Optimal Liquidation in a Mean-reverting Portfolio

TL;DR

This paper investigates an optimal liquidation strategy for a trader dealing with two mean-reverting assets, using advanced mathematical tools like HJB equations and FBSDEs to derive solutions and analyze performance.

Contribution

It introduces a novel approach to solve the liquidation problem with mean-reverting assets using FBSDEs, and provides a detailed analysis of the solution's properties and numerical performance.

Findings

HJB and FBSDE solutions are numerically close

The value function differs from the classical solution of the HJB

Numerical tests demonstrate the model's effectiveness

Abstract

In this work we study a finite horizon optimal liquidation problem with multiplicative price impact in algorithmic trading, using market orders. We analyze the case when an agent is trading on a market with two financial assets, whose difference of log-prices is modelled with a mean-reverting process. The agent's task is to liquidate an initial position of shares of one of the two financial assets, without having the possibility of trading the other stock. The criterion to be optimized consists in maximising the expected final value of the agent, with a running inventory penalty. The main result of this paper consists in finding a classical solution of the Hamilton-Jacobi-Bellman (HJB) equation associated to this problem, which is proved to not coincide with the value function. However, we find the value function as a solution to the forward-backward stochastic differential equation (FBSDE) associated to the problem. We provide numerical tests showing that the HJB and FBSDE solutions are close to each other and analysing performance of the described model. We also prove a verification theorem and a comparison principle for the viscosity solution to the HJB equation.