# Pairs Trading under Drift Uncertainty and Risk Penalization

## TL;DR

This paper develops a dynamic pairs trading strategy model under drift uncertainty, using stochastic filtering and risk penalization, providing optimal strategies and numerical insights for a two-state Markov chain scenario.

## Contribution

It introduces a novel approach combining stochastic filtering with risk penalization in pairs trading under Markovian drift uncertainty, solving for optimal strategies under partial information.

## Key findings

- Optimal dollar-neutral strategies characterized
- Certainty equivalence principle holds for the strategy
- Numerical analysis demonstrates the model's application

## Abstract

In this work, we study a dynamic portfolio optimization problem related to pairs trading, which is an investment strategy that matches a long position in one security with a short position in another security with similar characteristics. The relationship between pairs, called a spread, is modeled by a Gaussian mean-reverting process whose drift rate is modulated by an unobservable continuous-time, finite-state Markov chain. Using the classical stochastic filtering theory, we reduce this problem with partial information to the one with full information and solve it for the logarithmic utility function, where the terminal wealth is penalized by the riskiness of the portfolio according to the realized volatility of the wealth process. We characterize optimal dollar-neutral strategies as well as optimal value functions under full and partial information and show that the certainty equivalence principle holds for the optimal portfolio strategy. Finally, we provide a numerical analysis for a toy example with a two-state Markov chain.