5 research outputs found
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 is optimal for estimating the Hurst index ,
where is the noise intensity, is the sampling frequency and
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 , the market impact is
proportional to , where denotes the participation rate,
for 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 . In this
paper, we derive a consistent and asymptotically mixed normal estimator of
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