A Wiener Chaos Approach to Martingale Modelling and Implied Volatility Calibration

TL;DR

This paper introduces a flexible martingale model for asset prices using Wiener chaos expansion, enabling efficient calibration to implied volatility surfaces with demonstrated effectiveness on market data.

Contribution

It develops a novel Wiener chaos-based martingale model that simplifies calibration to implied volatility surfaces, combining theoretical formulation with practical implementation.

Findings

Model achieves fast calibration to implied volatility surfaces.

Numerical experiments show high flexibility and accuracy.

Effective on both simulated and real market data.

Abstract

Calibration to a surface of option prices requires specifying a suitably flexible martingale model for the discounted asset price under a risk-neutral measure. Assuming Brownian noise and mean-square integrability, we construct an over-parameterized model based on the martingale representation theorem. In particular, we approximate the terminal value of the martingale via a truncated Wiener--chaos expansion and recover the intermediate dynamics by computing the corresponding conditional expectations. Using the Hermite-polynomial formulation of the Wiener chaos, we obtain easily implementable expressions that enable fast calibration to a target implied-volatility surface. We illustrate the flexibility and expressive power of the resulting model through numerical experiments on both simulated and real market data.