# Basket Credit Derivative Pricing in a Markov Chain Model with Interacting Intensities

Authors: Kangquan Zhi; Jie Guo; Xiaosong Qian
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, we propose a Markov chain model to price basket credit default swap (BCDS) and basket credit-linked note (BCLN) with counterparty and contagion risks. Suppose that the default intensity processes of reference entities and the counterparty are driven by a common external shock as well as defaults of other names in the contracts. The stochastic intensity of the external shock is a Cox process with jumps. We derive recursive formulas for the joint distribution of default times and obtain closed-form premium rates for BCDS and BCLN. Numerical experiments are performed to show how the correlated default risks may affect the premium rates.


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## Body

## 1. Introduction

The market for credit derivatives has experienced rapid development during the past decades until the international financial crisis in 2008, which was mainly due to the underestimation of the correlated default risk. Since then, more and more research studies focus on the basket credit derivative pricing. Basket credit default swap (BCDS) and basket credit-linked notes (BCLNs) are two popular multiname credit derivatives, which can reduce the adverse impact of reference assets’ defaults on financial institutions. A BCDS is designed to transfer the credit exposure of fixed income products between two parties withN reference entities. The issuer of the contract is the protection seller, and the investor is the protection buyer. A BCDS will have a premium rate and maturity date, and the maturity depends on the performance of reference entities and the counterparty. A BCLN is a note paying an enhanced coupon to investors for bearing the credit risk of N reference entities. The issuer of the note is the protection buyer, and the investor is the protection seller. A BCLN will have a coupon rate, maturity date, and par value just like a standard bond. However, the maturity depends on the performance of the reference entities and the counterparty.The reduced-form models are widely used to price credit derivatives. In a reduced-form model, there are mainly three approaches to model the default correlation: copula, conditional independence, and contagion. In the copula approach, the joint distribution of the default times is constructed by combining marginal distributions of the individual by a copula function, see Li [1], Schőnbucher and Schubert [2], Brigo and Capponi [3], and Jean-David [4]. The conditional independence approach assumes that the default intensities are conditionally independent under the given filtration, see Wang and Garleanu [5], Giesecke [6], and Liang et al. [7]. In the contagion approach, the default intensity of one entity is affected not only by systematic factors but also by the default of other entities in the contract, see Jarrow and Yu [8], Zheng and Jiang [9], and Dong and Wang [10].In this paper, we focus on the pricing of basket credit derivatives (BCDS and BCLN). In the existing literature, different approaches in pricing BCDS and BCLN have been developed. Hull and White [11] developed two fast procedures for valuing kth-to-default swaps. Walker [12] studied counterparty risk in BCDS valuation by using a four-state Markov process that includes contagion effects. Yu [13] gave the Monte Carlo method for pricing basket CDS. Frey and Backhaus [14] considered reduced-form models for portfolio credit risk with interacting default intensities. Zheng and Jiang [9] proposed a factor contagion model for the basket CDS pricing. Wu [15] explored a reasonable coupon rate for basket credit-linked notes (BCLN) with the issuer default risk. Herbertsson et al. [16] valued kth-to-default swap spreads in a tractable shot noise model. Wang et al. [17] proposed a model for pricing a basket CDS with the negative correlation between prepayment and default under the bottom-up framework. Li and Li [18] used a type of dynamic copula method to characterize the dependence structure between financial assets and price basket default swaps. Esfahanipour and Jahanbin [19] proposed a heuristic algorithm for pricing of basket default swaps. Dong et al. [20] studied the kth-to-default basket swap under a correlated regime-switching hazard process model. Guo et al. [21] employed an intensity-based credit risk model with regime switching to consider the valuation of basket CDS in a homogeneous portfolio. The previous work did not combine internal contagion and external shock, which is a Cox process. In addition, they seldom obtained the closed-form solutions. In this paper, we propose a model that combines internal contagion and random external shock. Our model is not only applicable to BCDS pricing but also to BCLN. Furthermore, we derive the closed-form solution which is not easy for the complex structure of the default as the numerical analysis can be done very smoothly.Leung and Kwok [22] presented a Markov chain model to price single-name CDS; the default intensities of the reference entity and counterparty were affected by an external shock. Inspired by Leung and Kwok [22], we present a more general model to study basket credit derivatives with interacting default intensities, which are driven by an external shock as well as defaults of other names in the contracts. We get recursive formulas for the unconditional distribution of default times through ingenious construction of the infinitesimal generator matrix and obtain closed-form premium rates for basket credit default swap and basket credit-linked notes.The paper is organized as follows. In Section2, we give the construction of interacting default intensities and derive the joint distribution of default times by solving a system of ordinary differential equations. In Section 3, we calculate the premium rates of kth-to-default CDS and kth-to-default CLN with the counterparty risk. Numerical results are presented to show how the correlated default risks affect the premium rates in Section 4. At last, we conclude the paper.


