Abstract
In this study, on the basis of basket CDS, we take into account the complex correlations among market participants and choose to use the contagion model to price basket CDS. This study assumes that a basket of reference assets is defaultable and that defaults between two counterparties are contagious. Without taking into account the default of the CDS buyer, we assume that both the interest rate and the default intensity of the reference assets follow the Vasicek model. We obtain the related probability density function by the PDE method. Under the no-arbitrage principle, we set up several scenarios to get the basket CDS pricing formula. In the numerical analysis part, we verify the validity of the formula by Monte Carlo simulation, and discuss the influence of the main parameters on the formula.
1. Introduction
Since the end of the 20th century, the financial derivatives market has developed very rapidly. At present, it has become one of the most important financial innovations in the international financial market and a new tool to manage credit risks after loan transaction and asset securitization. Credit default swap (CDS) is one of the most important derivatives for managing credit risk in financial markets. The CDS market has expanded rapidly because of its ease of standardisation. The crisis brought by the rapid development of CDS, such as the US subprime mortgage crisis and the European debt crisis, made people realize that credit derivatives bring great risks as well as convenience. Therefore, the pricing of CDS has attracted the attention of many scholars and businessmen. There are two main forms of CDS contracts in the market: single name CDS and basket CDS. The difference between the two contracts is the number of reference assets. Nowadays, the relationship between market players is very complex. It is common for companies to hold shares and bonds with each other. Large companies, in particular, are often linked to more than one company, and defaults by large companies can cause serious chain reactions. Since the collapse of WorldCOM and Enron, fears of big company defaults have grown. Moreover, the subprime crisis has made people realize that the relevant default risk plays an important role in credit risk management.
At present, there are two main methods for the research of credit derivatives: one is the structured method pioneered by Black, Scholes, and Merton, and the other is the reduced method proposed by Jarrow and Turnbull, which corresponds to the structured model and the reduced model, respectively.
In the structured method, the assets of the reference firm are assumed to follow a stochastic process, and when the assets of the firm reach a predetermined boundary, the firm defaults. Examples of this approach include Black and Cox, Longstaff and Schwartz, and Leland and Toft. However, this method is often difficult to get clear results for solving the valuation problem of products with jump and diffusion risk. In contrast, the reduced method is easier to solve such problems. In the reduced method, the default time is modeled as the jump time of Poisson process, and the default is regarded as an exogenous event independent of the assets and capital structure of the reference company. Examples of this approach include Jarrow and Turnbull [1], Lando [2], Duffie and Singleton, Malherbe [3], among others. Jarrow and Turnbull first proposed the reduced model to describe the occurrence of exogenous default using Poisson process. Lando proposed a Cox procedure to describe the intensity of default and assumed that the risk-free interest rate satisfies the Vasicek model. Malherbe used Poisson processes to describe the default intensity and assumed that the default intensity is constant between defaults, but jumps may occur when defaults occur. Herbertsson and Rootzén [4] used matrix analytic method to derive closed-form expressions for basket CDS. In the current research, we tend to adopt the reduced method, because the structured method requires more company asset information and structural information, which is more complex to obtain. The reduced method treats default as an exogenous event independent of the company. In this way, we skip the process of obtaining relevant information about the company, which reduces the workload and simplifies the model.
With the rapid development of the financial market, the frequency of transactions increases, leading to the gradual accumulation of credit problems, that is, the total credit risk is gradually increasing. At the same time, we see that there are complex connections between companies, which means that defaults among companies are directly linked. Therefore, how to better manage contagion risk is an important subject of credit risk management. With the acceleration of global economic integration, the relationship between enterprises is getting closer and closer, so the research on basket CDS pricing has attracted more and more researchers. The key to basket CDS pricing is to obtain the relative default probability of multiple assets. In the existing literature, there are mainly three methods to model multiasset default risk: the copula function, the conditional independent model and the contagion model. Research shows that default dependence has a significant impact on CDS pricing. The credit contagion model has been proposed to explain the concentration of risk in large portfolios of defaultable securities. On the basis of this model, people realized that the existing model still has defects from the Korean bank crisis and other events, and then extended the model to introduce counterparty risk caused by counterparty default. Kijima and Muromachi [5] studied the pricing of basket CDS of the first default type on the basis of the reduced model, and obtained the analytical formula of the basket CDS price. Kijima and Muromachi [6] considered the pricing of the -to-default basket CDS, but they did not give a clear solution. Wu et al. [7] proposed a new model for evaluating the value of credit default swap contracts subject to multiple credit risks. They showed that default dependence has a significant impact on CDS pricing. Davis and Lo [8] first used the contagion model to describe the default risk of stock portfolios. Jarrow and Yu [9] introduced the concept of counterparty risk on the basis of contagion model and explained the impact of counterparty risk on CDS pricing. Hao and Wang [10] studied CDS pricing with contagion risk under fractional the Vasicek interest rate model. Huang and Song [11] priced basket CDS with counterparty risk under the multiname contagion model. Chen and Xing [12] assumed that the basket CDS and two counterparties are independent from each other, provided CDS pricing based on the Vasicek default intensity model and related closed-form solutions. Motivated by Chen and Xing, this study establishes a two-counterparty basket CDS pricing model with default contagion.
