On the roughness of paths and processes

Abstract

In recent years, a significant amount of the stochastic volatility literature has focused on modelling the ``roughness" or irregularity of the unobserved volatility time series and its effect on option pricing. In many models, roughness often takes the role of both the Hölder regularity of a trajectory as well as the covariance of the stochastic process. To extend the rough volatility literature, we contribute in two ways: (i) we extend the pathwise stochastic calculus to include integrators whose paths are rougher than the typical paths of Brownian motion, and (ii) we study two deep learning methods which allow us to determine the exact roughness of a given sample path.

For (i), we study the concept of $p$-variation of a continuous trajectory along a sequence of refining partitions, and the question of uniqueness when one changes the choice of refining partition sequence. We find a condition, termed $p$-roughness, which implies uniqueness of $p$-variation across a wide family of partition sequences. We then use this property to show that the rough pathwise Itô integral and the rough local time of a trajectory remain unchanged across this wide family of partition sequences.

For (ii), we study the roughness exponent of volatility under the risk-neutral and under the physical measure. Under the risk-neutral measure, we introduce a class of neural networks, called functional neural networks, which are able to learn interpolation schemes for the volatility surface. We use this method to show the recovery of stochastic volatility parameters on the rough Bergomi model. Furthermore, we consider the measurement of roughness under the physical measure for a class of models more general than fractional Brownian motion. We find that under this more general class, volatility roughness seems consistent with the roughness of volatility modelled as a fractional Brownian motion.