How to build a cross-impact model from first principles: theoretical requirements and empirical results

Abstract

Which models of the cross-impacts of trading financial instruments make sense?

Acknowledgments

The authors warmly thank J.P. Bouchaud, Z. Eisler, B. Tóth, M. Rosenbaum and A. Fosset for fruitful discussions. M. Tomas also acknowledges the support of the chairs Analytics and models for financial regulation, Deep finance and statistics and Machine learning and systematic methods.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 The whitening model is not independent of the symmetric factorization chosen for Σ and Ω. As convention, we will take the square root obtained by an orthogonal decomposition of each matrix and the square root of their eigenvalues.

2 The model proposed in Mastromatteo et al. 2017 is actually the response-based one, referred later as r-el⋆ model.

3 Note that this measure strongly penalizes models violating fragmentation invariance: errors along modes of zero risk should a-priori be enhanced by an infinite amount. In this study we have decided to clip the eigenvalues of Σ to a small, non-zero amount equal to ${10}^{-15}$.

4 For example, though the ${\mathcal{R}}^2\left(I_{\sigma }\right)$ score of the r-direct model is small, it explains about 35% of the variance of price changes of CRUDE0, roughly 20% for CRUDE1 but predicts incorrect price changes for the Calendar Spread CRUDE1_0.

5 To test this hypothesis, we estimated the empirical smallest eigenvalue of the covariance matrix for multiple futures contract as a function of relative tick size (not shown). If price changes of the Calendar Spread were given by the legs of the contract, this eigenvalue should be equal to zero. However, we found that as the tick size increases, so does the smallest eigenvalue away from zero. This thus validates our hypothesis and justifies the need for additional processing of futures data.

6 This is a simplification of the settlement rules to emphasize the expected value of the final settlement price. Further details about the final settlement price of E-MINI futures and 10-year US Treasury Note futures can be found in the CME Rulebook.

Additional information

Funding

This research was conducted within the Econophysics & Complex Systems Research Chair under the aegis of the Fondation du Risque, a joint initiative by the Fondation de l'École polytechnique, l'École polytechnique and Capital Fund Management.