Fractional Brownian motion with zero Hurst parameter: a rough volatility viewpoint

TL;DR

This paper investigates the limit of fractional Brownian motion as the Hurst parameter approaches zero, revealing convergence to a Gaussian distribution akin to a log-correlated field, with implications for rough volatility modeling.

Contribution

It introduces a natural limit for fractional Brownian motion as H approaches zero, connecting it to log-correlated Gaussian fields relevant for rough volatility models.

Findings

Fractional Brownian motion converges to a Gaussian distribution as H tends to zero.

The limiting process resembles a log-correlated random field.

Provides a new perspective on rough volatility modeling in finance.

Abstract

Rough volatility models are becoming increasingly popular in quantitative finance. In this framework, one considers that the behavior of the log-volatility process of a financial asset is close to that of a fractional Brownian motion with Hurst parameter around 0.1. Motivated by this, we wish to define a natural and relevant limit for the fractional Brownian motion when $H$ goes to zero. We show that once properly normalized, the fractional Brownian motion converges to a Gaussian random distribution which is very close to a log-correlated random field.