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Simulation of Fractional Brownian Motion and Related Stochastic Processes in Practice: A Straightforward Approach

Fractional Brownian motion, with its long-time correlated increments, has been applied in many fields in recent years. After volatility was shown to be rough by Gatheral, Jaisson, and Rosenbaum, fractional Brownian motion has gained popularity as a financial model. In this work, we revisit the definitions and properties of the univariate and multivariate fractional Brownian motions, and consider four simulation methods. We demonstrate the issues associated with applying the standard Euler scheme for simulating stochastic processes driven by fractional Brownian motion with $H < \frac{1}{2}$ (which we call the rough models). We then consider an approximate method for simulating such rough models based on the fast algorithm by Ma and Wu, which accounts for a 10x speedup. Finally, we consider applications of these methods to option pricing.

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