Skewness and kurtosis in pricing European and American options

This paper introduces models for pricing of European options when the future asset price structure is governed by a non-normal probability structure characterized by a mean, standard deviation (volatility), skewness, and kurtosis. The non-normal probability structure is derived from a normal probability structure in a particularly convenient way, the normal powers asymptotic expansion. Conditions on skewness and kurtosis are provided for the applicability of the pricing model. The pricing model reduces as a special case to the classical pricing model, i.e., skewness and kurtosis both zero. Prices are determined by simulation. We derive the differential characteristics of the option prices including the rates of change of the option with respect to skewness and kurtosis. Problems associated with estimation of implied skewness and kurtosis are discussed. Numerical difficulties with simultaneous estimating of implied volatility, skewness, and kurtosis suggest that implied skewness and kurtosis are very dynamic or are unstable. We provide a number of examples, graphical representations and applications. The behavior of the skewness-kurtosis pricing models is contrasted with the classical model. Graphical pricing methods are provided and discussed