Gaussian copula under multiscale volatility

Abstract

AbstractA copula is a multidimensional joint distribution function on random variables with uniform distribution. Gaussian copula is a special kind of copula that is widely applied in the study of financial markets. Although it is a very popular tool, Gaussian copula has an obvious shortcoming—it cannot show the tail dependence, which is an important property of the copula. The essence of tail dependence is the interdependence when the extreme events occur. Fouque and Zhou (Perturbed Gaussian copula. 2006; submitted) have discussed perturbed Gaussian copula under one stochastic volatility factor with a fast scale, and they used singular perturbation techniques to derive an approximate copula. Our work can be treated as the generalization and extension of the work of Fouque and Zhou (Perturbed Gaussian copula. 2006; submitted). In our paper, we will investigate the problem where the volatilities of goal variables are driven by two random variables, namely, it is a multiscale volatility model. In the model, we use a fast mean reverting Ornstein–Uhlenbeck process to depict the first scale, and a slow varying diffusion process to depict the second scale. We also use the regular and singular perturbation methods developed in the work of Fouque et al. (Multiscale Model. Simul. 2003; 2(1):22–42) to derive the approximate copula. We show that some tail dependence can be restored by Gaussian copula under multiscale stochastic volatility. Copyright © 2008 John Wiley & Sons, Ltd.