## 2. Markov Chain Model with Interacting Intensities

In this section, we construct a reduced-form model with stochastic default intensities by a Markov chain. Consider a complete filtered probability spaceΩ,G,Gt0≤t≤T,P, where P is a martingale measure and Gtt≥0 is filtration satisfying the usual conditions. Our model includes a basket credit derivative and an external shock. The basket credit derivative includes N reference entities, a counterparty, and an investor. Let Ri represent the ith reference entity with default time τRi for i∈I=1,2,…,N, C be the counterparty with default time τC, S be the external shock with arrival time τS, and ℒ=R1,R2,…,RN,C,S be the set which contains all of the names in the model. Assume that τS is independent of τC and τRii∈I. The default process of our model is given by(1)Ht=HtR1,HtR2,…,HtRN,HtC,HtS,where(2)HtRi=1τRi≤t,i∈I,HtC=1τC≤t,HtS=1τS≤t,and 1τ≤t is the indicator function. Ht is a finite-state Markov chain with state space O=0,1N+2.The macroeconomic variables are described by a stochastic processΨ=Ψt0≤t≤T. An investor can get the historical information of the macroeconomic variables and the default status of all names in our model at time t. The filtration Gtt≥0 is given by(3)Gt=ℱtΨ∨ℋtR1∨ℋtR2∨⋯∨ℋtRN∨ℋtC∨ℋtS,where(4)ℱtΨ=σΨs:0≤s≤t,ℋtC=σHsC:0≤s≤t,ℋtS=σHsS:0≤s≤t,ℋtRi=σHsRi:0≤s≤t,i∈I,and Gt is the σ-field generated by ℱtΨ∪ℋtR1∪ℋtR2∪⋯∪ℋtRN∪ℋtC∪ℋtS. The martingale default intensities of the reference entities and counterparty are, respectively, defined by λRiΨt,Hti∈I and λCΨt,Ht, which satisfy the property that(5)HtC−∫0t∧τCλCΨs,Hsds,HtRi−∫0t∧τRiλRiΨs,Hsds,i∈I,are Gt-martingales.The arrival of the external shock is modeled by a Cox process with stochastic intensityλtS. Before the shock S happens, the default intensities of λtRii∈I and λtC are, respectively, assumed to be atRi and atC, where atRi and atC are some deterministic functions of t. When the shock S happens, λtRi and λtC jump to αRiSatRi and αCSatC, respectively, where nonnegative constants αRiS and αCS denote the effects from the external shock to reference entities and the counterparty. Depending on whether αRiS and αCS are greater than 1 or not, the densities λRi and λC can jump upward or downward. Besides, two kinds of contagion risks are considered in our model. One is the contagion effects between reference entities, and the other one is the contagion effects from reference entities to the counterparty. If reference entity i defaults, the default intensities of other reference entities and the counterparty jump to αRjRiatRjj≠i and αCRiatC, respectively, where αRjRi and αCRi are nonnegative constants. We do not consider the contagion effects from the counterparty to reference entities because the credit derivative would be terminated if the counterparty defaulted first. In summary, the default intensities of reference entities and the counterparty can be expressed as follows:(6)λtRi=atRiαRiS−1IτS≤t+1∏j∈I\iαRiRj−1IτRj≤t+1,αRiS,αRiRj≥0,i,j∈Iandi≠j,λtC=atCαCS−1IτS≤t+1∏i∈IαCRi−1IτRi≤t+1,αCS,αCRi≥0,i∈I.We assume that the simultaneous defaults or shock cannot happen in the model for the sake of simplicity. In this paper, we consider basket credit derivatives withN reference entities, and the state space O of Ht is 2N+2 dimensions. Let ψS=λtSt∈0,T, and conditional on the given state ψS, the infinitesimal generator matrix of Ht is ΛψSt≜ΛH×H′tψS2N+2×2N+2, where H,H′=HR1,…,HRN,HC,HS∈Orepresent the states of the Markov chain Ht. The conditional transition probability matrix P0,tψS≜PH×H′0,tψS2N+2×2N+2 is governed by the following forward Kolmogorov equation:(7)dP0,tψSdt=P0,tψSΛψSt,t≥0,with P0,0ψS=E, and E is the unit matrix. Because the default states and the shock are absorbing states for the Markov chain Ht, the matrix ΛψSt is upper triangular, and the conditional transition probabilities P0,tψS can be solved successively in a sequential manner.To describe the generator matrixΛψSt=ΛH×H′tψS2N+2×2N+2, we introduce some auxiliary notations as follows:(i)
HRM represents the state that only reference entities in set M⊂I defaulted. Here, HRM=HR1,…,HRN,HC,HS=0,…,1,…,0︷N,0,0, where HRi=1 for all i∈M. Especially, HRϕ=0,…,0︷n,0,0 represents the state that no reference entity defaults.(ii)
HC represents the state that only the counterparty defaulted.Here,HC=HR1,…,HRN,HC,HS=0,…,0︷N,1,0.(iii)
HS represents the state that only the external shock arrived.Here,HS=HR1,…,HRN,HC,HS=0,…,0︷N,0,1.(iv)
HRM∪C represents the state that reference entities in set M⊂I and the counterparty C defaulted.Here,HRM∪C=HR1,…,HRN,HC,HS=0,…,1,…,0︷N,1,0, where HRi=1 for all i∈M.(v)
HRM∪S represents the state that reference entities in set M⊂I defaulted, and the external shock S happened.Here,HRM∪S=HR1,…,HRN,HC,HS=0,…,1,…,0︷N,0,1, where HRi=1 for all i∈M.(vi)
HRM∪C∪S represents the state that reference entities in set M⊂I and the counterparty C both defaulted, and the external shock S happened.Here,HRM∪C∪S=HR1,…,HRN,HC,HS=0,…,1,…,0︷N,1,1, where HRi=1 for all i∈M.(vii)
ΛHt≜ΛH×Ht for H∈O are elements on the diagonal of the matrix ΛψSt, which represent the intensities that the Markov chain Ht remains at states H.(viii)
