Analytical Pairs Trading Under Different Assumptions on the Spread and Ratio Dynamics

Abstract

We demonstrate how arbitrarily sized long/short baskets whose portfolio value is modelled with spread or ratio of any asset weighting can be treated as a sequential stopping problem. In particular, when the underlying data generating process follows an Ornstein-Uhlenbeck, Cox-Ingersoll-Ross or GARCH diffusion, we derive closed form expressions for mean and variance of trade time and trade return (with transaction costs). From these expressions the risk and return characteristics are examined when the objective functions aim to maximisise expected return and Sharpe ratio. The SDE where possible are estimated using exact maximum likelihood, Euler and Hermite Polynomial Expansion. The estimation parameters and their sensitivity can be used to rank potential trades and understand the risk/return profi le involved with this style of investment.