Perturbation Expansion for Option Pricing with Stochastic Volatility

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.

33 pages, 13 figures, LaTeX4

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Cited by in corpus (8)

• Renyi's information transfer between financial time series
• Superstatistical fluctuations in time series: Applications to share-price dynamics and turbulence
• Superpositions of Probability Distributions
• Superstatistics approach to path integral for a relativistic particle
• Tsallis thermostatics as a statistical physics of random chains
• Transitions between superstatistical regimes: validity, breakdown and applications
• Series representation of the pricing formula for the European option driven by space-time fractional diffusion
• A contribution to the systematics of stochastic volatility models