Fractional calculus and continuous-time finance

In this paper we present a rather general phenomenological theory of tick-b y-tick dynamics in financial markets. Many well-known aspects, such as the Levy scaling form, follow as particular cases of the theory. The theory ful ly takes into account the non-Markovian and non-local character of financia l time series. Predictions on the long-time behaviour of the waiting-time p robability density are presented. Finally, a general scaling form is given, based on the solution of the fractional diffusion equation. (C) 2000 Elsev ier Science B.V. All rights reserved.