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