The roughness exponent of a continuous function $x$ is defined as the number $R\in[0,1]$ for which the $p^{\text{th}}$ variation of $x$ is infinite if $p<1/R$ and zero if $p>1/R$. The roughness exponent characterizes the regularity of financial time series without any probabilistic assumptions and coincides with the traditional Hurst parameter for the sample paths of fractional Brownian motion. It was observed empirically in the seminal work by Gatheral, Jaisson, and Rosenbaum (2018) that the roughness exponent of realized volatility of many financial time series is rather small.


However, one difficulty in the roughness estimation process for the instantaneous volatility $\sigma_t$ is that the instantaneous volatility is observed only indirectly through its antiderivative, namely the quadratic variation $\langle\log S \rangle_t$ of the price trajectory $S_t$. To this end, we introduce a new estimator $\widehat{\mathscr{R}}_n$ for the roughness exponent of the instantaneous volatility that integrates the numerical differentiation of $\langle\log S \rangle_t$ with the estimation of the roughness exponent of the derivative $\sigma_t^2=\frac{d}{dt}\langle\log S \rangle_t$. In this talk, we will demonstrate the consistency of the estimator $\widehat{\mathscr{R}}_n$ for several classes of Gaussian processes, including fractional Brownian motion and fractional Ornstein--Uhlenbeck processes. Furthermore, this talk will highlight the underlying rationale of constructing the estimator $\widehat{\mathscr{R}}_n$, which is based on the robust approximation of the Faber--Schauder coefficients of $\sigma^2_t$.


This talk is based on the joint work with Alexander Schied.