A hyperbolic diffusion model for stock prices

Abstract

In the present paper we consider a model for stock prices which is a generalization of the model behind the BlackScholes formula for pricing European call options. We model the log-price as a deterministic linear trend plus a diusion process with drift zero and with a diusion coecient (volatil- ity) which depends in a particular way on the instantaneous stock price. It is shown that the model possesses a number of properties encountered in empiri- cal studies of stock prices. In particular the distribution of the adjusted log-price is hyperbolic rather than normal. The model is rather successfully tted to two dierent stock price data sets. Finally, the question of option pricing based on our model is discussed and comparison to the BlackScholes formula is made. The paper also introduces a simple general way of constructing a zero-drift diusion with a given marginal distribution, by which other models that are potentially useful in mathematical nance can be developed.