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Martingale expansion for stochastic volatility

Masaaki Fukasawa

TL;DR

This work develops a martingale expansion framework for continuous stochastic volatility models, yielding a rigorous first‑order perturbation expansion in a small parameter $\epsilon$ under minimal conditions and without relying on Malliavin calculus. By introducing and analyzing the pair $(X^\epsilon, Y^\epsilon)$, the authors show that the distribution of the log-price increment remains asymptotically normal, while a computable correction term involving $\mathsf{E}[Y|X=x]$ captures deviations from Black–Scholes. The main result provides an explicit expansion for option payoffs and for the implied total variance, linking the correction to conditional moments of the volatility process. An illustrative Bergomi-type example demonstrates applicability to practical SV models and yields a tractable expression for the skew via $\mathsf{E}[XY]$. Overall, the framework offers a direct, elementary route to first‑order SV corrections with potential impact on pricing and risk management in markets with stochastic volatility.

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.

Martingale expansion for stochastic volatility

TL;DR

This work develops a martingale expansion framework for continuous stochastic volatility models, yielding a rigorous first‑order perturbation expansion in a small parameter under minimal conditions and without relying on Malliavin calculus. By introducing and analyzing the pair , the authors show that the distribution of the log-price increment remains asymptotically normal, while a computable correction term involving captures deviations from Black–Scholes. The main result provides an explicit expansion for option payoffs and for the implied total variance, linking the correction to conditional moments of the volatility process. An illustrative Bergomi-type example demonstrates applicability to practical SV models and yields a tractable expression for the skew via . Overall, the framework offers a direct, elementary route to first‑order SV corrections with potential impact on pricing and risk management in markets with stochastic volatility.

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.
Paper Structure (6 sections, 2 theorems, 61 equations)

This paper contains 6 sections, 2 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Example
  4. Proofs
  5. Proof of Theorem \ref{['thm']}
  6. Proof of Corollary \ref{['cor1']}

Key Result

Theorem 1

If $Y^\epsilon$ is uniformly integrable and if $(X^\epsilon,Y^\epsilon)$ converges in law to such $(X,Y)$ that $x \mapsto \mathsf{E}[Y|X=x]$ is twice continuously differentiable with as $\epsilon \to 0$, then for any bounded Borel function $f$, where and $\phi$ is the standard normal density function.

Theorems & Definitions (6)

  • Theorem 1
  • Remark 1
  • Corollary 1
  • Remark 2
  • Remark 3
  • Remark 4