Unilateral counterparty risk valuation of CDS using a regime switching intensity model

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.

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 July 2013 Received in revised form 12 October 2013 Accepted 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 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 http://dx.doi.org/10.1016/j.spl.2013.11.001 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 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 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 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 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 Section gives some numerical results Section concludes 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, 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 T } pT (t ) + 1{t t }, CVAt = Et [pT (t ) − πT (t )] = Et [1{t : λit dt ≥ i P -martingales, it remains to show the last term vanishes t are both i i i 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  Tj { 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)∗ ∈ 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 fu du   t ∂F + ⟨Φ (t , T ), Q ∗ Xt ⟩ dt + e− fu du ⟨Φ (t , T ), dMt ⟩ ∂t Note that Zt is a bounded ℑXt martingale since ft > for any t > 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⟩ = ⟨ 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) 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 < s ≤ T , s P (τ1 > s, τ2 > s|ℑXT ) = e− (au +au )du  (4.1) For ≤ t < s ≤ T , P (τ1 > s, t < τ2 ≤ s|ℑXT ) = s  a3v e− t v s s au du− 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 s au du− 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 E [1{A} E [1{τ1 >s,τ2 >s} |ℑXT ]] = E [1{A} e− (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 < s ≤ T , the survival probability is defined by dP dP s   λ u du |ℑs = 1{τ1 >s} exp = ηs1 Furthermore, for any P -integrable random variable Y , the equality E [Y ηs1 ] = E [Y ] holds Therefore, changing measure P to P yields E [1{A} 1{τ1 >s,τ2 >s} ] = E [1{A} 1{τ2 >s} e− 1 s λ1u du X T ηs ] = E [1{τ2 >s} e− = E [E [1{τ2 >s} |ℑ ]e s − s au du {A} ] − = E [e s au du 1{A} ] s (au +au )du 1{A} ], where the third equality holds since e− au du 1{A} ∈ ℑXT , and the last equality holds because τ2 has the intensity a3u under P Then the fact that the distribution of Xt under P 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 we have E [1{A} E [1{τ1 >s,t s,t t ≥ 0, we have P (τ1 > s, t < τ2 ≤ s) = s  ⟨Ψ (0, v)diag(a3 (v))Ψ (v, s)1, X0 ⟩dv, (4.5) ⟨Ψ (0, v)diag(a1 (v))Ψ (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 s,t 0, the marginal distributions of τ1 and τ2 are given by P (τ1 > s) = ⟨Ψ (0, s)1, X0 ⟩ + s  ⟨Ψ (0, v)diag(a3 (v))Ψ (v, s)1, X0 ⟩dv (4.7) ⟨Ψ (0, v)diag(a1 (v))Ψ (v, s)1, X0 ⟩dv, (4.8) and P (τ2 > s) = ⟨Ψ (0, s)1, X0 ⟩ + s  respectively, where Ψ i (v, s), i = 1, 2, 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 not discuss them in detail 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= ⟨A1 (0, s)diag(L1 )a1 (s) + g1 (s), X0 ⟩ds , T ⟨A1 (0, s)1 + g2 (s), X0 ⟩ds (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) and g2 (s) = s  with L1 = − R1 ∈ RN Proof The expected present value of the swap premium payment over [t , T ] is κE T  e− s rv dv  1{τ1 >s} ds = κE T  e− s rv dv  E [1{τ1 >s,τ2 >s} |ℑXT ] + E [1{τ1 >s,τ2 ≤s} |ℑXT ] ds   0 =κ T    =κ T   ⟨A1 (0, s)1, X0 ⟩ +  ⟨A1 (0, s)1, X0 ⟩ +  a3v e− v s s av du− (ru +au )du− v au du v (au +au +ru )du  ds s E [e− v (ru +au +au )du |ℑXv ]]dv  s  =κ E [a3v e− T  s  =κ s  s E [e− (rv +av +av )dv ] + E ⟨diag(a3 (v))A2 (v, s)1, E [e− v (au +au +ru )du  Xv ]⟩dv ds  ds, T ⟨A1 (0, s)1 + g2 (s), X0 ⟩ds, 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 rv dv T  (1 − R1 (s))e− 1{τ1 ≤T } ] = E s rv dv T  E [(1 − R1 (s))e− = s 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 E e− (rv +av +av )dv ⟨diag(L1 )a1 (s), Xs ⟩ =  T   E ⟨diag(L1 )(a1 (s) + a2 (s)), Xs ⟩ + T  a3v e− 0 ⟨A1 (0, s)diag(L1 )a1 (s), X0 ⟩ds + = s  T  0 s  v s s au du− (ru +au )du− v au du s ⟨E [a3v e− dv  ds v (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, =  t ∧τ where the second equality holds because Hs1 − 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 < t ≤ s ≤ T and any ℑXT -measurable random variable Z , we have s 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 yields − s = 1{τ1 >t ,τ2 ≤t } E P [Ze− s 1{τ1 >t ,τ2 ≤t } Et [Z 1{τ1 >s} ] = 1{τ1 >t ,τ2 ≤t } E P [Ze 1 t (av +av 1{τ2 ≤v} )dv t (av +av 1{τ2 ≤v} )dv |ℑt ] |ℑXt , τ1 > t , τ2 ≤ t ] s E P [1{τ1 >t ,τ2 ≤t } Ze− t (av +av )dv |ℑXt ]  = 1{τ1 >t ,τ2 ≤t } E P [1{τ1 >t ,τ2 ≤t } |ℑXt ] s 1 E P [E P [1{τ2 ≤t } |ℑXT ]Ze− t (av +av )dv |ℑXt ]  = 1{τ1 >t ,τ2 ≤t } E P [1{τ2 ≤t } |ℑXt ] = 1{τ1 >t ,τ2 ≤t } E P [Ze− s t (av +av )dv |ℑXt ], s where the fourth equality holds since 1{τ1 >t } = under P and Ze− t (av +av )dv ∈ ℑXT , and the last equality holds because  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 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 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 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 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 − 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+ , , aN ) , where = 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 = − 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) 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  + ¯ (1 − R2 (s))Ps 1{τ1 ∧τ2 >s− } as ds s 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 − 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 Numerical results In this section, we shall present some numerical calculations based on the results derived in Section 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− ft dt |X0 = ei ] Figs and 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 = 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 The relation between q12 and κ, q21 = 0.2 Fig 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 and present the impacts of some model parameters on the CVA0 From them we can see that the CVA at time t = 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 = is higher when ξ0 = e2 Figs and 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 not incorporate the changes of market regimes into the pricing models, we shall underestimate or overestimate the spreads and the CVA 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 The relation between q12 and CVA0 , a3 (t ) = (0.005, 0.015)∗ Fig 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) References Buffington, J., Elliott, R.J., 2002 American options with regime switching Int J Theor Appl Finance 5, 497–514 Collin-Dufresne, P., Goldstein, R., Hugonnier, J., 2004 A general formula for valuing defaultable securities Econometrica 72, 1377–1407 Elliott, R.J., 1993 New finite dimensional filters and smoothers for noisily observed Markov chains IEEE Trans Inform Theory 39, 265–271 Elliott, R.J., Aggoun, L., Moore, J.B., 1994 Hidden Markov Models: Estimation and Control Springer-Verlag, Berlin, Heidelberg, New York Elliott, R.J, Siu, T.K., 2009 Robust optimal portfolio choice under Markovian regime-switching model Methodol Comput Appl Probab 11 (2), 145–157 Giesecke, K., Longstaff, F.A., Schaefer, S., Strebulaev, I., 2011 Corporate bond default risk: a 150-year perspective J Finance Econom 102, 233–250 Jarrow, R., Yu, F., 2001 Counterparty risk and the pricing of defaultable securities J Finance 56, 1765–1799 Shaked, M., Shanthikumar, J.G., 1987 The multivariate hazard construction Stochastic Process Appl 24, 241–258 Siu, T.K., 2010 A Markov regime switching marked point process for short rate analysis with credit risk Internat J Stoch Anal 18 Article ID 870516 Yu, F., 2007 Correlated defaults in intensity-based models Math Finance 17, 155–173 [...]... 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... 2001 Counterparty risk and the pricing of defaultable securities J Finance 56, 1765–1799 Shaked, M., Shanthikumar, J.G., 1987 The multivariate hazard construction Stochastic Process Appl 24, 241–258 Siu, T.K., 2010 A Markov regime switching marked point process for short rate analysis with credit risk Internat J Stoch Anal 18 Article ID 870516 Yu, F., 2007 Correlated defaults in intensity- based models... 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... 1994 Hidden Markov Models: Estimation and Control Springer-Verlag, Berlin, Heidelberg, New York Elliott, R.J, Siu, T.K., 2009 Robust optimal portfolio choice under Markovian regime- switching model Methodol Comput Appl Probab 11 (2), 145–157 Giesecke, K., Longstaff, F .A. , Schaefer, S., Strebulaev, I., 2011 Corporate bond default risk: a 150-year perspective J Finance Econom 102, 233–250 Jarrow, R., Yu,... Elliott, R.J., 2002 American options with regime switching Int J Theor Appl Finance 5, 497–514 Collin-Dufresne, P., Goldstein, R., Hugonnier, J., 2004 A general formula for valuing defaultable securities Econometrica 72, 1377–1407 Elliott, R.J., 1993 New finite dimensional filters and smoothers for noisily observed Markov chains IEEE Trans Inform Theory 39, 265–271 Elliott, R.J., Aggoun, L., Moore,... T.K., 2010 A Markov regime switching marked point process for short rate analysis with credit risk Internat J Stoch Anal 18 Article ID 870516 Yu, F., 2007 Correlated defaults in intensity- based models Math Finance 17, 155–173