Option Pricing Formulas based on a non-Gaussian Stock Price Model

TL;DR

This paper introduces a non-Gaussian stock price model using nonextensive thermodynamics to derive a generalized option pricing formula, providing a better fit to empirical data with a single volatility parameter.

Contribution

It develops a non-Gaussian extension of the Black-Scholes model based on nonextensive thermodynamics, offering closed-form solutions and improved empirical modeling.

Findings

The generalized model fits market data better than standard Black-Scholes.

Using q=1.5 captures empirical return distributions effectively.

A single volatility parameter suffices for accurate option pricing.

Abstract

Options are financial instruments that depend on the underlying stock. We explain their non-Gaussian fluctuations using the nonextensive thermodynamics parameter $q$. A generalized form of the Black-Scholes (B-S) partial differential equation, and some closed-form solutions are obtained. The standard B-S equation ($q=1$) which is used by economists to calculate option prices requires multiple values of the stock volatility (known as the volatility smile). Using $q=1.5$ which well models the empirical distribution of returns, we get a good description of option prices using a single volatility.