arXiv subjects

Grégoire Szymanski

Publications and source records attributed to Grégoire Szymanski.

11 recordsLinked to original sources

arXiv2026

Estimation of the invariant measure of a multidimensional diffusion from noisy observations

We introduce a new approach for estimating the invariant density of a multidimensional diffusion when dealing with high-frequency observations blurred by independent noise. We consider the intermediate regime, where observations occur at discrete time instances $kΔ_n$ for $k=0,\dots,n$, under the conditions $Δ_n\to 0$ and $nΔ_n\to\infty$. We construct a kernel density estimator based on preaveraged observations to reduce the effect of the noise and involves a two-step bias-correction procedure to appropriately account for the bias introduced by the pre-averaging. The rate of convergence of our estimator depends on both the anisotropic regularity of the density and the intensity of the noise. We establish conditions on the intensity of the noise that ensure the recovery of convergence rates similar to those achievable without any noise. Furthermore, we prove a Bernstein concentration inequality for our estimator, from which we derive an adaptive procedure for the kernel bandwidth selection.

math.ST↗

arXiv2026

Passive Market Impact: A Point Process Approach

While the market impact of aggressive orders has been extensively studied, the impact of passive orders, those executed through limit orders, remains less understood. The goal of this paper is to investigate passive market impact by developing a microstructure model connecting liquidity dynamics and price moves. A key innovation of our approach is to replace the traditional assumption of constant information content for each trade by a function that depends on the available volume in the limit order book. Within this framework, we explore scaling limits and analyze the market impact of passive metaorders. Additionally, we derive useful approximations for the shape of market impact curves, leading to closed-form formulas that can be easily applied in practice.

q-fin.MF↗

arXiv2026

Exact conditional simulation of Point processes: Application to pathwise market impact estimation

Market impact is defined as the difference between the observed price trajectory under a given execution strategy and the counterfactual trajectory that would have prevailed without it. Since this counterfactual is unobservable, estimating market impact requires simulating alternative paths under the same realized market randomness. We address this by studying the conditional simulation of point processes under perturbed intensities. Given an observed counting process whose intensity is determined by its own history, we characterize the conditional law of the latent Poisson random measure in a thinning representation. This yields an exact, event-driven algorithm that reconstructs counterfactual paths on a common randomness source, enabling rigorous pathwise market impact estimation for aggressive, passive, and mixed strategies.

q-fin.TR↗

arXiv2026

Trading with market resistance and concave price impact

We consider an optimal trading problem under a market impact model with endogenous market resistance generated by a sophisticated trader who (partially) detects metaorders and trades against them to exploit price overreactions induced by the order flow. The model features a concave transient impact driven by a power-law propagator with a resistance term responding to the trader's rate via a fixed-point equation involving a general resistance function. We derive a (non)linear stochastic Fredholm equation as the first-order optimality condition satisfied by optimal trading strategies. Existence and uniqueness of the optimal control are established when the resistance function is linear, and an existence result is obtained when it is strictly convex using coercivity and weak lower semicontinuity of the associated profit-and-loss functional. We also propose an iterative scheme to solve the nonlinear stochastic Fredholm equation and prove an exponential convergence rate. Numerical experiments confirm this behavior and illustrate optimal round-trip strategies under "buy" signals with various decay profiles and different market resistance specifications.

q-fin.TR↗

arXiv2026

A unified theory of order flow, market impact, and volatility

We propose a microstructural model for the order flow in financial markets that distinguishes between {\it core orders} and {\it reaction flow}, both modeled as Hawkes processes. This model has a natural scaling limit that reconciles a number of salient empirical properties: persistent signed order flow, rough trading volume and volatility, and power-law market impact. In our framework, all these quantities are pinned down by a single statistic $H_0$, which measures the persistence of the core flow. Specifically, the signed flow converges to the sum of a fractional process with Hurst index $H_0$ and a martingale, while the limiting traded volume is a rough process with Hurst index $H_0-1/2$. No-arbitrage constraints imply that volatility is rough, with Hurst parameter $2H_0-3/2$, and that the price impact of trades follows a power law with exponent $2-2H_0$. The analysis of signed order flow data yields an estimate $H_0 \approx 3/4$. This is not only consistent with the square-root law of market impact, but also turns out to match estimates for the roughness of traded volumes and volatilities remarkably well.

q-fin.ST↗

arXiv2025

Mean-Field Limits for Nearly Unstable Hawkes Processes

In this paper, we establish general scaling limits for nearly unstable Hawkes processes in a mean-field regime by extending the method introduced by Jaisson and Rosenbaum. Under a mild asymptotic criticality condition on the self-exciting kernels $\{ϕ^n\}$, specifically $\|ϕ^n\|_{L^1} \to 1$, we first show that the scaling limits of these Hawkes processes are necessarily stochastic Volterra diffusions of affine type. Moreover, we establish a propagation of chaos result for Hawkes systems with mean-field interactions, highlighting three distinct regimes for the limiting processes, which depend on the asymptotics of $n(1-\|ϕ^n\|_{L^1})^2$. These results provide a significant generalization of the findings by Delattre, Fournier and Hoffmann.

math.PR↗

arXiv2024

Statistical inference for rough volatility: Central limit theorems

In recent years, there has been a substantive interest in rough volatility models. In this class of models, 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.

math.ST↗

arXiv2024

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.

math.ST↗

arXiv2023

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 $\sqrtγ$, where $γ$ denotes the participation rate, for $γ$ large enough. Smaller participation rates induce a more linear dependence of the market impact in the participation rate.

q-fin.MF↗

arXiv2023

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 $n^{1/2} (ν_n Δ_n^{-H})^{-1/(2H+2K+1)}$ is optimal for estimating the Hurst index $H$, where $ν_n$ is the noise intensity, $Δ_n$ 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.

math.ST↗

arXiv2022

Optimal estimation of the rough Hurst parameter in additive noise

We estimate the Hurst parameter $H \in (0,1)$ of a fractional Brownian motion from discrete noisy data, observed along a high frequency sampling scheme. When the intensity $τ_n$ of the noise is smaller in order than $n^{-H}$ we establish the LAN property with optimal rate $n^{-1/2}$. Otherwise, we establish that the minimax rate of convergence is $(n/τ_n^2)^{-1/(4H+2)}$ even when $τ_n$ is of order 1. Our construction of an optimal procedure relies on a Whittle type construction possibly pre-averaged, together with techniques developed in Fukasawa et al. [Is volatility rough? arXiv:1905.04852, 2019]. We establish in all cases a central limit theorem with explicit variance, extending the classical results of Gloter and Hoffmann [Estimation of the Hurst parameter from discrete noisy data. The Annals of Statistics, 35(5):1947-1974, 2007].

math.ST↗