Asymptotic behaviour of randomised fractional volatility models

TL;DR

This paper investigates the asymptotic properties of small-noise fractional Brownian motion-driven diffusions with random initial conditions, extending large deviation results to characterize implied volatility in rough volatility models.

Contribution

It extends large deviation principles to fractional diffusions with random start points, providing new insights into small-time and tail behaviors in rough volatility models.

Findings

Characterizes small-time implied volatility estimates.

Provides tail behavior analysis of fractional diffusions.

Extends large deviation results for fractional Brownian motion.

Abstract

We study the asymptotic behaviour of a class of small-noise diffusions driven by fractional Brownian motion, with random starting points. Different scalings allow for different asymptotic properties of the process (small-time and tail behaviours in particular). In order to do so, we extend some results on sample path large deviations for such diffusions. As an application, we show how these results characterise the small-time and tail estimates of the implied volatility for rough volatility models, recently proposed in mathematical finance.