Perturbation Expansion for Option Pricing with Stochastic Volatility

TL;DR

This paper develops a perturbation expansion for option pricing under stochastic volatility modeled by a superposition of Gaussian distributions, providing explicit formulas and demonstrating near-optimal delta-hedging for specific market data.

Contribution

It introduces a novel perturbation expansion around Black-Scholes for options with stochastic volatility modeled by a superposition of Gaussians, incorporating non-Gaussian effects.

Findings

Derived an explicit analytic option pricing formula from the superposition model.

Showed the perturbation expansion accounts for non-Gaussian fluctuations via kurtosis.

Demonstrated near-optimal delta-hedging for Dow Jones Euro Stoxx 50 options.

Abstract

We fit the volatility fluctuations of the S&P 500 index well by a Chi distribution, and the distribution of log-returns by a corresponding superposition of Gaussian distributions. The Fourier transform of this is, remarkably, of the Tsallis type. An option pricing formula is derived from the same superposition of Black-Scholes expressions. An explicit analytic formula is deduced from a perturbation expansion around a Black-Scholes formula with the mean volatility. The expansion has two parts. The first takes into account the non-Gaussian character of the stock-fluctuations and is organized by powers of the excess kurtosis, the second is contract based, and is organized by the moments of moneyness of the option. With this expansion we show that for the Dow Jones Euro Stoxx 50 option data, a Delta-hedging strategy is close to being optimal.