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We revisit the problem of pricing options with historical volatilityestimators. We do this in the context of a generalized GARCH model withmultiple time scales and asymmetry. It is argued that the reason for theobserved volatility risk premium is tail risk aversion. We parametrize suchrisk aversion in terms of three coefficients: convexity, skew and kurtosis riskpremium. We propose that option prices under the real-world measure are notmartingales, but that their drift is governed by such tail risk premia. We thenderive a fair-pricing equation for options and show that the solutions can bewritten in terms of a stochastic volatility model in continuous time and undera martingale probability measure. This gives a precise connection between thepricing and real-world probability measures, which cannot be obtained usingGirsanov Theorem. We find that the convexity risk premium, not only shifts theoverall implied volatility level, but also changes its term structure.Moreover, the skew risk premium makes the skewness of the volatility smilesteeper than a pure historical estimate. We derive analytical formulas forcertain implied moments using the Bergomi-Guyon expansion. This allows for veryfast calibrations of the models. We show examples of a particular model whichcan reproduce the observed SPX volatility surface using very few parameters.

Option Pricing, Historical Volatility and Tail Risks