Perturbation expansion for option pricing with stochastic volatility

Petr Jizba, Hagen Kleinert and Patrick Haener

Physica A: Statistical Mechanics and its Applications, 2009, vol. 388, issue 17, 3503-3520

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 Δ-hedging strategy is close to being optimal.

Keywords: Black–Scholes formula; Volatility; Gamma distribution; Mellin transform (search for similar items in EconPapers)
Date: 2009
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Citations: View citations in EconPapers (5)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:388:y:2009:i:17:p:3503-3520

DOI: 10.1016/j.physa.2009.04.027

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