Mathieu Rosenbaum

Limit theorems for nearly unstable Hawkes processes

Because of their tractability and their natural interpretations in term of market quantities, Hawkes processes are nowadays widely used in high-frequency finance. However, in practice, the statistical estimation results seem to show that very often, only nearly unstable Hawkes processes are able to fit the data properly. By nearly unstable, we mean that the

Quarticity and other functionals of volatility : Efficient estimation

L^1

Simulating and Analyzing Order Book Data: The Queue-Reactive Model

norm of their kernel is close to unity. We study in this work such processes for which the stability condition is almost violated. Our main result states that after suitable rescaling, they asymptotically behave like integrated Cox-Ingersoll-Ross models. Thus, modeling financial order flows as nearly unstable Hawkes processes may be a good way to reproduce both their high and low frequency stylized facts. We then extend this result to the Hawkes-based price model introduced by Bacry et al. [Quant. Finance 13 (2013) 65-77]. We show that under a similar criticality condition, this process converges to a Heston model. Again, we recover well-known stylized facts of prices, both at the microstructure level and at the macroscopic scale.

Large tick assets: implicit spread and optimal tick size

We consider a multidimensional Ito semimartingale regularly sampled on [0,t] at high frequency

Improved Matrix Uncertainty Selector

1/\Delta_n

The characteristic function of rough Heston models

, with

Market microstructure: confronting many viewpoints

\Delta_n

Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes

going to zero. The goal of this paper is to provide an estimator for the integral over [0,t] of a given function of the volatility matrix. To approximate the integral, we simply use a Riemann sum based on local estimators of the pointwise volatility. We show that although the accuracy of the pointwise estimation is at most

On the Microstructural Hedging Error

\Delta_n^{1/4}

Asymptotic Lower Bounds for Optimal Tracking: a Linear Programming Approach

, this procedure reaches the parametric rate