Martingale expansion for stochastic volatility

TL;DR

This paper develops a martingale expansion framework tailored for continuous stochastic volatility models, providing refined distribution approximations that improve upon the normal approximation in financial modeling.

Contribution

It introduces a novel martingale expansion method applicable to stochastic volatility models with minimal assumptions, covering small volatility-of-volatility and fast mean-reversion scenarios.

Findings

Provides first-order perturbation expansions for stochastic volatility models.

Applicable to models with small volatility-of-volatility and fast mean-reversion.

Enhances distribution approximation accuracy beyond the central limit theorem.

Abstract

The martingale expansion provides a refined approximation to the marginal distributions of martingales beyond the normal approximation implied by the martingale central limit theorem. We develop a martingale expansion framework specifically suited to continuous stochastic volatility models. Our approach accommodates both small volatility-of-volatility and fast mean-reversion models, yielding first-order perturbation expansions under essentially minimal conditions.