At the same time, we also noticed the stochastic volatility model. He and Chen [13] proposed a new stochastic volatility model with long-term mean value on the basis of Heston volatility model, and obtained a closed pricing formula for European options. Subsequently, He and Lin [14, 15] introduced regime switching based on the He-Chen model, and proved that regime switching has a significant impact on option pricing. The introduction of regime switching into CDS pricing is also a hot topic in current research. Inspired by He and Chen [13, 16] and He and Lin [14, 15], our model can be further extended to a stochastic volatility model. The stochastic volatility model can describe the volatility aggregation phenomenon of default intensity, but it will be quite difficult to deduce it when applied to the basket CDS price. The reason is that when the volatility in the default intensity of the reference assets and two counterparties is random, the number of state variables will increase, which makes the calculation more difficult. The partial differential equation satisfying the relative default probability density does not necessarily have an analytical solution, so the CDS price can only be solved by Monte Carlo simulation and other numerical methods. At the same time, when the number of reference assets in the basket is small, regime switching can be further introduced to improve the accuracy of CDS pricing. However, when the number of reference assets is large, the parameters generated by regime switching will increase sharply, which will also cause computational difficulties, and can only be solved by some simulation methods.
The overall framework of this article is as follows: in Section 1, this article describes the development background of CDS, and then introduces two pricing methods of traditional CDS, and then leads to the research model of this study. In Section 2, we introduce some of the basic assumptions of this study. We assume that the reference assets are a basket of assets and two defaultable counterparties default is circular contagion. We also assume that both interest rate and the default intensities of the basket reference assets are follow the Vasicek model. In Section 3, we derive the survival probability densities of the basket reference assets without default and with default, and give a closed-form solution. Then, the joint distribution of two counterparties defaults is derived and their marginal distribution is further obtained. In Section 4, we set up some scenarios and get the pricing formula of basket CDS based on no-arbitrage principle. In Section 5, we verify the validity of basket CDS pricing formula through Monte Carlo simulation, and analyze the impact of main parameters on the formula. In Section 6, we summarize the relevant conclusions.
2. Assumptions
In this section, we will make assumptions about some of the variables in this article.
Traditional CDS contracts are signed with a counterparty. The potential danger of such contracts is that if the counterparty defaults after the reference asset defaults, the buyer of the protection will lose heavily. To avoid this loss, we consider entering into contracts with two counterparties. In this contract, the protection buyer pays the two counterparties if the reference asset does not default, and the nondefaulting counterparty pays the protection buyer the agreed compensation if the reference asset defaults. We assume that(1)CDS fees payments are continuous. Generally, CDS fees payments are discrete. However, counterparties usually sign CDS contracts with a lot of people, and for counterparties, they charge fees almost all the time. At the macro level, we can think of it as continuous.(2)The nominal principal is 1 dollar.
In the reduced model, we assume that is the filter space satisfying the general case. and is the event filed generated by sample space . is an equivalent martingale measure.Under this martingale measure, the discounted price of bond is a martingale, and is unique. represents the filter generated by the underlying random structure, which represents all the information available at a time. The conditional probability measure given by is denoted by , and the associated expectation is denoted by . The default time is a stopping time associated with the filter . For sufficiently small , if is an intensity process, satisfies the following equation:
We assume that the reference asset is a basket of bonds issued by different companies, each of which can default with the default intensity of . Now, we have two counterparties named and , and assume that both and can default with the default intensity of and , respectively. We assume that following the Vasicek process:
For and follow the following equation:where is the unique macrostate interest rate, and , as follows:where are positive constants; are real numbers and satisfy , ; represent the long-term mean values of and , respectively; denote the rate at which and return to the mean, respectively; are standard Brownian motions. When . When .
