Multiscale Volatility Model Research Articles

The Correction of Multiscale Stochastic Volatility to American Put Option: An Asymptotic Approximation and Finite Difference Approach

It has been found that the surface of implied volatility has appeared in financial market embrace volatility “Smile” and volatility “Smirk” through the long-term observation. Compared to the conventional Black-Scholes option pricing models, it has been proved to provide more accurate results by stochastic volatility model in terms of the implied volatility, while the classic stochastic volatility model fails to capture the term structure phenomenon of volatility “Smirk.” More attempts have been made to correct for American put option price with incorporating a fast-scale stochastic volatility and a slow-scale stochastic volatility in this paper. Given that the combination in the process of multiscale volatility may lead to a high-dimensional differential equation, an asymptotic approximation method is employed to reduce the dimension in this paper. The numerical results of finite difference show that the multiscale volatility model can offer accurate explanations of the behavior of American put option price.

Correlated Log-Normal Random Variables under a Multiscale Volatility Model

The study focuses on extending the fast mean-reversion volatility, which was developed by the author in a previous work, to the multiscale volatility model so that it can express a well-separated time scale. The leading-order term and first-order correction terms are analytically computed using the perturbation theory based on the Lie–Trotter operator splitting method. Finally, the study is concluded by deriving the numerical results that further validate the effectiveness of the model.

Analytic approximation of the transition density function under a multi-scale volatility model

The transition density function plays an important role in understanding and explaining the dynamics of the stochastic process. We propose an approach which can be used for the analytic approximation of the transition density related to a multi-scale stochastic volatility model. Using perturbation theory, we compute the leading-order term and the first-order correction terms. A numerical test also confirms the effectiveness of the model.

Gaussian copula under multiscale volatility

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.