[Paper Review] Optimal market making

Olivier Guéant|arXiv (Cornell University)|May 6, 2016

Economic theories and models6 citations

TL;DR

This paper presents a generalized stochastic optimal control framework for optimal market making, extending the Avellaneda-Stoikov model to include dynamic inventory risk management and multi-asset settings. It derives new closed-form approximations for optimal bid and ask quotes under various utility criteria, with key contributions including dimensionality reduction via change of variables and practical formulas validated on credit indices (CDX), showing strong performance even under model simplifications.

ABSTRACT

Market makers provide liquidity to other market participants: they propose prices at which they stand ready to buy and sell a wide variety of assets. They face a complex optimization problem with both static and dynamic components. They need indeed to propose bid and offer/ask prices in an optimal way for making money out of the difference between these two prices (their bid-ask spread). Since they seldom buy and sell simultaneously, and therefore hold long and/or short inventories, they also need to mitigate the risk associated with price changes, and subsequently skew their quotes dynamically. In this paper, (i) we propose a general modeling framework which generalizes (and reconciles) the various modeling approaches proposed in the literature since the publication of the seminal paper "High-frequency trading in a limit order book" by Avellaneda and Stoikov, (ii) we prove new general results on the existence and the characterization of optimal market making strategies, (iii) we obtain new closed-form approximations for the optimal quotes, (iv) we extend the modeling framework to the case of multi-asset market making and we obtain general closed-form approximations for the optimal quotes of a multi-asset market maker, and (v) we show how the model can be used in practice in the specific (and original) case of two credit indices.

Motivation & Objective

• To develop a unified, generalizable framework for optimal market making that reconciles and extends prior models in the literature.
• To prove existence and characterization of optimal strategies under general intensity functions and utility criteria, including CARA utility and risk-neutral cases.
• To derive new closed-form approximations for optimal bid and ask quotes, generalizing the Guéant–Lehalle–Fernandez-Tapia formulas to broader settings.
• To extend the model to multi-asset market making and analyze the impact of correlation between assets on optimal quotes.
• To validate the model empirically using real-world data from credit indices (IG and HY), demonstrating practical applicability and robustness.

Proposed method

• Formulates a stochastic optimal control problem for a market maker with dynamic inventory risk, using a Hamilton-Jacobi-Bellman (HJB) equation framework.
• Applies a change of variables to reduce the four-dimensional HJB equation to a system of ordinary differential equations (ODEs), enabling tractable solution.
• Uses implicit time discretization and Newton’s method to numerically solve the nonlinear ODE systems for optimal feedback control functions.
• Derives closed-form approximations for optimal quotes using asymptotic analysis and perturbation techniques, valid for small inventory positions.
• Extends the model to multi-asset market making by incorporating cross-asset correlation through a bivariate diffusion process.
• Validates the model on real data from CDX indices (IG and HY), comparing optimal quotes under full and simplified models.

Experimental results

Research questions

• RQ1 How can the optimal market making problem be generalized to include both static and dynamic trade-offs under general intensity functions?
• RQ2 What are the conditions under which the high-dimensional HJB equation can be reduced to a solvable system of ODEs?
• RQ3 How accurate are closed-form approximations for optimal quotes across different inventory levels and market conditions?
• RQ4 How does correlation between multiple assets affect the optimal bid and ask quotes in a multi-asset market making strategy?
• RQ5 To what extent does a simplified model (Model B) with reduced risk exposure still yield accurate and practical quotes compared to the full model (Model A)?

Key findings

• The dimensionality of the HJB equation is reduced by half through a change of variables, enabling efficient numerical solution of the optimal control problem.
• Closed-form approximations for optimal quotes are derived and shown to be highly accurate for small inventory positions, with deviations increasing for larger holdings.
• The simplified Model B, which ignores part of the risk, produces quotes nearly identical to those of the full Model A, validating its use as a practical approximation.
• For the IG and HY credit indices, the asymptotic regime for optimal quotes is reached after approximately one hour, indicating a stable long-term behavior.
• Correlation between assets has a strong influence: a long (or short) inventory in one asset leads to more conservative (or aggressive) quotes in the other, especially when correlation is high.
• The bid-ask spread and skew are not constant or linear, and the closed-form approximations capture these nonlinearities reasonably well for small inventories.