[Paper Review] High Frequency Market Making

René Carmona, Kevin Webster|arXiv (Cornell University)|Oct 21, 2012

Financial Markets and Investment Strategies16 references3 citations

TL;DR

This paper develops a stochastic control model for high-frequency market making under asymmetric information, using a codebook of client order types (implied alpha) to derive optimal limit-order strategies. It solves the market maker's problem via a stochastic partial differential equation (SPDE) framework, yielding tractable formulas for optimal quotes and limit-order book dynamics under identical client horizons or belief processes.

ABSTRACT

Since they were authorized by the U.S. Security and Exchange Commission in 1998, electronic exchanges have boomed, and by 2010 high frequency trading accounted for over 70% of equity trades in the US. Such markets are thought to increase liquidity because of the presence of market makers, who are willing to trade as counterparties at any time, in exchange for a fee, the bid-ask spread. In this paper, we propose an equilibrium model showing how such market makers provide liquidity. The model relies on a codebook for client trades, the implied alpha. After solving the individual clients optimization problems and identifying their implied alphas, we frame the market maker stochastic optimization problem as a stochastic control problem with an infinite dimensional control variable. Assuming either identical time horizons for all the clients, or a stochastic partial differential equation model for their beliefs, we solve the market maker problem and derive tractable formulas for the optimal strategy and the resulting limit-order book dynamics.

Motivation & Objective

• To model high-frequency market making as a stochastic control problem with infinite-dimensional control, capturing client order dynamics.
• To address adverse selection by modeling client trades through a codebook of implied alphas, reflecting private information and short-term views.
• To derive optimal market-making strategies that balance liquidity provision and risk exposure under asymmetric information.
• To establish limit-order book dynamics under stochastic belief processes using SPDEs.
• To provide tractable, closed-form solutions for optimal quotes and book shape under specific assumptions on client behavior.

Proposed method

• Formulates the market maker's problem as a stochastic control problem with an infinite-dimensional control variable representing the limit-order book.
• Introduces a codebook of client order types (implied alpha) to model private information and short-term trading views.
• Applies the Pontryagin maximum principle to derive necessary conditions for optimality, using adjoint equations and Hamiltonian maximization.
• Uses stochastic partial differential equations (SPDEs) to model the evolution of client beliefs and aggregate order flow.
• Imposes assumptions of identical time horizons or SPDE-driven belief processes to ensure solvability and tractability.
• Applies the law of large numbers to the empirical measure of client strategies, enabling mean-field approximation of the limit-order book.

Experimental results

Research questions

• RQ1 How can a market maker optimally set bid and ask quotes when facing clients with private information and short-term views?
• RQ2 What is the structure of the limit-order book that emerges from optimal market-making strategies under asymmetric information?
• RQ3 How do client order types (implied alpha) influence the dynamics of the market maker's optimal strategy?
• RQ4 Under what conditions can the market maker's stochastic control problem be solved explicitly using SPDEs?
• RQ5 What role does the distribution of client beliefs play in shaping the optimal limit-order book and the resulting bid-ask spread?

Key findings

• The optimal market-making strategy is derived as a function of the implied alpha and the dynamics of client beliefs, leading to a closed-form expression for the limit-order book.
• Under identical client horizons, the model yields explicit formulas for optimal bid and ask quotes and the resulting limit-order book shape.
• When client beliefs evolve according to an SPDE, the market maker's optimal strategy and book dynamics are characterized by a tractable system of equations.
• The model establishes a link between the liquidity cost curve and the distribution of client order types, showing how client behavior shapes market impact.
• The Hamiltonian maximization condition under the Pontryagin principle leads to a verification theorem ensuring optimality of the derived strategy.
• The law of large numbers applied to the empirical measure of client strategies enables a mean-field approximation, justifying the SPDE approach.