Asymptotic Efficiency for Fractional Brownian Motion with general noise

We investigate the Local Asymptotic Property for fractional Brownian models based on discrete observations contaminated by a Gaussian moving average process. We consider both situations of low and high-frequency observations in a unified setup and we show that the convergence rate $n^{1/2} (\nu_n \Delta_n^{-H})^{-1/(2H+2K+1)}$ is optimal for estimating the Hurst index $H$, where $\nu_n$ is the noise intensity, $\Delta_n$ is the sampling frequency and $K$ is the moving average order. We also derive asymptotically efficient variances and we build an estimator achieving this convergence rate and variance. This theoretical analysis is backed up by a comprehensive numerical analysis of the estimation procedure that illustrates in particular its effectiveness for finite samples

The two square root laws of market impact and the role of sophisticated market participants

The goal of this paper is to disentangle the roles of volume and of participation rate in the price response of the market to a sequence of transactions. To do so, we are inspired the methodology introduced in arXiv:1402.1288, arXiv:1805.07134 where price dynamics are derived from order flow dynamics using no arbitrage assumptions. We extend this approach by taking into account a sophisticated market participant having superior abilities to analyse market dynamics. Our results lead to the recovery of two square root laws: (i) For a given participation rate, during the execution of a metaorder, the market impact evolves in a square root manner with respect to the cumulated traded volume. (ii) For a given executed volume $Q$, the market impact is proportional to $\sqrt{\gamma}$, where $\gamma$ denotes the participation rate, for $\gamma$ large enough. Smaller participation rates induce a more linear dependence of the market impact in the participation rate

Statistical inference for rough volatility: Central limit theorems

In recent years, there has been substantive empirical evidence that stochastic volatility is rough. In other words, the local behavior of stochastic volatility is much more irregular than semimartingales and resembles that of a fractional Brownian motion with Hurst parameter $H<0.5$. In this paper, we derive a consistent and asymptotically mixed normal estimator of $H$ based on high-frequency price observations. In contrast to previous works, we work in a semiparametric setting and do not assume any a priori relationship between volatility estimators and true volatility. Furthermore, our estimator attains a rate of convergence that is known to be optimal in a minimax sense in parametric rough volatility models

STATISTICAL INFERENCE FOR ROUGH VOLATILITY: MINIMAX THEORY

Rough volatility models have gained considerable interest in the quantitative finance community in recent years. In this paradigm, the volatility of the asset price is driven by a fractional Brownian motion with a small value for the Hurst parameter H. In this work, we provide a rigorous statistical analysis of these models. To do so, we establish minimax lower bounds for parameter estimation and design procedures based on wavelets attaining them. We notably obtain an optimal speed of convergence of n −1/(4H+2) for estimating H based on n sampled data, extending results known only for the easier case H > 1/2 so far. We therefore establish that the parameters of rough volatility models can be inferred with optimal accuracy in all regimes