Modelling option-implied return distributions: A generalized log-logistic approximation

We propose a method to uncover risk-neutral return distributions of financial assets from option prices. The goal is to account for implied Black & Scholes volatility smiles by manipulating the form of the underlying asset's distribution. We start from a quasi-binomial option pricing model, in which an option's payoff is evaluated at the extremes of the expected range of the future price of the underlying asset. This quasi-binomial approach implies that the lognormal distribution is approximated by a suitable transformation of the log-logistic distribution. By generalizing the transformation of the log-logistic distribution, skewness and kurtosis effects can be incorporated. We show how option prices change relative to Black & Scholes prices when skewness and kurtosis effects are introduced.