Transient impact from the Nash equilibrium of a permanent market impact game

TL;DR

This paper demonstrates that transient market impact naturally emerges from a Nash equilibrium between traders in a market impact game, providing theoretical foundations for empirical transient impact models.

Contribution

It introduces a game-theoretic framework showing how transient impact arises from trader interactions and derives the impact decay kernel from equilibrium behavior.

Findings

Implied impact kernel can be uniquely derived from order flow and price response.

Multiple solutions exist for the impact decay kernel, including a linear form.

Theoretical explanation for the transient nature of market impact observed empirically.

Abstract

A large body of empirical literature has shown that market impact of financial prices is transient. However, from a theoretical standpoint, the origin of this temporary nature is still unclear. We show that an implied transient impact arises from the Nash equilibrium between a directional trader and one arbitrageur in a market impact game with fixed and permanent impact. The implied impact is the one that can be empirically inferred from the directional trader's trading profile and price reaction to order flow. Specifically, we propose two approaches to derive the functional form of the decay kernel of the Transient Impact Model, one of the most popular empirical models for transient impact, from the behaviour of the directional trader at the Nash equilibrium. The first is based on the relationship between past order flow and future price change, while in the second we solve an inverse optimal execution problem. We show that in the first approach the implied kernel is unique, while in the second case infinite solutions exist and a linear kernel can always be inferred.