A Simple Microstructural Explanation of Concave Pice Impact

Abstract

This article describes a simple model of market microstructure which explains a concave price impact. In the proposed model, the local relationship between the order flow and the fundamental price (i.e. the local price impact) is linear, which makes the model dynamically consistent. Nevertheless, the expected impact on midprice from a large sequence of co-directional trades is nonlinear and asymptotically concave. The main practical conclusion of the model is that, throughout a meta-order, the volumes at the best bid and ask prices change (on average) in favor of the executor. This conclusion, in turn, relies on two more concrete predictions of the model, one of which is tested using publicly available market data without the information about meta-orders. Introduction and main resultsThe term price impact often refers to the fact that a trade tends to move the asset price in its direction. However, this general observation has several more specific interpretations, and, as pointed out in [20], it is important to differentiate between various types of price impact. First, one may consider the local impact: i.e., the expected price change as a function of the volume of a single trade. The latter is only relevant in the markets where large trades occur often (e.g., OTC markets), but is not relevant, e.g., for the most popular stock exchanges. The second type of impact is the meta-order impact: i.e., the expected price change as a function of the volume of a sequence of co-directed trades (such sequence is called a meta-order). This type of impact is more relevant for public exchanges, where most participants willing to buy or sell a large quantity of the asset split this quantity into smaller pieces (child orders) and submit each of them separately. Within the meta-order impact, one can distinguish two sub-types, depending on whether the relative rate of the execution, or its total duration, is fixed. This paper studies the meta-order impact for a fixed execution rate (also known as the expected price trajectory).There exist various empirical studies that confirm the concavity of meta-order impact (see, e.g., [1,3,2,21,17,4]). They show that the expected price change as a function of traded volume is concave (see the left part of Figure 2), and some even claim a specific power (in particular, square-root) dependence of the impact on the volume. These empirical results motivated the search for a sound theoretical explanation of the concavity of price impact. The following four paragraphs describe the main types of explanations available to date.The first explanation is rather heuristic and has not been documented in the academic literature (to the author's best knowledge). It explains the concavity of price impact by the predictability of future prices and * The author thanks M. Mouyebe for conducting a numerical experiment whose results inspired this article. He thanks S. Jaimungal for providing the market data used herein. The author also thanks J.-P. Bouchaud, P. Ustinov, and K. Webster, for the useful discussion...

Power laws in market microstructure

“…1 See also Nadtochiy [19] for an explanation of concave price impact in a continuous-time setting. However, price is not determined in a rational expectations equilibrium setting in [19].…”

Power laws in market microstructure

“…See also Nadtochiy[19] for an explanation of concave price impact in a continuous-time setting. However, price is not determined in a rational expectations equilibrium setting in[19].2 This is in fact consistent with the assumption that the market makers are risk-neutral.…”

“…See also Nadtochiy[19] for an explanation of concave price impact in a continuous-time setting. However, price is not determined in a rational expectations equilibrium setting in[19].2 This is in fact consistent with the assumption that the market makers are risk-neutral. The more realistic case with risk-averse market makers that leads to the risk neutral measure being different than the physical one is left to future research.…”