Statistical Methods in Finance 2024

Constructing fake (fractional) Brownian Motion, and more general rough process

By: Purba Das
King's College, London

Modelling rough processes is often crucial in financial applications. Frequently people use Gaussian processes like Brownian motion, fractional Brownian motion, OU process to model financial quantities. As a consequence, in many financial applications like rough volatility literature, while calibrating the parameter H which represents the order/magnitude of roughness; a common assumption is different measures of roughness like Holder regularity, Hurst exponent, and Variation index are all the same.

We first construct processes which are statistically difficult to distinguish from (fractional) Brownian Motion, using a purely pathwise method. We further constructed processes where commonly used assumptions like 'Holder regularity is the same as Hurst exponent', and 'quadratic variation of a financial time series are independent of timestamps where the date is observed. As long as the time steps are going to zero' breaks down. These processes are constructed using generalized Schauder representation, in the spirit of Levy's construction of Brownian motion.