Dealing with the Inventory Risk. A solution to the market making problem

TL;DR

This paper models the market making problem as a stochastic control task, deriving solutions for optimal quotes considering inventory risk, and provides analytical approximations for practical implementation.

Contribution

It introduces a novel approach to solving the market making problem with inventory constraints using linear ODEs and spectral methods, extending prior stochastic control models.

Findings

Optimal quotes can be obtained from linear ODE systems.

Asymptotic behavior of quotes is characterized analytically.

Closed-form approximations for quotes are proposed.

Abstract

Market makers 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 at which they indeed provide liquidity, is challenged by the price risk they bear due to their inventory. In this paper, we consider a stochastic control problem similar to the one introduced by Ho and Stoll and formalized mathematically by Avellaneda and Stoikov. The market is modeled using a reference price $S_t$ following a Brownian motion with standard deviation $\sigma$, arrival rates of buy or sell liquidity-consuming orders depend on the distance to the reference price $S_t$ and a market maker maximizes the expected utility of its P&L over a finite time horizon. We show that the Hamilton-Jacobi-Bellman equations associated to the stochastic optimal control problem can be transformed into a system of linear ordinary differential equations and we solve the market making problem under inventory constraints. We also shed light on the asymptotic behavior of the optimal quotes and propose closed-form approximations based on a spectral characterization of the optimal quotes.