Share returns distribution: empirical observations and implications for options pricing

Underlying asset returns regularly depart from normal. In a Black-Scholes economy, asset prices are assumed lognormal and subsequently returns are normal. The literature regarding asset return behavior is extensive. In addition, the number of options pricing models, which take into account non-normality, discontinuities, and stochastic variables, are also extensive. These models are an effort at reconciling real world option prices with the assumptions in the Black-Scholes paradigm. This article contains a review of the literature regarding asset returns; alternative option pricing parameterizations, and recovering the implied risk-neutral distribution from listed options. Following this, theory takes precedence as a simple plain-vanilla option-pricing model, which incorporates the first-four moments of the risk-neutral density is explored. In the results section, S&P 500 Index market daily log returns are explored and univariate properties lead to a rejection of the hypothesis of normality. Then, using the simple four-moment model and skewness and kurtosis values consistent with those recovered from listed options by numerous researchers we imply the risk neutral densities associated with actual option prices on the S&P 500 Index. The implications are profound. It is very clear that the implied risk neutral densities are significantly different from the normal distribution which forms the basis for the Black-Scholes model.