3. Probability Density of Default
In this section, we will derive some relevant probability density formulas in preparation for deriving basket CDS prices.
Let denote the default times of the reference assets . Given , the default times are conditionally independent. The start time and the expiration time are denoted by and , respectively. Some theorems are given.
Theorem 1 (see [12]). Assuming that the reference assets do not default until time , then the survival probability density function satisfies the following PDE:
Proof. Because , we have . Through conditional independence, the joint survival probability density at time can be given as follows:With Feynman–Kac theorem, satisfies the previous PDE.
Corollary 1. Assuming that the reference assets do not default until time , then the survival probability density function has the following closed-form solution:where
Proof. According to ksendal [17], has a solution with the following form:Substitute the previous equation into the PDE of Theorem 1, two ODEs are obtained as follows:Solve the previous ODEs, obtain and .
Theorem 2 (see [12]). Assuming that among the reference assets , the first company to default is the company, and the default time is , we use denotes the default probability density at time , satisfies as follows:
Proof. Because , the probability density of the company first default at time is as follows:With Feynman–Kac theorem, satisfies the previous PDE.
Corollary 2. Assuming that among the reference assets , the first company to default is the company, and the default time is . denote the probability density of default at time , then has the following closed-form solution:where
and are given by Corollary 1.
Proof. According to ksendal, has a solution with the following form: and can be derived from Corollary 1. Put the previous equation, and substitute into the PDE of Theorem 2 to obtain two ODEs as follows:Solve the previous ODEs obtain and .
It is worth emphasizing that Theorems 1 and 2 refer to the study of Chen and Xing [12].
Now, we derive the probability density of no default and default of the reference assets. Next, we derive the probability density of two counterparties not defaulting, one of them defaulting, and both of them defaulting under the circular contagion model. First, we need to get the conditional marginal distributions of and at the default times and , respectively. To avoid the effects of circular defaults, we first calculate the conditional joint distribution of and . Define two new probability measures: Under the new measures, has intensity .
Theorem 3. Assuming that and are the default times of two counterparties and , respectively. The interest rate and the default intensities and satisfy the previous assumptions (3)–(5). If no default event occurs at time , then the conditional joint distribution of default times and on is as follows:
When ,
When ,
Proof. When ,Here, denotes the conditional expectation under measure , . When , the default intensity of under is , sothen,We substitute the previous two expressions into the joint distribution to obtain the joint distribution under the case . Similarly, we can obtain the joint distribution under the case , which is omitted here.
Corollary 3. Assuming that and are the default times of two counterparties and , respectively, and the default intensity and satisfy the previous assumptions. If no default event occurs at time , then the conditional marginal distribution of default times and on is as follows:Prove slightly.
4. Basket CDS Price
In this section, we will price the basket CDS under the no-arbitrage principle.
We assume that the CDS buyer holds bonds and the corresponding reference assets are reference assets . At the initial time, the buyer and two counterparties, and , enter into a CDS contract with a maturity of . The contract provides that the buyer will continue to pay and if no default occurs on the reference assets. In the event of a default of the reference assets, the buyer will cease to pay fees to and , and and will jointly pay compensation to the buyer. Here, we assume that the two counterparties and share the risk equally, that is, and pay half of the loss. This study argues that if the reference assets do not default, then the counterparties will not default first. Scenario 1: neither the reference assets nor the counterparties defaulted until the maturity date of ; Scenario 2: the company in the reference assets defaults first, and the default time is .
