Volatility Smile as Relativistic Effect
Physica A 475 (2017) 59-76
32 Pages Posted: 24 Aug 2016 Last revised: 3 Mar 2017
Date Written: January 29, 2017
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.
Keywords: volatility smile, fat tails, probability distribution, relativistic, Brownian motion, option pricing, Levy process, path integral, Hamiltonian, self-financing, Euclidean, Minkowski, rest energy, diffusion, stochastic, reparametrization invariance, tower law, martingale
JEL Classification: G00
Suggested Citation: Suggested Citation