Student's t-Distribution Based Option Sensitivities: Greeks for the Gosset Formulae

TL;DR

This paper introduces a Gosset approach for pricing European options using a Student's t-distribution, compares its greeks with Black-Scholes, and discusses implementation and implications for modeling fat-tailed returns.

Contribution

It extends the Black-Scholes model by incorporating Student's t-distribution, allowing for better modeling of fat-tailed return distributions and removing the need for known volatility.

Findings

Gosset and Black-Scholes greeks converge for large degrees of freedom ppa.

Gosset formulae do not require known volatility, unlike Black-Scholes.

Comparison shows differences in sensitivities for fat-tailed distributions.

Abstract

European options can be priced when returns follow a Student's t-distribution, provided that the asset is capped in value or the distribution is truncated. We call pricing of options using a log Student's t-distribution a Gosset approach, in honour of W.S. Gosset. In this paper, we compare the greeks for Gosset and Black-Scholes formulae and we discuss implementation. The t-distribution requires a shape parameter \nu to match the "fat tails" of the observed returns. For large \nu, the Gosset and Black-Scholes formulae are equivalent. The Gosset formulae removes the requirement that the volatility be known, and in this sense can be viewed as an extension of the Black-Scholes formula.