Martingale expansion for stochastic volatility

Paper: arXiv 2601.09324

Authors: Masaaki Fukasawa

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.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a strong theoretical contribution by developing a direct martingale expansion technique for continuous stochastic volatility models under minimal conditions, bypassing heavy Malliavin calculus. However, it is entirely theoretical with no empirical data, backtesting, or numerical implementation to validate the approximations.

Research Flowchart

  flowchart TD
    A["Research Goal: Refine marginal distribution approximations for continuous stochastic volatility models"] --> B["Inputs: Small vol-of-vol & fast mean-reversion parameters"]
    B --> C["Methodology: Develop Martingale Expansion beyond Central Limit Theorem"]
    C --> D["Computational Process: Derive first-order perturbation expansions under minimal conditions"]
    D --> E["Key Finding: Refined marginal distribution approximations for SV models"]