Credit Derivatives Pricing With The Copula Method

Credit derivatives has developed rapidly since its birth. As one of the main types of credit derivatives, Collateralised Debt Obligation (CDO) once occupied a large share of the market. However after the financial crisis in2008, credit derivatives which once had been traded very actively have been serious questioned. CDO also had a sharp drop in trading volume and almost disappeared, and until recent years, the situation has been better. As a main method for CDO pricing, copula approach has been discussed a lot by scholars. Especially after the financial crisis, the problems arising by traditional copula approach has attracted a lot of attention. Under this background, this paper give some research on the following questions.First we consider the problem for CDO pricing under the α-stable distribution. The financial crisis show that the tail of the traditional Gaussian copula is not heavy enough, which leads to underestimate the extreme events of default. So we choose the a-stable copula to create a copula model which fit the empirical market quote better, because of its heavy-tailed property and the freedom for the parameters. Moreover recovery rates tend to be inversely related to default rates in reality. So we also consider the situation that the recovery rate is stochastic. According to the comparison, the performance of the α-stable copula is indeed better than other copulas.Second we research the problem for CDO pricing with the dynamic copula. In most of the literature, CDO pricing is in a static copula model. So it can not express the time-varying characteristic of CDO. Therefore, we assume the recovery rate and the default rate are dynamic correlated, and then give the CDO pricing model with this dynamic copula. Because the conditional Chebyshev Law of Large Number only makes an estimate of the loss distribution, but can not tell us how much the error is. So we introduce the the saddle-point approximation approach to estimate the loss distribution and also the errors.In the end, we made a brief introduction to the implied copula. Different from the general copula approaches, instead of specifying a copula, we get the default probability directly from the market quotes, and there is a copula corresponding to it. However, from the perspective of model implementation there is no need to determine what this copula is. So we call it implied copula. In this paper, we introduce the method that how to use this implied copula to price the CDO, and give some cases that this method may be extended. We hope we can keep on researching it in the future.