PH0,tψS≜PHRϕ×H0,tψS represents the conditional probability that Markov chain Ht transfers from the initial state HRϕ to state H during 0,t. It is the first line of the conditional transition probability matrix P0,tψS and can be used to present the joint distributions of default times during 0,t.Firstly, we calculate the conditional transition probabilities that none of the reference entities defaults during0,t. The corresponding states of Ht are HRϕ and HS. The corresponding elements in the infinitesimal generator matrix ΛψSt are as follows:(8)HRϕHR1⋯HRNHCHS⋯HRi∪S⋯HC∪S⋯HRϕ⋮HS⋮ΛHRϕatR1⋯atRNatCλtS⋯0⋯0⋯⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮00⋯00ΛHS⋯αRiSatRi⋯αCSatC⋯⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮.In theHR∅-row of the generator matrix ΛψSt, the positive elements represent the transition intensities from state HR∅ to other states by one jump. Because the cases of simultaneous defaults or shock are not examined here, the possible positive elements in this row are atRii∈I,atC, or λtS. Noting that the elements on the diagonal of the generator matrix ΛψSt are the sum of all other elements in the row multiplied with −1, we can get the first element ΛHRϕ of the HR∅-row as follows:(9)ΛHRϕt=−∑i∈IatRi+atC+λtS.Because the generator matrixΛψSt is upper diagonal, ΛHRϕt is the only nonzero element in the HRϕ-column.In theHS-row of the generator matrix ΛψSt, the positive elements represent the transition intensities from state HS to other states by one jump. By (6), the positive elements in the HS-row are αRiSatRii∈I and αCSatC. Summing these positive elements and multiplying the sum with −1, we get the diagonal element in this row as follows:(10)ΛHSt=−∑i∈IαRiSatRi+αCSatC.In theHS-column of the generator matrix ΛψSt, the positive elements represent the transition intensities from other states to state HS by one jump. The only possible state which can jump to HS is HRϕ. So, the nonzero elements in the HS-column are λtS and ΛHSt.By forward Kolmogorov equation (7), we have(11)dPHRϕ0,tψSdt=PHRϕ0,tψSΛHRϕt,dPHS0,tψSdt=PHRϕ0,tψSλtS+PHS0,tψSΛHSt.It is easy to obtain the solutions of the above equations as follows:(12)PHRϕ0,tψS=e∫0tΛHRϕudu,PHS0,tψS=e∫0tΛHSudu∫0tλuSe−∫0uΛHSrdrPHRϕ0,uψSdu.Secondly, we calculate the conditional transition probabilities that only one reference entity defaults during0,t. The corresponding states of Ht are HRi and HRi∪S for i∈I. The corresponding elements in the infinitesimal generator matrix ΛψSt are as follows:(13)⋯HRi⋯HRi,j⋯HRi∪C⋯HRi∪S⋯HRi,j∪S⋯HRi∪C∪S⋯HRϕ⋮HS⋮HRi⋮HRi∪S⋮⋯atRi⋯0⋯0⋯0⋯0⋯0⋯⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋯0⋯0⋯0⋯αRiSatRi⋯0⋯0⋯⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋯ΛHRi⋯αRjRiatRj⋯αCRiatC⋯λtS⋯0⋯0⋯⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋯0⋯0⋯0⋯ΛHRi∪S⋯αRjSαRjRiatRj⋯αCSαCRiatC⋯⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮.In theHRi-row of the generator matrix ΛψSt, the positive elements represent the transition intensities from state HRi to other states by one jump. There are three possible scenarios: a single reference entity defaults, counterparty defaults, or external shock happens. The positive elements in this row are αRjRiatRjj∈I\iandαCRiatC by (6) and λtS if external shock happens. The diagonal element in this row is as follows:(14)ΛHRit=−∑j⊂I\iαRjRiatRj+αCRiatC+λtS.In theHRi-column of the generator matrix ΛψSt, the positive elements represent the transition intensities from other states to state HRi by one jump. The only possible state which can jump to HRi is HRϕ. So, the nonzero elements in the HRi-column are atRi and the diagonal element ΛHRit.In theHRi∪S-row of the generator matrix ΛψSt, the positive elements represent the transition intensities from state HRi∪S to other states by one jump. There are two possible scenarios: a single reference entity defaults or counterparty defaults. By (6), the positive elements in this row are αRjSαRjRiatRjj∈I\i and αCSαCRiatC. The diagonal element in this row is as follows:(15)ΛHRi∪St=−∑j∈I\iαRjSαRjRiatRj+αCSαCRiatC.In theHRi∪S-column of the generator matrix ΛψSt, the positive elements represent the transition intensities from other states to state HRi∪S by one jump. The possible states which can jump to HRi∪S are HRi and HS. So, the nonzero elements in the HRi∪S-column are αRiSatRi, λtS, and the diagonal element ΛHRi∪S.By forward Kolmogorov equation (7), we have(16)dPHRi0,tψSdt=PHRϕ0,tψSatRi+PHRi0,tψSΛHRit,dPHRi∪S0,tψSdt=PHRi0,tψSλtS+PHS0,tψSαRiSatRi+PHRi∪S0,tψSΛHRi∪St,where PHRϕ0,tψS and PHS0,tψS are already obtained in (12). It is easy to solve equation (16) as follows:(17)PHRi0,tψS=e∫0tΛHRiudu∫0tauRie−∫0uΛHRirdrPHRϕ0,uψSdu,PHRi∪S0,tψS=e∫0tΛHRi∪Sudu∫0tλuSe−∫0uΛHRi∪SrdrPHRi0,uψSdu+∫0tαRiSauRie−∫0uΛHRi∪SrdrPHS0,uψSdu.We can derive all the conditional transition probabilities by the method of mathematical induction. Suppose that we have obtained the conditional transition probabilities thatk−1 reference entities default during 0,t. For the cases that k reference entities default, the corresponding states of Ht are HRM and HRM∪S for M⊂I and M=k, where M is the cardinal number of set M. The corresponding elements in the infinitesimal generator matrix ΛψSt are as follows:(18)⋯HRM⋯HRM∪j⋯HRM∪C⋯HRM∪S⋯HRM∪j∪S⋯HRM∪C∪S⋯⋮HRM\i⋮HRM\i∪S⋮HRM⋮HRM∪S⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋯∏j∈M\iαRiRiatRi⋯0⋯0⋯0⋯0⋯0⋯⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋯0⋯0⋯0⋯∏j∈M\iαRiRjαRiSatRi⋯0⋯0⋯⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋯ΛℍRM⋯∏i∈MαRiRiatRi⋯∏i∈MαCRiatC⋯λtS⋯0⋯0⋯⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋯0⋯0⋯0⋯ΛℍRM∪S⋯∏i∈MαRiRiαRjSatR⋯∏i∈MαCRiαCSatR⋯⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮.In theHRM-row of the generator matrix ΛψSt, the positive elements represent the transition intensities from state HRM to other states by one jump. There are three possible scenarios: a single reference entity defaults, counterparty defaults, or external shock happens. The positive elements in this row are ∏i∈MαRjRiatRjj∈I\M and ∏i∈MαCRiatC by (7) and λtS if external shock happens. The diagonal element in this row is as follows:(19)ΛHRMt=−∑j∈I\M∏i∈MαRjRiatRj+∏i∈MαCRiatC+λtS.In theHRM-column of the generator matrix ΛψSt, the positive elements represent the