Optimal incentives in a limit order book: a SPDE control approach

TL;DR

This paper models the limit order book as an SPDE and develops an optimal control framework for exchanges to design incentives that shape liquidity, providing theoretical characterization and numerical analysis of incentive strategies.

Contribution

It introduces a novel SPDE control approach to optimize exchange incentives for liquidity provision, with explicit solutions and numerical insights.

Findings

Closed-form incentives derived at order book limits

Numerical analysis of incentive impact on order book shape

Sensitivity analysis of incentives to market parameters

Abstract

With the fragmentation of electronic markets, exchanges are now competing in order to attract trading activity on their platform. Consequently, they developed several regulatory tools to control liquidity provision / consumption on their liquidity pool. In this paper, we study the problem of an exchange using incentives in order to increase market liquidity. We model the limit order book as the solution of a stochastic partial differential equation (SPDE) as in [12]. The incentives proposed to the market participants are functions of the time and the distance of their limit order to the mid-price. We formulate the control problem of the exchange who wishes to modify the shape of the order book by increasing the volume at specific limits. Due to the particular nature of the SPDE control problem, we are able to characterize the solution with a classic Feynman-Kac representation theorem. Moreover, when studying the asymptotic behavior of the solution, a specific penalty function enables the exchange to obtain closed-form incentives at each limit of the order book. We study numerically the form of the incentives and their impact on the shape of the order book, and analyze the sensitivity of the incentives to the market parameters.