European option pricing using Gumbel distribution

Abstract

A risk-neutral-density (RND) function plays a key role in determining option prices accurately. The RND function is useful in understanding the negative skewness and the excess kurtosis of the stock returns. RND models bases on the log-normal distribution (for example, Black–Scholes (BS) model (Black, F and M Scholes (1973). The pricing of options and corporate liabilities, The Journal of Political Economy  3, 637–654)) are less accurate to capture the negative skewness and the excess kurtosis of stock returns (Markose, S and A Alentorn (2011). The generalized extreme value distribution, implied tail index, and option pricing, The Journal of Derivatives 18(3), 35–60). The Extreme Value Theory (EVT) is useful in addressing the tail behavior of stock returns. The tail shape parameter in a Generalized Extreme Value (GEV) distribution class helps control tails’ size and shape. This paper model, the RND function in pricing the European call option, is modeled using a GEV distribution class, namely, a Gumbel distribution. It then compares GEV distribution option values with the BS values. It observes that the BS model underprices the call option values near maturity compared to those obtained by the GEV model. The Gumbel distribution for stock returns is useful in reducing the pricing bias of the BS model.