Asset Price Modeling: From Fractional to Multifractional Processes

This contribution surveys the main characteristics of two stochastic processes that generalize the fractional Brownian motion: the multifractional Brownian motion and the multifractional processes with random exponent. A special emphasis will be devoted to the meaning and to the applications that they can have in finance. If fractional Brownian motion is by now very well-known and studied as a model of the price dynamics, multifractional processes are yet widely unknown in the field of quantitative finance, mainly because of their nonstationarity. Nonetheless, in spite of their complex structure, such processes deserve consideration for their capability to seize the stylized facts that most of the current models cannot account for. In addition, their functional parameter provides an insightful and parsimonious interpretation of the market mechanism, and is able to unify in a single model two opposite approaches such as the theory of efficient markets and the behavioral finance.

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