Optimal closing of a pair trade with a model containing jumps

TL;DR

This paper extends the optimal closing problem for pair trades to models with jumps, using Ornstein-Uhlenbeck processes driven by finite activity Lévy processes, and provides theoretical and numerical analysis.

Contribution

It generalizes previous models by incorporating jumps via Lévy processes and offers verification theorems and error estimates for the associated free boundary problem.

Findings

Verification theorem established for the jump model

Numerical methods with proven error bounds

Insights from numerical simulations on optimal closing strategies

Abstract

A pair trade is a portfolio consisting of a long position in one asset and a short position in another, and it is a widely applied investment strategy in the financial industry. Recently, Ekstr\"om, Lindberg and Tysk studied the problem of optimally closing a pair trading strategy when the difference of the two assets is modelled by an Ornstein-Uhlenbeck process. In this paper we study the same problem, but the model is generalized to also include jumps. More precisely we assume that the above difference is an Ornstein-Uhlenbeck type process, driven by a L\'evy process of finite activity. We prove a verification theorem and analyze a numerical method for the associated free boundary problem. We prove rigorous error estimates, which are used to draw some conclusions from numerical simulations.