A possible way of estimating options with stable distributed underlying asset prices

Abstract

Option pricing theory is considered when the underlying asset price satisfies a stochastic differential equation which is driven by random motions generated by stable distributions. The properties of the stable distributions are discussed and their connection with the theory of fractional Brownian motion is noted. This approach attempts to generalize the classical Black–Scholes formulation, to allow for the presence of fat tails in the distribution of log prices which leads to a diffusion equation involving fractional Brownian motion. The resulting option pricing via a hedging strategy approach is independently derived by constructing a backward Kolmogorov equation for a simple trinomial model where the probabilities are assumed to satisfy a particular fractional Taylor series due to Dzherbashyan and Nersesyan. To effect this development, some knowledge of fractional integration and differentiation is required so this is briefly reviewed. Consideration is also given to a different hedging strategy approach leading to a fractional Black–Scholes equation involving the market price of risk. Modification to the model is also considered such as the impact of transaction costs. A simple example of American options is also considered.

Keywords:

• stable distributions
• fractional Taylor series
• fractional Black–Scholes equation