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

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.


Introduction
e 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 with N reference entities. e 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. e 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. e 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]. e 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 fourstate 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-todefault 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. e 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. e paper is organized as follows. In Section 2, 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.

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 space (Ω, G, G t 0≤t≤T , P), where P is a martingale measure and G t t≥0 is filtration satisfying the usual conditions. Our model includes a basket credit derivative and an external shock. e basket credit derivative includes N reference entities, a counterparty, and an investor. Let R i represent the ith reference entity with default time τ R i for i ∈ I � 1, 2, . . . , N { }, C be the counterparty with default time τ C , S be the external shock with arrival time τ S , and L � R 1 , R 2 , . . . , R N , C, S be the set which contains all of the names in the model. Assume that τ S is independent of τ C and τ R i (i ∈ I). e default process of our model is given by where . e macroeconomic variables are described by a stochastic process Ψ � (Ψ t ) 0≤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. e filtration (G t ) t≥0 is given by where and G t is the σ-field generated by e martingale default intensities of the reference entities and counterparty are, respectively, defined by λ R i (Ψ t , H t )(i ∈ I) and λ C (Ψ t , H t ), which satisfy the property that are G t -martingales. e arrival of the external shock is modeled by a Cox process with stochastic intensity λ S t . Before the shock S happens, the default intensities of λ R i t (i ∈ I) and λ C t are, respectively, assumed to be a R i t and a C t , where a R i t and a C t are some deterministic functions of t. When the shock S happens, λ R i t and λ C t jump to α S R i a R i t and α S C a C t , respectively, where nonnegative constants α S R i and α S C denote the effects from the external shock to reference entities and the counterparty. Depending on whether α S R i and α S C are greater than 1 or not, the densities λ R i and λ C 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 α C 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: 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 with N reference entities, and the state space T] , and conditional on the given state ψ S , the infinitesimal e conditional transition probability matrix P(0, t | ψ S ) ≜ (P H×H′ (0, t | ψ S )) 2 N+2 ×2 N+2 is governed by the following forward Kolmogorov equation: with P(0, 0 | ψ S ) � E, and E is the unit matrix. Because the default states and the shock are absorbing states for the Markov chain H t , the matrix Λ [ψ S ] (t) is upper triangular, and the conditional transition probabilities P(0, t | ψ S ) can be solved successively in a sequential manner.
To describe the generator matrix , which represent the intensities that the Markov chain H t remains at states H.

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In the H R ∅ -row of the generator matrix Λ [ψ S ] (t), the positive elements represent the transition intensities from state H R ∅ 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 a R i t (i ∈ I), a C t , or λ S t . Noting that the elements on the diagonal of the generator matrix Λ [ψ S ] (t) are the sum of all other elements in the row multiplied with −1, we can get the first element Λ H R ϕ of the H R ∅ -row as follows: Because the generator matrix In the H S -row of the generator matrix Λ [ψ S ] (t), the positive elements represent the transition intensities from state H S to other states by one jump. By (6), the positive elements in the H S -row are α S R i a R i t (i ∈ I) and α S C a C t . Summing these positive elements and multiplying the sum with −1, we get the diagonal element in this row as follows: In the H S -column of the generator matrix Λ [ψ S ] (t), the positive elements represent the transition intensities from other states to state H S by one jump. e only possible state which can jump to H S is H R ϕ . So, the nonzero elements in the H S -column are λ S t and Λ H S (t). By forward Kolmogorov equation (7), we have It is easy to obtain the solutions of the above equations as follows: In the H R i { } -row of the generator matrix Λ [ψ S ] (t), the positive elements represent the transition intensities from state H R i { } to other states by one jump. ere are three possible scenarios: a single reference entity defaults, counterparty defaults, or external shock happens. e positive elements in this row are α (6) and λ S t if external shock happens. e diagonal element in this row is as follows: In , the positive elements represent the transition intensities from state H R i { } ∪ S to other states by one jump. ere are two possible scenarios: a single reference entity defaults or counterparty defaults. By (6), the positive elements in this e diagonal element in this row is as follows:  (7), we have where P H R ϕ (0, t | ψ S ) and P H S (0, t | ψ S ) are already obtained in (12). It is easy to solve equation (16) as follows: We can derive all the conditional transition probabilities by the method of mathematical induction. Suppose that we have obtained the conditional transition probabilities that k − 1 reference entities default during e corresponding elements in the infinitesimal generator matrix Λ [ψ S ] (t) are as follows: In the H R M -row of the generator matrix Λ [ψ S ] (t), the positive elements represent the transition intensities from state H R M to other states by one jump.
ere are three possible scenarios: a single reference entity defaults, counterparty defaults, or external shock happens. e positive elements in this row are ( i∈M α (7) and λ S t if external shock happens. e diagonal element in this row is as follows:

