Pricing European Options with a Log Student's t-Distribution: a Gosset Formula

TL;DR

This paper introduces a method for pricing European options using a log Student's t-distribution, capturing fat tails in return distributions, and generalizes the Black-Scholes model for non-normal returns.

Contribution

It presents a novel Gosset formula for European option pricing based on the log Student's t-distribution, extending the Black-Scholes framework to fat-tailed return distributions.

Findings

The Gosset formula accurately prices options with fat-tailed return distributions.

The approach reduces to Black-Scholes for normally distributed returns.

It provides a flexible framework for using alternative distributions in option pricing.

Abstract

The distribution of the returns for a stock are not well described by a normal probability density function (pdf). Student's t-distributions, which have fat tails, are known to fit the distributions of the returns. We present pricing of European call or put options using a log Student's t-distribution, which we call a Gosset approach in honour of W.S. Gosset, the author behind the nom de plume Student. The approach that we present can be used to price European options using other distributions and yields the Black-Scholes formula for returns described by a normal pdf.