A Wiener Chaos Approach to Martingale Modelling and Implied Volatility Calibration

TL;DR

The paper introduces a Wiener chaos martingale model that represents the discounted asset price under a risk-neutral measure via a truncated Wiener chaos expansion, yielding a universal and arbitrage-free framework. Calibration to the implied-volatility surface is achieved by learning the chaos coefficients, with efficient pricing via Monte Carlo or quadrature methods and variance-reduction techniques. The model demonstrates strong fit to Heston and rough Heston surfaces and shows robust out-of-sample pricing for path-dependent options, as well as good performance on real SPX data. The approach provides a flexible, time-consistent alternative to parametric models, capable of capturing a wide range of dynamics while enabling scalable calibration and hedging in practice.

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.

Figures (9)

• Figure 1: Implied-volatility surfaces at the calibrated maturities for the Heston model (top left) and the corresponding absolute-error surfaces (top right), obtained with the piecewise-constant and Legendre bases. The bottom panel shows the implied-volatility smiles. The MAE is 7.23 bp for the piecewise-constant basis and 8.80 bp for the Legendre basis.
• Figure 2: Implied-volatility surfaces at the non-calibrated maturities for the Heston model (top left) and the corresponding absolute-error surfaces (top right), obtained with the piecewise-constant and Legendre bases. The bottom panel shows the implied-volatility smiles. The MAE is 16.62 bp for the piecewise-constant basis and 34.95 bp for the Legendre basis.
• Figure 3: Forward starting call option comparison, price (top) and Implied Volatility (bottom).
• Figure 4: Down-and-Out call option comparison, price (top) and Implied Volatility (bottom).
• Figure 5: Floating strike lookback call options comparison, price (top) and Implied Volatility (bottom).

Theorems & Definitions (17)

• Theorem 2.1
• Theorem 2.2
• Definition 2.3
• Definition 3.1: Wiener chaos martingale model
• Remark 3.2
• Proposition 3.3
• proof
• Remark 3.4
• Proposition 3.5
• proof