A characterisation of cross-impact kernels

Mathieu Rosenbaum; Mehdi Tomas

doi:10.3934/fmf.2022005

Article Contents

2022, Volume 1, Issue 4: 491-523. Doi: 10.3934/fmf.2022005 open access

This issue Previous Article Generalized Cox model for default times Next Article On asymptotically arbitrage-free approximations of the implied volatility

A characterisation of cross-impact kernels

Mathieu Rosenbaum^1, and
Mehdi Tomas^1,2, ,

1.
CMAP UMR CNRS 7641, Ecole Polytechnique, 91128 Palaiseau Cedex, France
2.
LadHyX UMR CNRS 7646, Ecole Polytechnique, 91128 Palaiseau Cedex, France

^*Corresponding author: Mehdi Tomas
^*Corresponding author: Mehdi Tomas

The authors thank Michael Benzaquen, Iacopo Mastromatteo and Michele Vodret for helpful discussions and comments.

Received: July 2021

Revised: April 2022

Early access: October 2022

Published: December 2022

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Abstract

Trading a financial asset pushes its price as well as the prices of other assets, a phenomenon known as cross-impact. We consider a general class of kernel-based cross-impact models and investigate suitable parametrisations for trading purposes. We focus on kernels that guarantee that prices are martingales and anticipate future order flow (martingale-admissible kernels) and those that ensure there is no possible price manipulation (no-statistical-arbitrage-admissible kernels). We determine the overlap between these two classes and provide formulas for calibration of cross-impact kernels on data. We illustrate our results using SP500 futures data.

Keywords:

Mathematics Subject Classification: Primary: 60G55, 60G44; Secondary: 62M10.

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Figure 1. Order flow auto-covariance $ \Omega $

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Figure 2. Boundary values of $ K^1 $ and $ K^2 $

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Figure 3. Values of the $ K^1 $ kernel

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Figure 4. Values of the $ K^2 $ kernel

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Figure 5. Example of predicted prices from $ K^1 $ and $ K^2 $

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