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.
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 with 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 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 th-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 th-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 th-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 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 th-to-default CDS and th-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 space , where is a martingale measure and is filtration satisfying the usual conditions. Our model includes a basket credit derivative and an external shock. The basket credit derivative includes reference entities, a counterparty, and an investor. Let represent the th reference entity with default time for , be the counterparty with default time , be the external shock with arrival time , and be the set which contains all of the names in the model. Assume that is independent of and . The default process of our model is given bywhereand is the indicator function. is a finite-state Markov chain with state space .
The macroeconomic variables are described by a stochastic process . 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 is given bywhereand is the -field generated by . The martingale default intensities of the reference entities and counterparty are, respectively, defined by and , which satisfy the property thatare -martingales.
The arrival of the external shock is modeled by a Cox process with stochastic intensity . Before the shock happens, the default intensities of and are, respectively, assumed to be and , where and are some deterministic functions of . When the shock happens, and jump to and , respectively, where nonnegative constants and denote the effects from the external shock to reference entities and the counterparty. Depending on whether and are greater than or not, the densities and 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 defaults, the default intensities of other reference entities and the counterparty jump to and , respectively, where and 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 reference entities, and the state space of is dimensions. Let , and conditional on the given state , the infinitesimal generator matrix of is , where represent the states of the Markov chain . The conditional transition probability matrix is governed by the following forward Kolmogorov equation:with , and is the unit matrix. Because the default states and the shock are absorbing states for the Markov chain , the matrix is upper triangular, and the conditional transition probabilities can be solved successively in a sequential manner.
To describe the generator matrix , we introduce some auxiliary notations as follows:(i) represents the state that only reference entities in set defaulted. Here, , where for all . Especially, represents the state that no reference entity defaults.(ii) represents the state that only the counterparty defaulted. Here, .(iii) represents the state that only the external shock arrived. Here, .(iv) represents the state that reference entities in set and the counterparty defaulted. Here, , where for all .(v) represents the state that reference entities in set defaulted, and the external shock happened. Here, , where for all .(vi) represents the state that reference entities in set and the counterparty both defaulted, and the external shock happened. Here, , where for all .(vii) for are elements on the diagonal of the matrix , which represent the intensities that the Markov chain remains at states .(viii) represents the conditional probability that Markov chain transfers from the initial state to state during . It is the first line of the conditional transition probability matrix and can be used to present the joint distributions of default times during .
Firstly, we calculate the conditional transition probabilities that none of the reference entities defaults during . The corresponding states of are and . The corresponding elements in the infinitesimal generator matrix are as follows:
In the -row of the generator matrix , the positive elements represent the transition intensities from state 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 , or . Noting that the elements on the diagonal of the generator matrix are the sum of all other elements in the row multiplied with , we can get the first element of the -row as follows:
Because the generator matrix is upper diagonal, is the only nonzero element in the -column.
In the -row of the generator matrix , the positive elements represent the transition intensities from state to other states by one jump. By (6), the positive elements in the -row are and . Summing these positive elements and multiplying the sum with , we get the diagonal element in this row as follows:
In the -column of the generator matrix , the positive elements represent the transition intensities from other states to state by one jump. The only possible state which can jump to is . So, the nonzero elements in the -column are and .
By forward Kolmogorov equation (7), we have
It is easy to obtain the solutions of the above equations as follows:
Secondly, we calculate the conditional transition probabilities that only one reference entity defaults during . The corresponding states of are and for . The corresponding elements in the infinitesimal generator matrix are as follows:
In the -row of the generator matrix , the positive elements represent the transition intensities from state 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 by (6) and if external shock happens. The diagonal element in this row is as follows:
In the -column of the generator matrix , the positive elements represent the transition intensities from other states to state by one jump. The only possible state which can jump to is . So, the nonzero elements in the -column are and the diagonal element .
In the -row of the generator matrix , the positive elements represent the transition intensities from state 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 and . The diagonal element in this row is as follows:
In the -column of the generator matrix , the positive elements represent the transition intensities from other states to state by one jump. The possible states which can jump to are and . So, the nonzero elements in the -column are , , and the diagonal element .
By forward Kolmogorov equation (7), we havewhere and 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 reference entities default during . For the cases that reference entities default, the corresponding states of are and for and , where is the cardinal number of set . The corresponding elements in the infinitesimal generator matrix are as follows:
In the -row of the generator matrix , the positive elements represent the transition intensities from state 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 and by (7) and if external shock happens. The diagonal element in this row is as follows:
In the -column of the generator matrix , the positive elements represent the transition intensities from other states to state by one jump. The possible states which can jump to are for . So, the nonzero elements in this column are by (6) and the diagonal element .
In the -row of the generator matrix , the positive elements represent the transition intensities from state 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 and . The diagonal element in this row is obtained as follows:
In the -column of the generator matrix , the positive elements represent the transition intensities from other states to state by one jump. The possible states which can jump to are and . So, the nonzero elements in this column are , , and the diagonal element .
By forward Kolmogorov equation (7), we havewhere and 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.
Proposition 1. (i) contains only one random term which is outside the integral symbol; (ii) each term in contains only one random term which is inside the integral symbol.
Proof. We use the induction method to prove this proposition.(i)Firstly, : just contains one random term owning the path of . Secondly, : contains one random term outside the integral. Assume that (i) holds for . When , In the integral of (25), the only random term contained in offsets the term in . Then, contains only one random term from outside the integral in (25), and (i) holds for .(ii)Firstly, : contains one random term inside the integral. Secondly, : The two terms in the right side of (27) contain only one random term (or ) inside the integral.Assume that (ii) holds for . When ,By , only contains one random term which is outside the integral symbol. Then, the first term in contains only one random term . By induction, each term in contains only one random term which is in the integral symbol. Then, each term in the sum part of equation (28) contains only one random term in the integral. In summary, (ii) also holds when .
