Closed-form Solutions of Relativistic Black-Scholes Equations

TL;DR

This paper introduces a relativistic extension of the Black-Scholes option pricing model, incorporating a speed limit parameter that improves empirical fit and converges to the classical model as the limit increases.

Contribution

It presents a novel closed-form solution to a relativistic version of the Black-Scholes equation, integrating a new parameter for information transfer speed c.

Findings

The new formula converges to Black-Scholes as c approaches infinity.

It better captures the volatility smile and skew observed in markets.

Provides an alternative distribution family for stock prices.

Abstract

Drawing insights from the triumph of relativistic over classical mechanics when velocities approach the speed of light, we explore a similar improvement to the seminal Black-Scholes (Black and Scholes (1973)) option pricing formula by considering a relativist version of it, and then finding a respective solution. We show that our solution offers a significant improvement over competing solutions (e.g., Romero and Zubieta-Martinez (2016)), and obtain a new closed-form option pricing formula, containing the speed limit of information transfer c as a new parameter. The new formula is rigorously shown to converge to the Black-Scholes formula as c goes to infinity. When c is finite, the new formula can flatten the standard volatility smile which is more consistent with empirical observations. In addition, an alternative family of distributions for stock prices arises from our new formula, which offer a better fit, are shown to converge to lognormal, and help to better explain the volatility skew.