Option smiles, optimally hedged Monte-Carlo and historical option pricing
Dr Jean-Philippe Bouchaud, Service the Physique de l'Etat Condense', CEA Saclay and Science & Finance, Paris

Abstract:
Price fluctuations are well known to be non Gaussian, with fat tails, long range volatility correlations and return-volatility correlations. This has the consequence of making real world options smiled. We explain intuitively how the characteristics of the smile (curvature, skewness) are related to these statistical anomalies. We discuss then a `hedged' Monte-Carlo method to price financial derivatives, which allows to determine simultaneously the optimal hedge, that takes into account all the non trivial features of price changes. The explicit accounting of the hedging cost naturally converts the objective probability into the `risk-neutral' one. This allows a consistent use of purely historical time series to price derivatives and obtain their residual risk.