  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.
