Pricing credit derivatives with a copula-based actuarial model for credit risk

Credit derivatives are financial contracts whose pay-off are contingent on the creditworthiness of some counterparts. As was pointed out in some recent works (Mashal & Naldi (2002), Meneguzzo & Vecchiato (2002)), they have become in recent years the main tool for transferring and hedging credit risk. The most complicated of such instruments are the multinames ones. Indeed, these instruments are not quoted (market prices are not available). Besides, we do not posses closed forms for their pricing: we must necessarily set up a Monte Carlo simulation procedure. The key to perform this task consists in modelling correctly multiple defaults. A dependence structure using copulas methods was first set up by Li (2000). In this paper, Li considers time-until-default for each obligor and model their dependence structure through a Student t-copula. Other papers which take into account a copula dependence structure are due to Cherubini & Luciano (2002, 2004), Galiani (2003), Gregory & Laurent (2002), Li describes a default for a single obligor through the so-called survival function S(t) " Pr T # t! which represents the probability that this counterpart attains age t and is the time-until-default. Li also assumes that the hazard rate function is constant, . This means that the survival time is exponentially distributed with constant parameter . Other features of this model are the following: credit migrations at the end of the time horizon were not taken into account and recovery rates in default situations are assumed deterministic. This model has been resumed by Mashal & Naldi with the intent to price particular multinames credit derivatives such as nth-to-default baskets. Their model is a hybrid of the well-known structural and reduced form approaches for modelling defaults. After simulating a large number of multivariate times-until-default, one deduces pay-off for our derivative. Finally, the pricing is estimated using standard risk-neutral pricing technology (by assuming complete markets and no-arbitrage hypothesis). The credit risk model for the underlying portfolio, already developed in Masala, Menzietti & Micocci (2004), follows a general credit risk framework: hazard rates are random variables whose values follow gamma distributions coherently with Credit Risk Plus (1997), Micocci (2000), Burgisser, Kurth & Wagner (2001) and Menzietti (2002); recovery rates themselves are supposed to be stochastic as in Gupton, Finger & Bathia (1997), and following a Beta distribution, moreover credit migrations are allowed. This feature becomes very important when we treat credit derivatives whose payoff depends on credit spread. The paper is structured as follows. Section 2 presents the model for default and credit migration; the section is divided in subsections facing the problems of time-until-default, the hazard rate function and the recovery rates, the credit migration and the exposure valuation, the loss distribution. Section 3 introduces some basket credit derivatives with numerical applications. Section 4 concludes