An empirical model of volatility of returns and option pricing

This paper reports several entirely new results on financial market dynamics and option pricing. We observe that empirical distributions of returns are much better approximated by an exponential distribution than by a Gaussian. This exponential distribution of asset prices can be used to develop a new pricing model for options (in closed algebraic form) that is shown to provide valuations that agree very well with those used by traders. We show how the Fokker–Planck formulation of fluctuations can be used with a local volatility (diffusion coefficient) to generate an exponential distribution for asset returns, and also how fat tails for extreme returns are generated dynamically by a simple generalization of our new volatility model. Nonuniqueness in deducing dynamics from empirical data is discussed and is shown to have no practical effect over time scales much less than one hundred years. We derive an option pricing pde and explain why it is superfluous, because all information required to price options in agreement with the delta-hedge is already included in the Green function of the Fokker–Planck equation for a special choice of parameters. Finally, we also show how to calculate put and call prices for a stretched exponential returns density.