ABSTRACT
In this chapter, we introduce the fractional Brownian motion (fBm), a Gaussian self-similar process with stationary increments that allows the increments to be correlated and whose behaviour depends on its Hurst parameter. We study the main properties of this model, as the fact that it is not a martingale, the regularity of its paths, and its long and short memory properties. A truncated version of the fBm, the Riemann-Liouville fractional Brownian motion (RLfBm) has a simpler representation in terms of the Brownian motion and has been widely used in the construction of fractional volatilities. We discuss how to simulate these processes and we study their applications in finance, both in the description of the asset prices behaviour and in the modeling of the volatility process. We also see that the Malliavin derivative of most fractional volatility models can be easily computed, as we see in the cases of fractional Ornstein-Uhlenbeck volatilities and the rough Bergomi model.
