A lognormal type stochastic volatility model with quadratic drift

TL;DR

This paper introduces a new one-factor stochastic volatility model with quadratic drift, ensuring mean reversion and martingale property, and develops an accurate option pricing method using polynomial expansions.

Contribution

The paper proposes a novel stochastic volatility model with quadratic drift and connects it to polynomial diffusions for improved option pricing accuracy.

Findings

Steady-state volatility follows a Generalized Inverse Gaussian distribution.

Quadratic drift prevents moment explosions and maintains martingale property.

Polynomial expansion method yields highly accurate option prices.

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.