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

Abstract

Cross-impact, namely the fact that on average buy (sell) trades on a financial instrument induce positive (negative) price changes in other correlated assets, can be measured from abundant, although noisy, market data. In this paper we propose a principled approach that allows to perform model selection for cross-impact models, showing that symmetries and consistency requirements are particularly effective in reducing the universe of possible models to a much smaller set of viable candidates, thus mitigating the effect of noise on the properties of the inferred model. We review the empirical performance of a large number of cross-impact models, comparing their strengths and weaknesses on a number of asset classes (futures, stocks, calendar spreads). Besides showing which models perform better, we argue that in presence of comparable statistical performance, which is often the case in a noisy world, it is relevant to favor models that provide ex-ante theoretical guarantees on their behavior in limit cases. From this perspective, we advocate that the empirical validation of universal properties (symmetries, invariances) should be regarded as holding a much deeper epistemological value than any measure of statistical performance on specific model instances.

A characterisation of cross-impact kernels

“…The price-covariance matrix Σ, not shown here but reported in [24], shows strong correlation (≈ 90%) between the two maturities. This is natural since both futures have the same underlying.…”

“…With these conventions, Ω(0) represents the trade covariance, so that Ω 11 (0) is the instantaneous variance of signed order flow on the front month future SPMINI and Ω 22 (0) the instantaneous variance of signed order flow on SPMINI3. These quantities reflect the liquidity of the underlying assets since they increase with the daily traded volume [24]. The figure shows that the frontmonth maturity SPMINI is approximately 10 times more liquid than the SPMINI3, which highlights that most trading occurs on the leading month contract.…”

“…The two futures selected are the lead month future, referred to as SPMINI, and the next upcoming future, referred to as SPMINI3. This data has been explored in a previous study and we refer the reader to [24] for more details into the underlying data and processing methodology. 1 We begin by reporting the relevant empirical observable of our system, namely the reduced covariance measure Ω and the price-covariance matrix Σ.…”

“…Thus, it ignores a potentially important aspect of price formation. As many assets are simultaneously traded in financial markets, this element is required to generate complete market dynamics and, ultimately, answer the question of how markets digest liquidity.To solve this issue, a recent strand of works [1,5,21,22,24,25,29,31] has studied cross-impact, which describes how transactions on a universe of instruments drive their prices. This paper contributes to the literature by characterising cross-impact models which lead to well-behaved market dynamics.…”

“…To solve this issue, a recent strand of works [1,5,21,22,24,25,29,31] has studied cross-impact, which describes how transactions on a universe of instruments drive their prices. This paper contributes to the literature by characterising cross-impact models which lead to well-behaved market dynamics.…”