  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.
