Short-time expansion of characteristic functions in a rough volatility setting with applications

TL;DR

This paper develops a higher-order asymptotic expansion for the characteristic function of Itô semimartingale increments over short times, accommodating rough paths and jumps, with applications to estimating the Hurst parameter from options.

Contribution

It introduces a novel asymptotic expansion that captures the effects of rough volatility and jumps, enabling new nonparametric estimation methods.

Findings

Derived a higher-order expansion for characteristic functions.

Applied the expansion to estimate the Hurst parameter.

Demonstrated the method's effectiveness in rough volatility settings.

Abstract

We derive a higher-order asymptotic expansion of the conditional characteristic function of the increment of an It\^o semimartingale over a shrinking time interval. The spot characteristics of the It\^o semimartingale are allowed to have dynamics of general form. In particular, their paths can be rough, that is, exhibit local behavior like that of a fractional Brownian motion, while at the same time have jumps with arbitrary degree of activity. The expansion result shows the distinct roles played by the different features of the spot characteristics dynamics. As an application of our result, we construct a nonparametric estimator of the Hurst parameter of the diffusive volatility process from portfolios of short-dated options written on an underlying asset.