Mathematical Problems in Engineering
In the H R M -column of the generator matrix Λ ere are two possible scenarios: a single reference entity defaults or counterparty defaults. By (6), the positive elements in this t . e diagonal element in this row is obtained as follows: In the H R M ∪ S -column of the generator matrix Λ By forward Kolmogorov equation (7), we have are known by induction. It is easy to get the solutions of equation (21) as follows: In order to calculate the unconditional transition probabilities, we need the following proposition. Proof. We use the induction method to prove this proposition.
(i) Firstly, |M| � 0: P H R ϕ (0, t | ψ S ) just contains one random term owning the path of λ S t .

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Assume that (i) holds for |M| � k − 1. When |M| � k, In the integral of (25), the only random term (26)

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inside the integral. Secondly, |M| � 1: e two terms in the right side of (27) contain only one Assume that (ii) holds for |M| � k − 1. When |M| � k,  Mathematical Problems in Engineering According to Proposition 1 and Fubini's theorem, we can take the expectation E ψ S [·] of equation (22) and obtain the unconditional transition probabilities as follows: where E ψ S [·] is the expectation taken over the path of , we adopt the affine diffusion process with the jump for λ S t 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. e stochastic differential equation of λ S t is given by where Z t is a standard Brownian motion, J t is a pure jump process, and ΔJ t denotes the jump of J t at time t. Here, J t is taken to be independent of Z t 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 where Mathematical Problems in Engineering Replacing the conditional expectations in (29) with equation (31), we obtain the unconditional transition probabilities.

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 is 1, T is the maturity date of the contracts, c k is the premium rate of kth-to-default credit derivatives, τ k 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.

kth-to-Default CDS.
e cash flow of a kth-to-default CDS is presented in Figure 1.
(i) e protection buyer pays constant premium rate c k during the life of the kth-to-default CDS (ii) If the kth default in the pool occurs before the maturity, the protection buyer gives a recovery payoff ρ, and the contract is terminated e expected cash flow of kth-to-default CDS is as follows (see Zheng and Jiang [9]): and the premium rate is given by To calculate the premium rate for the kth-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 where t )]dt corresponds to the case otherwise. So, the expected present value of the compensation payment paid by the protection seller within (t, t + dt] is Over the entire period [0, T], the expected present value of the compensation payment paid by the protection seller is e kth-to-default CDS contract will be terminated if more than k − 1 reference entities default. e corresponding states to continue the contract are H R M and H R M ∪ S (M ⊂ I, 0 ≤ |M| ≤ k − 1). e expected premium payment paid by the protection buyer within (t, t + dt] is given by where Over the entire period [0, T], the expected present value of the premium payment paid by the protection buyer is By (34), the kth-to-default CDS premium rate is given by

