Merton-like Theoretical Frame for Fractional Brownian Motion in Finance

Abstract

Despite the classical hypothesis states that the asset returns are (log-Normally) identically and independently distributed, in many financial market is detectable significative empirical evidence that there are dependence inside such returns. From a distributional point of view, this dependence can be modelled by the so-calledfractionalBrownian (fB) motion which is a Gaussian stochastic process whose increments are (long-term) dependent with each other. Although there exists an increasing empirical literature about this topic, from a theoretical standpoint there is not an equivalent number of results concerning with the relationships between the fB motion and the financial markets.

Starting from these remarks, in this work we propose a Merton-like system of economic-financial assumptions on the dynamical behaviour of financial asset price by which it is possible to deduce the consistency between the fB motion and the discrete-time trading. Moreover, we also prove the “convergence” of the fB motion to the standard Brownian (sB) one when the discrete-time trading tends to the continuous-time one.