Market impact with multi-timescale liquidity

Abstract

Finite-memory effects on the dynamics of the latent order book can be accounted for by allowing finite cancellation and deposition rates within a continuous reaction-diffusion set-up

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Erratum

Acknowledgements

We thank J. Bonart, A. Darmon, J. de Lataillade, J. Donier, Z. Eisler, A. Fosset, S. Gualdi, I. Mastromatteo, M. Rosenbaum and B. Tóth for extremely fruitful discussions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 A ‘meta-order’ (or parent order) is a bundle of orders corresponding to a single trading decision. A meta-order is typically traded incrementally through a sequence of child orders.

2 The variable x denotes the reservation price relative to the ‘fair’ price such that the true reservation price reads . We here assume that the fair price encodes all informational aspects of prices and itself performs (on short time scales) an additive random walk with diffusivity coefficient D.

3 Note that working at constant implies .

4 Indeed, the price trajectory during a meta-order execution results from the combination of a square-root (see table 2) and the wander about of the fair price (see Sect. 1). The deterministic square-root signal is detectable if it exceeds the noise level, that is which precisely corresponds to . In the regime (equivalent to ) the signal is hidden in the overall noise, which explains why square-root trajectories are seldom observed in real price time series.

5 In full generality, the diffusion coefficient should be allowed to depend on . This can be simply implemented by changing the distribution of frequencies as: .

6 Note that rigorously speaking, one should also introduce a low-frequency cut-off to ensure the existence of a stationary state of the order book in the absence of meta-order. Otherwise, when and the system does not reach a stationary state (see the end of section 2 and Benzaquen and Bouchaud (2018) for a further discussion of this point).

7 Taking into account the H(t) contribution turns out not to change the following scaling argument.

8 Here again, we assume that is much larger than the implicit low-frequency cut-off .

9 Note that is very close to the empirical impact results reported by Almgren et al. (2005) and Brokmann et al. (2015) in the case of equities, for which is usually close to 1/2.

11 Equivalently, rapid and long execution is only consistent with large meta-order volume (combining and implies ).