Efficient algorithms for basket default swap pricing with multivariate Archimedean copulas
Introduction
Modeling dependence structure of credit risks is a crucial problem for pricing portfolio credit derivatives. Judging by the collapse of many financial institutions in the global financial crisis that began in 2007, we may conclude that credit risk modeling has failed. For example, the Gaussian copula-based model ignores fat tails, and the one factor model does not consider the correlation risk among assets belonging to different industry sectors or subportfolios. They can underestimate the price of portfolio credit derivatives. To capture the richer and more realistic correlation structures in multiple default risks, we adopt two kinds of multivariate Archimedean copulas: the exchangeable Archimedean copula and the nested Gumbel copula. In particular, by using the nested copula we specify the multilevel dependence structure of the underlying portfolio composed of several subportfolios.
Our main contribution is summarized as follows: We develop the importance sampling algorithms for efficient evaluation of basket default swaps whose dependence structures are based on the factor copula model. Algorithm 3 employs exchangeable Archimedean copulas, and Algorithm 4 employs nested Gumbel copulas. Our algorithms for basket default swap pricing produce remarkably small variances compared to the crude Monte Carlo simulation. Furthermore, the performance becomes better as the relative frequency of the desired event becomes smaller.
The investors of basket default swaps are exposed to correlation risk which is the tendency of simultaneous defaults in a portfolio. Basket default swaps provide insurance to the protection buyer who wants to remove credit risks from the protection seller who wants to take the credit risks. In the th-to-default swap, the protection buyer is compensated for loss amount of the th defaulted asset but not for any subsequent defaulted assets, so its contingent claim depends on time to the th defaulted event. Default correlation of securities in the underling portfolio influences the price of basket default swaps.
Copula functions are defined by multivariate distribution functions of standard uniform random variables. Using a Gaussian copula function Li (2000) firstly introduced a mathematical model to build a multivariate distribution of survival times for the valuation of portfolio credit derivatives. Many researchers have introduced analytical approaches based on copula functions. Andersen et al. (2003) suggested a recursion technique to derive the loss distribution. Hull and White (2004) proposed another recursion method for valuation of th-to-default-swaps. Gregory and Laurent (2005) provided a semi-analytic form for the loss distributions using the fast Fourier transform and probability generating function. Frey and McNeil (2003) modified the Gaussian copula model, Mashal and Naldi (2002) used student -copulas, and Schönbucher and Schubert (2001) used Archimedean copulas. Choe and Jang (in print) derived the th default time distribution using order statistic analysis based on a one factor copula model.
Multivariate Archimedean copulas provide plenty of dependence structures. Nested Archimedean copulas are constructed by combining two or more Archimedean copulas with another Archimedean copula. They provide a model for multi-level dependence structures such as dependence of dependence. For instance, an underlying portfolio of a credit derivative may be divided into subportfolios by criteria such as credit ratings, industrial sectors or regional sectors. Assets in a subportfolio are usually more strongly correlated than assets belonging to different subportfolios. Moreover, there exist different dependence structures for each subportfolio. By building several layered Archimedean copulas, we provide a more intuitive and flexible model for default correlation and overcome the drawback of an exchangeable property of other copulas. For the application to finance, Savu and Trede (2006) showed the density of a nested Archimedean copula and applied this to asset returns. Hofert and Scherer (2008) proposed the algorithms for the calibration of pricing of portfolio credit derivatives.
Monte Carlo simulation requires an efficient scheme for rare events such as default occurrence of highly credible bonds or contracts with short maturity. Importance sampling is a powerful variance reduction technique for improving the efficiency of extreme events in Monte Carlo simulations. In pricing th-to-default swaps, only the samples of the th default before or at contribute to the contract payoff. Joshi and Kainth (2004) introduced an importance sampling algorithm which forces all paths to produce a predetermined number of defaults. Their method depends on the order in which the default times are generated, which was improved by Chen and Glasserman (2008). Chiang et al. (2007) introduced a new algorithm for importance sampling.
This paper is organized as follows: Section 2 provides sampling algorithms for exchangeable Archimedean copulas and nested Archimedean copulas. Section 3 presents analytical approaches for the valuation of basket default swaps and reviews the importance sampling for the th-to-default swaps. Section 4 introduces an importance sampling procedure in Algorithm 3 employing exchangeable Archimedean copulas. Section 5 introduce an another importance sampling procedure in Algorithm 4 employing nested Gumbel copulas. Section 6 illustrate the numerical results for th-to-default swaps produced by our proposed estimators, which are compared with the crude Monte Carlo simulation to justify efficiency and accuracy. Section 7 concludes the paper.
Section snippets
Sampling of Archimedean copulas
The Laplace transform can be used in sampling multivariate Archimedean copulas, which was recognized by Marshall and Olkin (1988). For more information see Joe (1997), McNeil et al. (2005) and Whelan (2004). McNeil (2008) extended the works in Joe (1997) to the case of nested Archimedean copulas.
Basket default swap pricing
In this section we give a brief review of the valuation of th-to-default swaps under the dependence assumption of underlying default times, and describe an importance sampling procedure for pricing them.
Importance sampling for Archimedean copulas
In order to select an appropriate new probability and a likelihood ratio for in Lemma 4, the implied distribution of has to be specified. Instead of computing the distribution analytically, we build an alternative event which has the same probability as .
We establish new algorithms for importance sampling in Monte Carlo simulation to compute the basket default swaps where the dependence structures of are given by exchangeable Archimedean copulas. The following
Importance sampling for nested Gumbel copulas
As the construction of the underlying portfolios of credit derivatives, we choose the partially nested structure in Eq. (6) with Gumbel families because it is intuitive to capture the hierarchical structure in the underlying portfolio.
If the assets in the underlying portfolio are composed by different industrial sectors or credit ratings, then we consider several subportfolios in the underlying portfolio. Correlation structures of assets in the same subportfolio are modeled by inner copulas
Numerical simulations
In this section, we demonstrate the accuracy and the efficiency of the theoretical results. Example 8, Example 11 show the results in Proposition 5, which are and for any maturities, and also converges to 1 as the maturity goes to infinity.
Secondly, Example 9, Example 12 show that the efficiency of our proposed estimators increases as the ranking of default increases. These results indicate our algorithms are more efficient when the frequency of the th default event becomes
Concluding remarks
In this paper, we proposed algorithms for importance sampling to price basket default swaps by employing the exchangeable Archimedean copula and nested Gumbel copulas. Applying the hierarchical property of the nested copula, we establish the dependence structure to inner copulas and an outer copula of the underlying portfolio which is composed of several subportfolios.
Using sampling methods in Marshall and Olkin (1988) and Joe (1997) for exchangeable and nested Archimedean copulas, we build new
Acknowledgements
The authors wish to thank the referees for valuable suggestions.
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