Multifractional processes in finance

There is a growing consensus that fundamental financial theory based on the assumption that markets are complete is not sustainable when financial markets become increasingly complex. Traditional models fail to capture many of the stylized facts and biases identified by recent financial developments that have sought to explain financial prices when markets are incomplete. These approaches, and in particular behavioral finance, in turn, do not formalize and quantify the assumptions their approaches are based on, which are required for financial calculations and assets pricing. One of the open questions is therefore whether a model exists that is able to deduce the overall equilibrium stated by the current paradigm as a sequence of balancing disequilibria. To this aim, a class of stochastic models – the multifractional processes – is suggested and presented in this paper. Multifractional processes are defined as a generalization of fractional Brownian motion, providing a parsimonious mechanisms for modeling real financial markets. This approach includes temporary departures from equilibrium triggered by investors’ biases.