kth-to-Default CLN.
e cash flow of a kth-to-default CLN is presented in Figure 2.
(i) e protection seller pays a nominal amount at the initial date (ii) e protection buyer pays constant premium rate c k during the life of the kth-to-default CLN (iii) If the kth default in the pool occurs before the maturity, the protection buyer gives a recovery payoff ρ, and the contract is terminated (iv) If the kth default in the pool does not occur before the maturity, the protection buyer returns the nominal amount e expected cash flow of the kth-to-default CLN is as follows: e premium rate is given by Here, the method for calculating the default probabilities is the same as the method for the kth-to-default CDS case.
Over the whole period [0, T], the expected present value of the recovery payment paid by the CLN issuer is where e expected present value of the premium payment paid by the CLN issuer is where e expectation of the payment of nominal amount at maturity of the contract is By (43), the kth-to-default CLN premium rate is given by

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.

kth-to-Default CDS.
e parameters of the model are assumed to be ρ � 0.4, For convenience of calculation, we assume We take a basket CDS with ten reference entities, for example. Table 1 lists the kth-to-default CDS premium rates for k � 1, . . . , 10. e CDS premium rates decrease as k increases because the default probability of k reference entities decreases with increasing of k. Lower default probability leads to lower premium rates. e premium rates tend to 0 with increasing of k.
We analyze the effects of the default correlation on kthto-default CDS premium rates for k � 1, 4. e numerical calculation shows that the defaults of other reference entities have no effects on the 1st-to-default CDS. e 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. Figure 3 illustrates that kth-to-default CDS premium rates are increasing functions of α S R i (i ∈ I) because the reference entities become more risky with higher α S R i , and a higher default risk leads to a higher premium rate. Figure 4 illustrates that kth-to-default CDS premium rates are decreasing functions of α S C because the counterparty becomes more risky with higher α S C , and a higher counterparty risk leads to a lower premium rate. Figure 5 illustrates that kth-to-default CDS (k ≥ 2) premium rates are increasing functions of α because the reference entities become more risky with high α R i R j , and a higher default risk leads to a higher premium rate. e contagion risk has no effect on 1th-to-default CDS because the contract will be terminated if any reference entity defaulted. Figure 6 illustrates that kth-to-default CDS (k ≥ 2) premium rates are decreasing functions of α R i C (i ∈ I) because the counterparty becomes more risky with high α R i C , and a higher counterparty risk leads to a lower premium rate. e contagion risk has no effect on 1st-to-default CDS because the contract will be terminated if any reference entity defaulted.
For convenience of calculation, we assume We take a basket CLN with ten reference entities, for example. Table 2 lists the kth-to-default CLN premium rates for k � 1, . . . , 10. e premium rates of the BCLN on k 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. e effects of the default correlation on kth-to-default CLN premium rates are demonstrated for k � 1, 4. e numerical calculation shows that the defaults of other reference entities have no effects on the 1st-to-default CLN. e 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. Figure 7 illustrates that the premium rates of BCLN are increasing functions of α S R i (i ∈ I), which is the same as the BCDS case. Figure 8 illustrates that kth-to-default CLN premium rates are increasing functions of α S C which is different from the BCDS case because the counterparty in the BCLN is the protection buyer. Figure 9 illustrates that the kth-to-default (k ≥ 2) CLN premium rates are increasing functions of α R i R j (i ∈ I, j ∈ I∖ i { }), which is the same as the BCDS case. e contagion risk also has no effect on the 1st-to-default CLN. Figure 10 illustrates that kth-to-default (k ≥ 2) CLN premium rates are increasing functions of α R i C (i ∈ I), which is different from the BCDS case because the counterparty in the BCLN is the protection buyer. e contagion risk also has no effect on the 1st-to-default CLN.

Conclusion
In this paper, we use a Markov chain model with interacting intensities to compute the kth-to-default CDS and kth-todefault 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 2 N+2 × 2 N+2 dimension, we obtain the joint distribution of default times. en, the formulas of kth-to-default CDS and kth-todefault 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-todefault 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.
Data Availability e data in this paper can be used publicly.

Conflicts of Interest
e authors declare that they have no conflicts of interest.