Volatility Smile as Relativistic Effect

TL;DR

This paper introduces a relativistic extension of the Black-Scholes model, providing a new probability distribution that captures fat tails and volatility smiles through a parameter analogous to the speed of light.

Contribution

It presents an explicit, normalized distribution formula based on relativistic Brownian motion, extending the Black-Scholes model with a new parameter for relativistic effects in finance.

Findings

Distribution obeys the tower law and is suitable for option pricing.

Relativistic effects produce fat tails and volatility smiles.

Model generalizes Black-Scholes with a new diffusion parameter.

Abstract

We give an explicit formula for the probability distribution based on a relativistic extension of Brownian motion. The distribution 1) is properly normalized and 2) obeys the tower law (semigroup property), so we can construct martingales and self-financing hedging strategies and price claims (options). This model is a 1-constant-parameter extension of the Black-Scholes-Merton model. The new parameter is the analog of the speed of light in Special Relativity. However, in the financial context there is no "speed limit" and the new parameter has the meaning of a characteristic diffusion speed at which relativistic effects become important and lead to a much softer asymptotic behavior, i.e., fat tails, giving rise to volatility smiles. We argue that a nonlocal stochastic description of such (Levy) processes is inadequate and discuss a local description from physics. The presentation is intended to be pedagogical.