Paper: arXiv 2307.15300

Authors: Ruyi Liu, Jingzhi Tie, Zhen Wu, Qing Zhang

Abstract

The focus of this paper is on identifying the most effective selling strategy for pairs trading of stocks. In pairs trading, a long position is held in one stock while a short position is held in another. The goal is to determine the optimal time to sell the long position and repurchase the short position in order to close the pairs position. The paper presents an optimal pairs-trading selling rule with trading constraints. In particular, the underlying stock prices evolve according to a two dimensional geometric Brownian motion and the trading permission process is given in terms of a two-state {trading allowed, trading not allowed} Markov chain. It is shown that the optimal policy can be determined by a threshold curve which is obtained by solving the associated HJB equations (quasi-variational inequalities). A closed form solution is obtained. A verification theorem is provided. Numerical experiments are also reported to demonstrate the optimal policies and value functions.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a strong theoretical contribution using stochastic control, Hamilton-Jacobi-Bellman (HJB) equations, and verification theorems to establish an optimal pairs-trading exit strategy under Markovian regime constraints. However, it is primarily theoretical with a focus on numerical experiments rather than rigorous empirical backtesting against real historical market data.

Research Flowchart

  flowchart TD
    A[Goal: Determine Optimal Selling Rule<br/>for Pairs Trading with Constraints] --> B[Model Setup: Asset Prices via Geometric Brownian Motion<br/>& Trading Availability via 2-State Markov Chain]
    B --> C[Formulate Stochastic Control Problem<br/>using Hamilton-Jacobi-Bellman Equations]
    C --> D[Solve Quasi-Variational Inequalities<br/>to Obtain Closed-Form Threshold Curve]
    D --> E[Prove Verification Theorem & Run Numerical Experiments]
    E --> F[Outcome: Analytical Optimal Policy &<br/>Characterized Value Functions]