Calibration of Stochastic Volatility Models: A Tikhonov Regularization Approach

Abstract

We aim to calibrate stochastic volatility models from option prices. We develop a Tikhonov regularization approach to recover the risk ne-utral drift term and diffusion term of the volatility (or variance) process which are presumed to be a deterministic function of instantaneous volatility (or v-ariance) and time, as well as the correlation coefficient as a function of time. In contrast to the most existing literature, we do not assume that these terms in the general stochastic volatility model have special structures. A modified Dupire's equation associated with stochastic volatility models is first proposed, which allows us to formulate the calibration problem as a standard inverse problem of partial differential equations. Then a Tikhonov regularization method can be used to recover these three terms respectively. The necessary condition that the optimal solution satisfies is derived for each term. We further simplified these necessary conditions. Then we propose a gradient descent algorithm to numerically find the optimal solution. Our algorithm can be applied to calibrate general stochastic volatility models. The numerical test and market test are presented to demonstrate the efficiency of our numerical algorithm.