(1)Neither nor is in default;(2) or defaults;(3)Both and have defaulted.Next, we calculate the discounted value of the buyer’s payment to , denoted by . First, analyze the discounted value of the amount counterparty receives and uses for compensation under different circumstances. Scenario 1: according to Corollary 1, the discounted value of the fees received by counterparty is as follows: Because the reference assets are not in default, the counterparty will not pay compensation to the buyer until the maturity date . Scenario 2: according to Corollary 2, the discounted value of the fees received by counterparty is as follows:
(1)At this point, the counterparty has not defaulted and will compensate the buyer. Because two counterparties share the risk, according to Corollaries 2 and 3, the discounted value of the fees counterparty pays to the buyer is as follows: where is the bond recovery rate.(2)When the counterparty defaults, counterparty will not pay compensation to the buyer.Based on the no-arbitrage principle, the fees received by the counterparty should be equal to the cost of compensation:
To get
Similarly, suppose that the buyer pays for , obtained as follows:
5. Numerical Analysis
In this section, we will show the impacts of the main parameters on the basket CDS price. Before this, we first use Monte Carlo simulation to verify the validity of the formula. Because CDS price and have the similar structure, we mainly take as the analysis object in this section. We can see from the structure of that if we want to get , we need to get the associated probability density , , and marginal probability distribution . Here, we use Monte Carlo simulation to get the values of all three. In Section 2, we assume that the default intensities of reference assets follow the Vasicek model. Here, we use the following formula to simulate their default intensities:where is the standard normal distribution. By simulating the default intensity, we can calculate the relative probability density and get the price of CDS. In Monte Carlo simulation, we set the time step as 100 and the number of simulations as 10000. From Table 1, we can see that the difference between simulation results and formula derivation results is less than 1%. Therefore, the formula is valid for calculating CDS prices. Similarly, the validity of can be proved.
We compare the two counterparties model with the single counterparty model in the following numerical analysis. Here, we set to get a simplified single counterparty model. We use the following parameters in the numerical analysis, .
Figure 1 shows the impact of different reference assets quantities on basket CDS price. We set . From Figure 1, it can be seen that basket CDS price first rose rapidly and then leveled off. Because when the amount of assets in the basket increases, the default event is more likely to occur, making the default probability of reference assets greater. When is large enough, adding only causes the default probability to approach 1 infinitely. So, as the probability of default increases, the CDS buyer should pay more for the contract. At the same time, we can see that there is no significant difference between the CDS price of the single counterparty model and the two counterparties model at the beginning. However, as increases, the difference becomes apparent. The reason is that the more reference assets, the more significant the risk sharing effect. It also suggests that the two counterparties do act to share the risk of default, reducing the cost of CDS.
Figure 2 shows the impact of initial default intensity on basket CDS price. We set . We can see from Figure 2 that basket CDS price increase as the initial default intensity increases. At the same time, it can be seen that, similar to the impact of the reference assets quantities on the CDS price, the price of basket CDS first rose rapidly and then stabilized. The difference is that default intensity has a more direct impact on CDS price, so you can see Figure 2 flattening out more quickly than Figure 1. The size of the initial default intensity directly reflects the willingness of CDS buyers to pay fees. The CDS price on the single counterparty model is higher than on the two counterparties model, further evidence that risk is being shared.
Figure 3 shows the impact of recovery rates on basket CDS price. We set . It can be seen from Figure 3 that CDS price rapidly decreases to 0 (when R = 1) with the increase of . The CDS price of the single counterparty model is higher than that of the two counterparties model at the beginning, and then gradually overlap. The reason is that the recovery rate , reflects the likely losses of the buyer of the CDS. The greater the recovery rate, the smaller the loss. With the increase of , the default risk continues to decrease, resulting in a smaller share of risks and a smaller gap between the two models.
Figure 4 shows the impact of correlation coefficient between reference assets on basket CDS price. We set . It can be seen from the Figure 4 that the basket CDS price decreases with the increase of correlation coefficient. Because as the correlation coefficient increases, a change in one company in the reference assets may cause another company to default, making CDS buyers less willing to pay more for the contract. At the same time, the price of single counterparty is higher than that of two counterparties because the risk is better shared between the two counterparties as the probability of default on the reference assets increases.
6. Conclusion
Based on the reduced model, a basket of reference assets and two default counterparties are established in this study. In the model, we assume that the default intensity of reference assets and macro variables (interest rates) follow the Vasicek model. The event of default is contagious between two counterparties. In Section 3, we use the PDE method to obtain an approximate closed solution of the conditional survival probability density with and without default of the reference asset, and derive the marginal conditional probability distribution of the counterparty default. In Section 4, we construct several scenarios and obtain the CDS price of the model based on the principle of no arbitrage. In Section 5, we verify the validity of the price formula through Monte Carlo simulation, and analyze the influence of main parameters on the price formula. We also construct a simplified single counterparty model and compare it with our model to get relevant conclusions.
Data Availability
Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research is supported by the National Natural Science Foundation of China (grant nos. 62261049 and 12261080), Natural Science Foundation of Gansu Province (grant no. 20JR10RA085), and Higher Education Innovation Fund of Gansu Provincial Department of Education (grant no. 2022A-017).