Valuing European Put Options under Skewness and Increasing [Excess] Kurtosis

Abstract

To capture the impact of skewness and increase kurtosis on Black’s [1] European put values, we first substitute a Gram-Charlier (GC) distribution and next a Johnson distribution for Black’s Gaussian one. We introduce next each distribution in the option payoff and develop until the closedform expression of each put is arrived at. Finally, we estimate by simulations GC, Johnson and Black put options, choosing the latter one as benchmark. Simulation estimates encompassing both skewness and kurtosis show that, for at-the-money (ATM) or slightly in-the-money put values, 1) Black’s overvaluation with respect to Johnson puts is very significant and 2) its undervaluation with respect to GC ones remains moderate. Yet, by using the same skewness values for both GC and Johnson puts, we highlight the differences induced by increasing kurtosis between the two models. In this case, the GC overvaluation for ATM values is explained by value differences in the put time component. Yet, while both Black and GC values exhibit significant time decay close to expiry, Johnson’s ones remain stable up to maturity.