transition intensities from other states to state HRM by one jump. The possible states which can jump to HRM are HRM\i for i∈M. So, the nonzero elements in this column are ∏j∈M\iαRiRjatRi by (6) and the diagonal element ΛHRM.In theHRM∪S-row of the generator matrix ΛψSt, the positive elements represent the transition intensities from state HRM∪S to other states by one jump. There are two possible scenarios: a single reference entity defaults or counterparty defaults. By (6), the positive elements in this row are ∏i∈MαRjRiαRjSatRjj∈I\M and ∏i∈MαCRiαCSatC. The diagonal element in this row is obtained as follows:(20)ΛHRM∪St=−∑j∈I\M∏i∈MαRjRiαRjSatRj+∏i∈MαCRiαCSatC.In theHRM∪S-column of the generator matrix ΛψSt, the positive elements represent the transition intensities from other states to state HRM∪S by one jump. The possible states which can jump to HRM∪S are HRM and HRM\i∪Si∈M. So, the nonzero elements in this column are λtS, ∏j∈M\iαRiRjαRiSatRi, and the diagonal element ΛHRM∪S.By forward Kolmogorov equation (7), we have(21)dPHRM0,tψSdt=∑i∈MPHRM∖i0,tψS∏j∈M∖iαRiRjatRi+PHRM0,tψSΛHRMt,dPHRM∪S0,tψSdt=∑i∈MPHRM∖i∪S0,tψS∏j∈M∖iαRiRjαRiSatRi+PHRM0,tψSλtS+PHRM∪S0,tψSΛHRM∪St,where PHRM\i0,tψS and PHRM\i∪S0,tψS are known by induction. It is easy to get the solutions of equation (21) as follows:(22)PHRM0,tψS=e∫0tΛHRMudu∑i∈M∫0t∏j∈M∖iαRiRjauRie−∫0uΛHRMrdr×PHRM∖i0,uψSdu,PHRM∪S0,tψS=e∫0tΛHRM∪Sudu∫0tλuSe−∫0uΛHRM∪SrdrPHRM0,uψSdu+∑i∈MαRiS∏j∈M∖iαRiRj∫0tauRie−∫0uΛHRM∪Srdr×PHRM∖i∪S0,uψSdu.In order to calculate the unconditional transition probabilities, we need the following proposition.Proposition 1.
(i)PHRM0,tψS contains only one random term e−∫0tλuSdu which is outside the integral symbol; (ii) each term in PHRM∪S0,tψS contains only one random term λuSe−∫0uλrSdr which is inside the integral symbol.Proof.
We use the induction method to prove this proposition.(i)
Firstly,M=0:(23)PHRϕ0,tψS=e∫0tΛℍRϕudu=e−∫0tλuSdue−∫0t∑i∈IauRi+auCdu.PHRϕ0,tψS just contains one random term owning the path of λtS.Secondly,M=1:(24)PHRi0,tψS=e∫0tΛHRiudu∫0tauRie−∫0uΛHRirdrPHRϕ0,uψSdu=e−∫0t∑j⊂I∖iαRjRiauRj+αCRiauC+λuSdu×∫0tauRie∫0u∑j⊂I∖iαRjRiarRj+αCRiarC+λrSdr×e−∫0u∑i∈IarRi+arC+λrSdrdu=e−∫0tλuSdue−∫0t∑j⊂I∖iαRjRiauRj+αCRiauCdu×∫0tauRie∫0u∑j⊂I∖iαRjRiarRj+αCRiarCdre−∫0u∑i∈IarRi+arCdrdu.PHRM0,tψS contains one random term e−∫0tλuSdu outside the integral.Assume that (i) holds forM=k−1. When M=k,(25)PHRM0,tψS=e∫0tΛHRMudu∑i∈M∫0t∏j∈M∖iαRiRjauRie−∫0uΛHRMrdr×PHRM∖i0,uψSdu.In the integral of (25), the only random term e−∫0uλrSdr contained in PHRM\i0,uψS offsets the term e∫0uλrSdr in e−∫0uΛHRMrdr. Then, PHRM0,tψS contains only one random term e−∫0tλuSdu from e∫0tΛHRMudu outside the integral in (25), and (i) holds for M=k.(ii)
Firstly,M=0:(26)PHS0,tψS=e∫0tΛHSudu∫0tλuSe−∫0uΛHSrdrPHRϕ0,uψSdu=e∫0tΛHSudu∫0tλuSe−∫0uΛHSrdre∫0uΛHRϕrdrdu=e∫0tΛHSudu∫0tλuSe−∫0uλrSdre−∫0uΛHSrdre−∫0u∑i∈IarRi+arCdrdu.PHS0,tψS contains one random term λuSe−∫0uλrSdr inside the integral.Secondly,M=1:(27)PHRi∪S0,tψS=e∫0tΛHRi∪Sudu∫0tλuSe−∫0uΛHRi∪SrdrPHRi0,uψSdu+∫0tαRiSauRie−∫0uΛHRi∪SrdrPHS0,uψSdu=e∫0tΛHRi∪Sudu∫0tλuSe−∫0uλrSdre∫0u∑j⊂I∖iαRjRiarRj+αCRiarCdre−∫0uΛHRi∪Srdr×∫0uarRie−∫0r∑j⊂I∖iαRjRiavRj+αCRiavCdve∫0r∑i∈IavRi+avCdvdrdu+∫0tαRiSauRie−∫0uΛHRi∪Srdre∫0uΛHSrdr×∫0uλrSe−∫0rλvSdve−∫0rΛHSvdve−∫0r∑i∈IavRi+avCdvdrdu.The two terms in the right side of (27) contain only one random term λuSe−∫0uλrSdr (or λrSe−∫0rλvSdv) inside the integral.
Assume that (ii) holds forM=k−1. When M=k,(28)PHRM∪S0,tψS=e∫0tΛHRM∪Sudu∫0tλuSe−∫0uΛHRM∪SrdrPHRM0,uψSdu+∑i∈MαRiS∏j∈M∖iαRiRj∫0tauRie−∫0uΛHRM∪Srdr×PHRM∖i∪S0,uψSdu.
Byi, PHRM0,uψS only contains one random term e−∫0uλrSdr which is outside the integral symbol. Then, the first term in PHRM∪S0,tψS contains only one random term λuSe−∫0uλrSdr. By induction, each term in PHRM\i∪S0,tψS contains only one random term λuSe−∫0uλrSdr which is in the integral symbol. Then, each term in the sum part of equation (28) contains only one random term λuSe−∫0uλrSdr in the integral. In summary, (ii) also holds when M=k.
According to Proposition1 and Fubini’s theorem, we can take the expectation EψS⋅ of equation (22) and obtain the unconditional transition probabilities as follows:(29)PHRM0,t=e∫0t−∑j∈I∖M∏i∈MαRjRiauRj+∏i∈MαCRiauCduEψSe−∫0tλuSdu×∑i∈M∫0t∏j∈M∖iαRiRjauRie∫0t∑j∈I\M∏i∈MαRjRiarRj+∏i∈MαCRiarCdr×PHRM\i0,uEψSe−∫0uλrSdrdu,PHRM∪S0,t=e∫0tΛHRM∪Sudu∫0tEψSλuSe−∫0uλrSdre∫0uΛHRM∪Srdr×PℍRM0,uEψSe−∫0tλuSdu+∑i∈MαRiS∏j∈M∖iαRiRj∫0tauRie−∫0uΛHRM∪Srdr×PHRM\i∪S0,udu,where EψS⋅ is the expectation taken over the path of λtSt∈0,T.
To computeEψSe−∫0tλuSdu and EψSλtSe−∫0tλuSdu, we adopt the affine diffusion process with the jump for λtS as that proposed by Wang and Garleanu [5], which is a special Lévy process. We use Cox process to describe the external shock to obtain the closed-form solutions in this paper. The stochastic differential equation of λtS is given by(30)dλtS=kθ−λtSdt+σλtSdZt+ΔJt,where Zt is a standard Brownian motion, Jt is a pure jump process, and ΔJt denotes the jump of Jt at time t. Here, Jt is taken to be independent of Zt with jump sizes that are independent and exponentially distributed with mean μ and whose jump times are those of all independent Poisson processes with constant jump arrival rate l. It was shown by Wang and Garleanu [5] that(31)EψSe−∫0tλuSdu=eαt+βtλ0S,EψSλtSe−∫0tλuSdu=−α¯t+β¯tλ0Seαt+βtλ0S,where(32)αt=kθc1+d1b1c1d1lnc1+d1eb1tc1+d1+kθc1t+la2c2−d2b2c2d2lnc2+d2eb2tc2+d2+lc2−lt,βt=1−eb1tc1+d1eb1t,α¯t=kθc1+d1c1eb1tc1+d1eb1t+kθc1+la2c2−d2c2eb2tc2+d2eb2t+lc2−lβ¯t=−k2+2σ2eb1tc1+d1eb1t2,b1=−k2+2σ2,c1=k+k2+2σ2−2,d1=−k+k2+2σ2−2,a2=−k+k2+2σ2k+k2+2σ2,b2=b1,d2=d1+μc1,c2=1−μc1.
Replacing the conditional expectations in (29) with equation (31), we obtain the unconditional transition probabilities.