According to Proposition 1 and Fubini’s theorem, we can take the expectation of equation (22) and obtain the unconditional transition probabilities as follows:where is the expectation taken over the path of .
To compute and , we adopt the affine diffusion process with the jump for 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 is given bywhere is a standard Brownian motion, is a pure jump process, and denotes the jump of at time . Here, is taken to be independent of with jump sizes that are independent and exponentially distributed with mean and whose jump times are those of all independent Poisson processes with constant jump arrival rate . It was shown by Wang and Garleanu [5] thatwhereReplacing 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 is , is the maturity date of the contracts, is the premium rate of th-to-default credit derivatives, is the th default time of reference entities, constant is the risk-free interest rate, and constant is the recovery rate.
3.1. kth-to-Default CDS
The cash flow of a th-to-default CDS is presented in Figure 1.(i)The protection buyer pays constant premium rate during the life of the th-to-default CDS(ii)If the th default in the pool occurs before the maturity, the protection buyer gives a recovery payoff , and the contract is terminated
The expected cash flow of th-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 th-to-default CDS, there are two possible scenarios during : nonoccurrence of the external shock or occurrence of . If reference entities have defaulted during and another reference entity defaults during , the probability of such occurrence is given bywhere corresponds to nonoccurrence of and corresponds to the case otherwise. So, the expected present value of the compensation payment paid by the protection seller within is
Over the entire period , the expected present value of the compensation payment paid by the protection seller is
The th-to-default CDS contract will be terminated if more than reference entities default. The corresponding states to continue the contract are and . The expected premium payment paid by the protection buyer within is given bywhere
Over the entire period , the expected present value of the premium payment paid by the protection buyer is
By (34), the th-to-default CDS premium rate is given by
3.2. kth-to-Default CLN
The cash flow of a th-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 rate during the life of the th-to-default CLN(iii)If the th default in the pool occurs before the maturity, the protection buyer gives a recovery payoff , and the contract is terminated(iv)If the th default in the pool does not occur before the maturity, the protection buyer returns the nominal amount
The expected cash flow of the th-to-default CLN is as follows:
The premium rate is given by
Here, the method for calculating the default probabilities is the same as the method for the th-to-default CDS case.
Over the whole period , the expected present value of the recovery payment paid by the CLN issuer iswhere
The expected present value of the premium payment paid by the CLN issuer iswhere
The 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
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
For convenience of calculation, we assume
We take a basket CDS with ten reference entities, for example. Table 1 lists the th-to-default CDS premium rates for .
The CDS premium rates decrease as increases because the default probability of reference entities decreases with increasing of . Lower default probability leads to lower premium rates. The premium rates tend to with increasing of .
We analyze the effects of the default correlation on th-to-default CDS premium rates for .
The numerical calculation shows that the defaults of other reference entities have no effects on the st-to-default CDS. The premium rates of th-to-default CDS have similar trends on the default correlation for . We just choose immediate number to illustrate the trends.
Figure 3 illustrates that th-to-default CDS premium rates are increasing functions of because the reference entities become more risky with higher , and a higher default risk leads to a higher premium rate.
(a)
(b)
Figure 4 illustrates that th-to-default CDS premium rates are decreasing functions of because the counterparty becomes more risky with higher , and a higher counterparty risk leads to a lower premium rate.
(a)
(b)
Figure 5 illustrates that th-to-default CDS premium rates are increasing functions of because the reference entities become more risky with high , and a higher default risk leads to a higher premium rate. The contagion risk has no effect on th-to-default CDS because the contract will be terminated if any reference entity defaulted.
(a)
(b)
Figure 6 illustrates that th-to-default CDS premium rates are decreasing functions of because the counterparty becomes more risky with high , and a higher counterparty risk leads to a lower premium rate. The contagion risk has no effect on st-to-default CDS because the contract will be terminated if any reference entity defaulted.
(a)
(b)
4.2. kth-to-Default CLN
The parameters of the model are assumed to be
For convenience of calculation, we assume
We take a basket CLN with ten reference entities, for example. Table 2 lists the th-to-default CLN premium rates for .
The premium rates of the BCLN on have similar trends as the premium rate of BCDS. However, the premium rates tend to the risk-free interest rate with the increase of .
The effects of the default correlation on th-to-default CLN premium rates are demonstrated for . The numerical calculation shows that the defaults of other reference entities have no effects on the st-to-default CLN. The premium rates of the th-to-default CLN have similar trends on the default correlation for . We just choose immediate number to illustrate the trends.
Figure 7 illustrates that the premium rates of BCLN are increasing functions of , which is the same as the BCDS case.
(a)
(b)
Figure 8 illustrates that th-to-default CLN premium rates are increasing functions of which is different from the BCDS case because the counterparty in the BCLN is the protection buyer.
(a)
(b)
Figure 9 illustrates that the th-to-default CLN premium rates are increasing functions of , which is the same as the BCDS case. The contagion risk also has no effect on the st-to-default CLN.
(a)
(b)
Figure 10 illustrates that th-to-default CLN premium rates are increasing functions of , 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 st-to-default CLN.
(a)
(b)
5. Conclusion
In this paper, we use a Markov chain model with interacting intensities to compute the th-to-default CDS and th-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 dimension, we obtain the joint distribution of default times. Then, the formulas of th-to-default CDS and th-to-default CLN premium rates are calculated. In numerical analysis, we show that the premium rates decrease as 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.
Data Availability
The data in this paper can be used publicly.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (nos. 11671291 and 11901420) and Humanities and Social Science Research Projects in the Ministry of Education (no. 20YJAZH025).