Multiscale estimation of processes related to the fractional Black-Scholes equation

We consider a fractional-order differential equation involving fractal activity time to represent the stochastic behaviour of a log-price process of an underlying asset. The log-price process is defined in terms of fractional integration of the fractional derivative of Brownian motion on fractal time. A stable solution to the extrapolation and filtering problems associated is obtained in terms of covariance vaguelette functions (Angulo and Ruiz-Medina 1999). A simulation study is carried out to illustrate the methodology presented. Copyright Physica-Verlag 2003