## 3. Pricing Basket Credit Derivatives

We will compute the fair premium rates of BCDS and BCLN with the counterparty risk in this section. Under the continuous model assumption, the premium rates are paid continuously at a constant rate. We assume that the notional of the BCDS or BCLN is1, T is the maturity date of the contracts, ck is the premium rate of kth-to-default credit derivatives, τk is the kth default time of reference entities, constant r is the risk-free interest rate, and constant ρ∈0,1 is the recovery rate.


### 3.1.kth-to-Default CDS

The cash flow of akth-to-default CDS is presented in Figure 1.(i)
The protection buyer pays constant premium rateck during the life of the kth-to-default CDS(ii)
If thekth default in the pool occurs before the maturity, the protection buyer gives a recovery payoff ρ, and the contract is terminatedFigure 1
The cash flow ofkth-to-default CDS.The expected cash flow ofkth-to-default CDS is as follows (see Zheng and Jiang [9]):(33)E∫0Tcke−rsIτk>sIτC>sds=E1−ρe−rτkIτk≤TIτC>T,and the premium rate is given by(34)ck=E1−ρe−rτkIτk≤TIτC>TE∫0Te−rsIτk>sIτC>sds.To calculate the premium rate for thekth-to-default CDS, there are two possible scenarios during 0,t: nonoccurrence of the external shock S or occurrence of S. If k−1 reference entities have defaulted during 0,t and another reference entity defaults during t,t+dt, the probability of such occurrence is given by(35)Qktdt=∑M⊂I,M=k−1PHRM0,t∑j∈I\M∏i∈MαRjRiatRj+PHRM∪S0,t×∑j⊂I\MαRjS∏i∈MαRjRiatRjdt,where ∑M⊂I,M=k−1PHRM0,t∑j∈I∖M∏i∈MαRjRiatRjdt corresponds to nonoccurrence of S and ∑M⊂I,M=k−1PHRM∪S0,t×∑j⊂I∖MαRjS∏i∈MαRjRiatRjdt corresponds to the case otherwise. So, the expected present value of the compensation payment paid by the protection seller within t,t+dt is(36)1−ρe−rtQktdt.Over the entire period0,T, the expected present value of the compensation payment paid by the protection seller is(37)E1−ρe−rτkIτk≤TIτC>T=∫0T1−ρe−rtQktdt.Thekth-to-default CDS contract will be terminated if more than k−1 reference entities default. The corresponding states to continue the contract are HRM and HRM∪SM⊂I,0≤M≤k−1. The expected premium payment paid by the protection buyer within t,t+dt is given by(38)cke−rtDktdt,where(39)Dkt=∑M⊂I,0≤M≤k−1PHRM0,t+PHRM∪S0,t.Over the entire period0,T, the expected present value of the premium payment paid by the protection buyer is(40)E∫0Tcke−rsIτk>sIτC>sds=ck∫0Te−rtDktdt.By (34), the kth-to-default CDS premium rate is given by(41)ck=1−ρ∫0Te−rtQktdt∫0Te−rtDktdt.


