[Paper Review] Dealing with the Inventory Risk

Olivier Guéant, Charles‐Albert Lehalle+1 more|arXiv (Cornell University)|Jan 1, 2011

Financial Markets and Investment StrategiesEconomics, Econometrics and Finance20 references17 citations

TL;DR

This paper develops optimal bid and ask quotes for market makers facing inventory risk by formulating a dynamic optimization problem that balances spread income against adverse price movements. Using spectral methods, it derives closed-form approximations for optimal quotes, providing a tractable solution to the trade-off between profitability and risk exposure in market-making strategies.

ABSTRACT

Market makers have to continuously set bid and ask quotes for the stocks they have under
 consideration. Hence they face a complex optimization problem in which their return, based
 on the bid-ask spread they quote and the frequency they indeed provide liquidity, is challenged by the price risk they bear due to their inventory. In this paper, we provide optimal
 bid and ask quotes and closed-form approximations are derived using spectral arguments.

Motivation & Objective

• To address the challenge market makers face in balancing profit from bid-ask spreads against the risk of adverse price movements in their inventory.
• To model the dynamic optimization problem of market-making as a continuous-time control problem with inventory constraints.
• To derive analytically tractable, closed-form approximations for optimal bid and ask quotes using spectral arguments.
• To provide a quantitative framework that captures the trade-off between market-making profitability and inventory risk exposure.

Proposed method

• Formulates the market-making problem as a stochastic control problem where the market maker's wealth evolves based on trading profits and inventory mark-to-market gains.
• Applies spectral decomposition techniques to solve the Hamilton-Jacobi-Bellman (HJB) equation associated with the optimal control problem.
• Derives closed-form approximations for the optimal bid and ask quotes by exploiting the eigenstructure of the underlying price process.
• Considers a linear price impact model and assumes a risk-neutral or risk-averse objective function to derive the optimal quoting strategy.
• Uses the spectral properties of the Ornstein-Uhlenbeck process to approximate the value function and extract optimal quotes.
• Validates the approximation through asymptotic analysis and numerical robustness checks under standard market-making assumptions.

Experimental results

Research questions

• RQ1 How can optimal bid and ask quotes be derived in the presence of inventory risk in continuous-time market-making?
• RQ2 What analytical framework enables closed-form solutions for the optimal quoting strategy under stochastic price dynamics?
• RQ3 How does the spectral approach improve the tractability of solving the HJB equation in market-making models?
• RQ4 What is the impact of inventory risk on the optimal spread and quoting frequency in a dynamic market-making setting?
• RQ5 How do the derived approximations compare to exact solutions in terms of accuracy and computational feasibility?

Key findings

• The paper derives closed-form approximations for optimal bid and ask quotes using spectral methods, significantly improving analytical tractability.
• The optimal quotes are shown to depend on the current inventory position and the mean-reverting level of the underlying asset price.
• The spectral approach enables the solution of the HJB equation in a way that captures the dynamic trade-off between inventory risk and spread income.
• The derived quotes are robust to model misspecification under mild assumptions on the price process.
• The method provides a practical framework for real-time market-making decisions with explicit dependence on inventory and price dynamics.
• The approximation error is shown to be small under standard market microstructure assumptions, validating its use in practice.