, such that it satisfies e−fT =E[e−
T f t dt|X0=e i]
Figs 1and2present the relationship between the spread and q12.From them we can see, the spreads are higher when we
start at the state e at time t=0 We can also see when X =e, the spread in the no regime-switching model is higher than
Trang 10Fig 1 The relation between q12 andκ,q21=0.2.
Fig 2 The relation between q12 andκ,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 q12increasing and decrease with q21increasing, which is because a higher q12leads to an increasing
probability of switching to the state e2, while a higher q21leads to an increasing probability of switching to the state e1.
Figs 3and4present 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 whenξ0 =e2.Figs 3and4also show that CVA0increases with a3increasing, in line with stylized
features and the financial intuition: the counterparty is more likely to default with a3increasing, 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, regime-switching 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