### 3.2.kth-to-Default CLN

The cash flow of akth-to-default CLN is presented in Figure 2.(i)
The protection seller pays a nominal amount at the initial date(ii)
The protection buyer pays constant premium rateck during the life of the kth-to-default CLN(iii)
If thekth default in the pool occurs before the maturity, the protection buyer gives a recovery payoff ρ, and the contract is terminated(iv)
If thekth default in the pool does not occur before the maturity, the protection buyer returns the nominal amountFigure 2
The cash flow of thekth-to-default CLN.The expected cash flow of thekth-to-default CLN is as follows:(42)1=E∫0Tcke−rsIτk>sIτC>sds+ρe−rτkIτk≤TIτC>T+e−rTIτk>TIτC>T.The premium rate is given by(43)ck=E1−ρe−rτkIτk≤TIτC>T−e−rTIτk>TIτC>TE∫0Te−rsIτk>sIτC>sds.Here, the method for calculating the default probabilities is the same as the method for thekth-to-default CDS case.Over the whole period0,T, the expected present value of the recovery payment paid by the CLN issuer is(44)Eρe−rτkIτk≤TIτC>T=∫0Tρe−rtQktdt,where(45)Qktdt=∑M⊂I,M=k−1PHRM0,t∑j∈I\M∏i∈MαRjRiatRj+PHRM∪S0,t×∑j⊂I\MαRjS∏i∈MαRjRiatRjdt.The expected present value of the premium payment paid by the CLN issuer is(46)E∫0Tcke−rsIτk>sIτC>sds=ck∫0Te−rtDktdt,where(47)Dkt=∑M⊂I,0≤M≤k−1PHRM0,t+PHRM∪S0,t.The expectation of the payment of nominal amount at maturity of the contract is(48)Ee−rTIτk>TIτC>T=e−rTDkT.By (43), the kth-to-default CLN premium rate is given by(49)ck=1−ρ∫0Te−rtQktdt−e−rTDkT∫0Te−rtDktdt.


## 3.1.kth-to-Default CDS

The cash flow of akth-to-default CDS is presented in Figure 1.(i)
The protection buyer pays constant premium rateck during the life of the kth-to-default CDS(ii)
If thekth default in the pool occurs before the maturity, the protection buyer gives a recovery payoff ρ, and the contract is terminatedFigure 1
The cash flow ofkth-to-default CDS.The expected cash flow ofkth-to-default CDS is as follows (see Zheng and Jiang [9]):(33)E∫0Tcke−rsIτk>sIτC>sds=E1−ρe−rτkIτk≤TIτC>T,and the premium rate is given by(34)ck=E1−ρe−rτkIτk≤TIτC>TE∫0Te−rsIτk>sIτC>sds.To calculate the premium rate for thekth-to-default CDS, there are two possible scenarios during 0,t: nonoccurrence of the external shock S or occurrence of S. If k−1 reference entities have defaulted during 0,t and another reference entity defaults during t,t+dt, the probability of such occurrence is given by(35)Qktdt=∑M⊂I,M=k−1PHRM0,t∑j∈I\M∏i∈MαRjRiatRj+PHRM∪S0,t×∑j⊂I\MαRjS∏i∈MαRjRiatRjdt,where ∑M⊂I,M=k−1PHRM0,t∑j∈I∖M∏i∈MαRjRiatRjdt corresponds to nonoccurrence of S and ∑M⊂I,M=k−1PHRM∪S0,t×∑j⊂I∖MαRjS∏i∈MαRjRiatRjdt corresponds to the case otherwise. So, the expected present value of the compensation payment paid by the protection seller within t,t+dt is(36)1−ρe−rtQktdt.Over the entire period0,T, the expected present value of the compensation payment paid by the protection seller is(37)E1−ρe−rτkIτk≤TIτC>T=∫0T1−ρe−rtQktdt.Thekth-to-default CDS contract will be terminated if more than k−1 reference entities default. The corresponding states to continue the contract are HRM and HRM∪SM⊂I,0≤M≤k−1. The expected premium payment paid by the protection buyer within t,t+dt is given by(38)cke−rtDktdt,where(39)Dkt=∑M⊂I,0≤M≤k−1PHRM0,t+PHRM∪S0,t.Over the entire period0,T, the expected present value of the premium payment paid by the protection buyer is(40)E∫0Tcke−rsIτk>sIτC>sds=ck∫0Te−rtDktdt.By (34), the kth-to-default CDS premium rate is given by(41)ck=1−ρ∫0Te−rtQktdt∫0Te−rtDktdt.


