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