| Since the volatility fluctuations of the S&P 500 index is fitted well by a Gamma distribution, the distribution of the returns is a corresponding superposition of Gaussian distributions. The Fourier transform of this has, remarkably, the Tsallis form. The option pricing formula is derives from the same superposition of Black-Scholes expressions. For this superposition I derive a perturbation expansion around the Black-Scholes formula with the excess kurtosis and moneyness as expansion parameters. |
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