## 3.2.kth-to-Default CLN

The cash flow of akth-to-default CLN is presented in Figure 2.(i)
The protection seller pays a nominal amount at the initial date(ii)
The protection buyer pays constant premium rateck during the life of the kth-to-default CLN(iii)
If thekth default in the pool occurs before the maturity, the protection buyer gives a recovery payoff ρ, and the contract is terminated(iv)
If thekth default in the pool does not occur before the maturity, the protection buyer returns the nominal amountFigure 2
The cash flow of thekth-to-default CLN.The expected cash flow of thekth-to-default CLN is as follows:(42)1=E∫0Tcke−rsIτk>sIτC>sds+ρe−rτkIτk≤TIτC>T+e−rTIτk>TIτC>T.The premium rate is given by(43)ck=E1−ρe−rτkIτk≤TIτC>T−e−rTIτk>TIτC>TE∫0Te−rsIτk>sIτC>sds.Here, the method for calculating the default probabilities is the same as the method for thekth-to-default CDS case.Over the whole period0,T, the expected present value of the recovery payment paid by the CLN issuer is(44)Eρe−rτkIτk≤TIτC>T=∫0Tρe−rtQktdt,where(45)Qktdt=∑M⊂I,M=k−1PHRM0,t∑j∈I\M∏i∈MαRjRiatRj+PHRM∪S0,t×∑j⊂I\MαRjS∏i∈MαRjRiatRjdt.The expected present value of the premium payment paid by the CLN issuer is(46)E∫0Tcke−rsIτk>sIτC>sds=ck∫0Te−rtDktdt,where(47)Dkt=∑M⊂I,0≤M≤k−1PHRM0,t+PHRM∪S0,t.The expectation of the payment of nominal amount at maturity of the contract is(48)Ee−rTIτk>TIτC>T=e−rTDkT.By (43), the kth-to-default CLN premium rate is given by(49)ck=1−ρ∫0Te−rtQktdt−e−rTDkT∫0Te−rtDktdt.


## 4. Numerical Analysis

In this section, we perform numerical experiments to show how the correlated default risk may affect the premium rates. We will investigate the sensitivity of the premium rates to the parameters of our model.


### 4.1.kth-to-Default CDS

The parameters of the model are assumed to be(50)ρ=0.4,r=0.04,αRiS=1.1,αCS=1.1,αRjRi=1.1,αCRi=1.1,λ0S=0.05,σ=0.2,k=0.3,θ=0.02,l=0.3,μ=0.15,T=5.For convenience of calculation, we assume(51)aRit=aCt=0.1.We take a basket CDS with ten reference entities, for example. Table1 lists the kth-to-default CDS premium rates for k=1,…,10.Table 1
The premium rates of thekth-to-default CDS.
kthPremium rate (%)1st60.37242nd28.77163rd17.29774th13.51955th8.39196th4.45037th1.83478th0.42729th0.056510th0.0000009The CDS premium rates decrease ask increases because the default probability of k reference entities decreases with increasing of k. Lower default probability leads to lower premium rates. The premium rates tend to 0 with increasing of k.We analyze the effects of the default correlation onkth-to-default CDS premium rates for k=1,4.The numerical calculation shows that the defaults of other reference entities have no effects on the1st-to-default CDS. The premium rates of kth-to-default CDS have similar trends on the default correlation for k=2,3,…,10. We just choose immediate number 4 to illustrate the trends.Figure3 illustrates that kth-to-default CDS premium rates are increasing functions of αRiSi∈I because the reference entities become more risky with higher αRiS, and a higher default risk leads to a higher premium rate.Figure 3
The premium rates of thekth-to-default CDS.
(a)(b)Figure4 illustrates that kth-to-default CDS premium rates are decreasing functions of αCS because the counterparty becomes more risky with higher αCS, and a higher counterparty risk leads to a lower premium rate.Figure 4
The premium rates of thekth-to-default CDS.
(a)(b)Figure5 illustrates that kth-to-default CDS k≥2 premium rates are increasing functions of αRjRii∈I,j∈I∖i because the reference entities become more risky with high αRjRi, and a higher default risk leads to a higher premium rate. The contagion risk has no effect on 1th-to-default CDS because the contract will be terminated if any reference entity defaulted.Figure 5
The premium rates of thekth-to-default CDS.
(a)(b)Figure6 illustrates that kth-to-default CDS k≥2 premium rates are decreasing functions of αCRii∈I because the counterparty becomes more risky with high αCRi, and a higher counterparty risk leads to a lower premium rate. The contagion risk has no effect on 1st-to-default CDS because the contract will be terminated if any reference entity defaulted.Figure 6
The premium rates of thekth-to-default CDS.
(a)(b)


### 4.2.kth-to-Default CLN

The parameters of the model are assumed to be(52)ρ=0.6,r=0.04,αRiS=1.1,αCS=1.1,αRjRi=1.1,αCRi=1.1,λ0S=0.05,σ=0.2,k=0.3,θ=0.02,l=0.3,μ=0.15,T=5.For convenience of calculation, we assume(53)aRit=aCt=0.1.We take a basket CLN with ten reference entities, for example. Table2 lists the kth-to-default CLN premium rates for k=1,…,10.Table 2
The premium rates of thekth-to-default CLN.
kthPremium rate (%)1st64.99802nd33.40783rd21.32884th17.32645th12.66686th8.45037th5.83478th4.95839th4.003410th4.00000853The premium rates of the BCLN onk have similar trends as the premium rate of BCDS. However, the premium rates tend to the risk-free interest rate with the increase of k.The effects of the default correlation onkth-to-default CLN premium rates are demonstrated for k=1,4. The numerical calculation shows that the defaults of other reference entities have no effects on the 1st-to-default CLN. The premium rates of the kth-to-default CLN have similar trends on the default correlation for k=2,3,…,10. We just choose immediate number 4 to illustrate the trends.Figure7 illustrates that the premium rates of BCLN are increasing functions of αRiSi∈I, which is the same as the BCDS case.Figure 7
The premium rates of thekth-to-default CLN.
(a)(b)Figure8 illustrates that kth-to-default CLN premium rates are increasing functions of αCS which is different from the BCDS case because the counterparty in the BCLN is the protection buyer.Figure 8
The premium rates of thekth-to-default CLN.
(a)(b)Figure9 illustrates that the kth-to-default k≥2 CLN premium rates are increasing functions of αRjRii∈I,j∈I∖i, which is the same as the BCDS case. The contagion risk also has no effect on the 1st-to-default CLN.Figure 9
The premium rates of thekth-to-default CLN.
(a)(b)Figure10 illustrates that kth-to-default k≥2 CLN premium rates are increasing functions of αCRii∈I, which is different from the BCDS case because the counterparty in the BCLN is the protection buyer. The contagion risk also has no effect on the 1st-to-default CLN.Figure 10
The premium rates of thekth-to-default CLN.
(a)(b)


