A Lognormal Type Stochastic Volatility Model With Quadratic Drift

Abstract

This paper presents a novel one-factor stochastic volatility model where the instantaneous volatility of the asset log-return is a diffusion with a quadratic drift and a linear dispersion function. The instantaneous volatility mean reverts around a constant level, with a speed of mean reversion that is affine in the instantaneous volatility level. The steady-state distribution of the instantaneous volatility belongs to the class of Generalized Inverse Gaussian distributions. We show that the quadratic term in the drift is crucial to avoid moment explosions and to preserve the martingale property of the stock price process. Using a conveniently chosen change of measure, we relate the model to the class of polynomial diffusions. This remarkable relation allows us to develop a highly accurate option price approximation technique based on orthogonal polynomial expansions. * We thank Damien Ackerer for helpful comments. † New York University, Tandon School of Engineering ‡É cole Polytechnique Fédérale de Lausanne (EPFL) and Swiss Finance Institute 1 With lognormal type of diffusion we mean a diffusion σt with d[σ, σ]t = ν 2 σ 2 t dt, for some ν > 0. 2 See Christoffersen et al. (2010) for the S&P500 index, Andersen et al. (2001) for individual stocks in the DJIA index, and Andersen et al. (2001) for foreign exchange markets.