Perturbed Gaussian Copula

Gaussian copula is by far the most popular copula used in the financial industry in default dependency modeling. However, it has a major drawback — it does not exhibit tail dependence, a very important property for copula. The essence of tail dependence is the interdependence when extreme events occur, say, defaults of corporate bonds. In this paper we show that some tail dependence can be restored by introducing stochastic volatility on a Gaussian copula. Using perturbation methods we then derive an approximate copula — called perturbed Gaussian copula in this paper. A copula is a joint distribution function of uniform random variables. Sklar’s Theorem states that for any multivariate distribution, the univariate marginal distributions and the dependence structure can be separated. The dependence structure is completely determined by the copula. It then implies that one can “borrow” the dependence structure, namely the copula, of one set of dependent random variables and exchange the marginal distributions for a totally different set of marginal distributions. An important property of copula is its invariance under monotonic transformation. More precisely, if gi is strictly increasing for each i, then (g1(X1), g2(X2), . . . , gn(Xn)) have the same copula as (X1,X2, . . . ,Xn). From the above discussion, it is not hard to see that copula comes in default dependency modeling very naturally. For a much detailed coverage on copula, including the precise format of Sklar’s Theorem, as well as modeling default dependency by way of copula, the readers are referred to Schonbucher (2003) [5]. Let (Z1, . . . , Zn) be a normal random vector with standard normal marginals and correlation matrix R, and Φ(·) be the standard normal cumulative distribution function. Then the joint distribution function of (Φ(Z1), . . . ,Φ(Zn)) is called the Gaussian copula with correlation matrix R. Work supported by NSF grant DMS-0455982 Department of Statistics and Applied Probability, University of California, Santa Barbara, CA 93106-3110, [email protected]. Mathematics Department, North Carolina State University, Raleigh, NC 27695, [email protected]. 1 Gaussian copula is by far the most popular copula used in the financial industry in default dependency modeling. This is basically because of two reasons. Firstly it is easy to simulate. Secondly it requires the “right” number of parameters — equal to the number of correlation coefficients among the underlying names. However, Gaussian copula does not exhibit any tail dependence, a very important property for copula. The essence of tail dependence is the interdependence when extreme events occur, say, defaults of corporate bonds. In fact, this is considered as a major drawback of Gaussian copula. On the other hand, by introducing stochastic volatility into the classic Black-Scholes model, Fouque, Papanicolaou and Sircar (2000) [1], by way of singular perturbation method, gave a satisfactory answer to the “smile curve” problem of implied volatilities in the financial market, leading to a pricing formula which is in the form of a robust simple correction to the classic Black-Scholes constant volatility formula. Furthermore, an application of this perturbation method to defaultable bond pricing has been studied by Fouque, Sircar and Solna (2005) [3]. By fitting real market data, they concluded that the method works fairly well. An extension to multi-name first passage models is proposed by Fouque, Wignall and Zhou (2006) [4]. In this paper we will show the effect of stochastic volatility on a Gaussian copula. Specifically, in Section 1, we first set up the stochastic volatility model and state out the objective — the transition density functions. Then by singular perturbation, we obtain approximate transition density functions. In order to make them true probability density functions, we introduce the transformation 1 + tanh(·). In Section 2, we study this new class of approximate copula density functions, first analytically and then numerically. Section 3 concludes this paper.