Optimal market making (1605.01862v5)

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.

• The paper presents a unified modeling framework that extends previous methods by incorporating general intensity functions for stochastic market processes.
• It derives closed-form approximations for optimal bid-ask quotes while reducing the dimensionality of the Hamilton-Jacobi-Bellman equations for efficiency.
• The research extends its findings to multi-asset market making, offering practical insights into managing correlated asset portfolios and liquidity risk.

Optimal Market Making: A Comprehensive Analysis

The paper explores a sophisticated domain of quantitative finance, concentrating on market making strategies that provide liquidity in financial markets. At its core, the research seeks to address the optimization challenges faced by market makers—entities or algorithms that continuously quote buy (bid) and sell (ask) prices for various assets in order to profit from the bid-ask spread while managing inventory risk due to price fluctuations.

Key Contributions

1. Modeling Framework: The paper presents a generalized modeling framework that unifies previous approaches, expanding on the seminal work by Avellaneda and Stoikov. This framework accounts for stochastic processes that are exogenous to the market maker's actions and includes general intensity functions instead of the standard exponential intensity functions.
2. Existence and Characterization of Optimal Strategies: The research establishes new general results concerning the existence and structure of optimal market making strategies. It identifies the Hamilton-Jacobi-BeLLMan (HJB) equations involved and reduces their dimensionality, which facilitates more efficient computations.
3. Closed-form Approximations: New closed-form approximations for optimal bid and ask quotes are derived. These approximations extend the Guéant-Lehalle-Fernandez-Tapia formulas to accommodate a broader class of intensity functions and differing optimization criteria within market making.
4. Multi-Asset Market Making: The framework is extended to multi-asset market making, and closed-form approximations for the optimal quotes in this setting are developed. This addresses the practical need for managing portfolios with correlated assets, making the findings especially valuable for practitioners handling large portfolios.
5. Application to Credit Indices: The paper applies the theoretical findings to the case of two credit indices—CDX.NA.IG and CDX.NA.HY. This practical application illustrates the importance of considering correlation in multi-asset frameworks and demonstrates the model's ability to handle real-world cases effectively.

Numerical Results

The paper provides numerical results showcasing the accuracy of the closed-form approximations. The approximations are shown to be robust for small inventory levels, but less so for larger inventories. It emphasizes that while the actual quotes computed numerically can deviate from the approximations, especially under high volatility, the approximations remain useful, indicating a trade-off between computational efficiency and accuracy.

Practical and Theoretical Implications

• Risk Management: The insights into inventory risk management can enhance the strategies employed by market makers, particularly in minimizing unwanted exposure due to volatile price swings or execution uncertainty.
• Market Stability: Understanding and implementing optimal market making strategies can contribute to more stable financial markets by improving liquidity and ensuring more orderly price discovery processes.
• Future Directions: There are opportunities to extend this research, particularly concerning refining models to account for adversarial trading environments, exploring different asset classes, and further integrating machine learning approaches to adaptively update strategies in real-time.

In conclusion, the paper advances the understanding of market making through a robust theoretical framework, practical numerical solutions, and applications to contemporary financial instruments. It provides a stepping stone for further academic inquiry and industry adoption in the increasingly algorithm-driven landscape of financial markets.