Option pricing based on the generalised Tukey distribution

There is good empirical evidence to show that the financial series, whether stocks or indices, currencies or interest rates do not follow the log-normal random walk underlying the Black-Scholes model, which is the basis for most of the theory of options valuation. This article presents a derivation to determine the price of a derivative when the underlying stock's distribution under normality assumption is not valid, using the density function associated with the Tukey's g-h family of generalised distributions, which has tails heavier than the normal distribution. Using the Tukey's g-h family of generalised distributions, we approximate asset price distribution and in the process include both the skewness and kurtosis of the underlying stock's distribution to obtain the impact of these measures on the option pricing. We also obtain the price of the European option to different log-symmetrical and these prices are illustrated with suitable examples. We have also obtained explicit formula for option valuation with two additional parameters g and h relative to the Black-Scholes model, providing control over skewness and kurtosis respectively.