Fractal scaling and Black-Scholes: the full story

Considerable econometric research has been devoted to the quest for suitable classes of models which capture the essential statistical properties of stock and stock index returns. While stock returns are themselves uncorrelated, squared returns and absolute returns do exhibit long-range dependence, the effects of which may be evident for more than a decade. Fractals are indeed present, but at a more subtle level which is not captured by the Black-Scholes model. In this article, the authors demonstrate this effect and they describe a new model involving a simple change to the Black-Scholes stock price which is capable of reproducing the main statistical features of stock, stock index and foreign-exchange rate data. Since Black and Scholes postulated a lognormal model for stock prices (and by implication a normal model for stock returns), intensive econometric studies have established the essential ways in which statistical properties of real returns differ from those implied by the Black-Scholes model.