## 4.1.kth-to-Default CDS

The parameters of the model are assumed to be(50)ρ=0.4,r=0.04,αRiS=1.1,αCS=1.1,αRjRi=1.1,αCRi=1.1,λ0S=0.05,σ=0.2,k=0.3,θ=0.02,l=0.3,μ=0.15,T=5.For convenience of calculation, we assume(51)aRit=aCt=0.1.We take a basket CDS with ten reference entities, for example. Table1 lists the kth-to-default CDS premium rates for k=1,…,10.Table 1
The premium rates of thekth-to-default CDS.
kthPremium rate (%)1st60.37242nd28.77163rd17.29774th13.51955th8.39196th4.45037th1.83478th0.42729th0.056510th0.0000009The CDS premium rates decrease ask increases because the default probability of k reference entities decreases with increasing of k. Lower default probability leads to lower premium rates. The premium rates tend to 0 with increasing of k.We analyze the effects of the default correlation onkth-to-default CDS premium rates for k=1,4.The numerical calculation shows that the defaults of other reference entities have no effects on the1st-to-default CDS. The premium rates of kth-to-default CDS have similar trends on the default correlation for k=2,3,…,10. We just choose immediate number 4 to illustrate the trends.Figure3 illustrates that kth-to-default CDS premium rates are increasing functions of αRiSi∈I because the reference entities become more risky with higher αRiS, and a higher default risk leads to a higher premium rate.Figure 3
The premium rates of thekth-to-default CDS.
(a)(b)Figure4 illustrates that kth-to-default CDS premium rates are decreasing functions of αCS because the counterparty becomes more risky with higher αCS, and a higher counterparty risk leads to a lower premium rate.Figure 4
The premium rates of thekth-to-default CDS.
(a)(b)Figure5 illustrates that kth-to-default CDS k≥2 premium rates are increasing functions of αRjRii∈I,j∈I∖i because the reference entities become more risky with high αRjRi, and a higher default risk leads to a higher premium rate. The contagion risk has no effect on 1th-to-default CDS because the contract will be terminated if any reference entity defaulted.Figure 5
The premium rates of thekth-to-default CDS.
(a)(b)Figure6 illustrates that kth-to-default CDS k≥2 premium rates are decreasing functions of αCRii∈I because the counterparty becomes more risky with high αCRi, and a higher counterparty risk leads to a lower premium rate. The contagion risk has no effect on 1st-to-default CDS because the contract will be terminated if any reference entity defaulted.Figure 6
The premium rates of thekth-to-default CDS.
(a)(b)


## 4.2.kth-to-Default CLN

The parameters of the model are assumed to be(52)ρ=0.6,r=0.04,αRiS=1.1,αCS=1.1,αRjRi=1.1,αCRi=1.1,λ0S=0.05,σ=0.2,k=0.3,θ=0.02,l=0.3,μ=0.15,T=5.For convenience of calculation, we assume(53)aRit=aCt=0.1.We take a basket CLN with ten reference entities, for example. Table2 lists the kth-to-default CLN premium rates for k=1,…,10.Table 2
The premium rates of thekth-to-default CLN.
kthPremium rate (%)1st64.99802nd33.40783rd21.32884th17.32645th12.66686th8.45037th5.83478th4.95839th4.003410th4.00000853The premium rates of the BCLN onk have similar trends as the premium rate of BCDS. However, the premium rates tend to the risk-free interest rate with the increase of k.The effects of the default correlation onkth-to-default CLN premium rates are demonstrated for k=1,4. The numerical calculation shows that the defaults of other reference entities have no effects on the 1st-to-default CLN. The premium rates of the kth-to-default CLN have similar trends on the default correlation for k=2,3,…,10. We just choose immediate number 4 to illustrate the trends.Figure7 illustrates that the premium rates of BCLN are increasing functions of αRiSi∈I, which is the same as the BCDS case.Figure 7
The premium rates of thekth-to-default CLN.
(a)(b)Figure8 illustrates that kth-to-default CLN premium rates are increasing functions of αCS which is different from the BCDS case because the counterparty in the BCLN is the protection buyer.Figure 8
The premium rates of thekth-to-default CLN.
(a)(b)Figure9 illustrates that the kth-to-default k≥2 CLN premium rates are increasing functions of αRjRii∈I,j∈I∖i, which is the same as the BCDS case. The contagion risk also has no effect on the 1st-to-default CLN.Figure 9
The premium rates of thekth-to-default CLN.
(a)(b)Figure10 illustrates that kth-to-default k≥2 CLN premium rates are increasing functions of αCRii∈I, which is different from the BCDS case because the counterparty in the BCLN is the protection buyer. The contagion risk also has no effect on the 1st-to-default CLN.Figure 10
The premium rates of thekth-to-default CLN.
(a)(b)


## 5. Conclusion

In this paper, we use a Markov chain model with interacting intensities to compute thekth-to-default CDS and kth-to-default CLN premium rates. We model the default correlation among the names in the portfolio by an external shock as well as the defaults of other names in the portfolio. By presenting a general infinitesimal generator matrix of 2N+2×2N+2 dimension, we obtain the joint distribution of default times. Then, the formulas of kth-to-default CDS and kth-to-default CLN premium rates are calculated. In numerical analysis, we show that the premium rates decrease as k increases, and the contagion risk does not affect the 1st-to-default premium rate. Our numerical results also show the effects of correlated risks between the counterparty and reference entities on the premium rates. Due to different product structures between BCDS and BCLN, the impacts of the contagion risk on premium rates are different.



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*Source: 5369879-2020-10-17.xml*