This PhD thesis studies the detection problem associated with pairs trading strategies. We model the difference between the prices of the two underlying stocks in pairs trading strategies as an OrnsteinUhleneck process based on its meanreverting property. The meanreversion rate (and the meanreverting level) of the underlying OrnsteinUhleneck process is assumed to have a change at some random/unobservable time. We consider the problem of detecting the random/unobservable time as accurately as possible. The problem is formulated as a quickest detection problem. We consider the most random scenario where the unobservable/random time is (i) exponentially distributed and (ii) independent from the underlying OrnsteinUhleneck process prior to its change in the meanreversion rate. We formulate the quickest detection problem as a Lagrangian of the probability of false alarm and the expected detection delay. By changing the probability measures we transform the quickest detection problem to a twodimensional Lagrange formulated optimal stopping problem. We reduce the underlying process to its canonical form and transform the optimal stopping problem correspondingly. We also decouple the coefficients on the diffusion term of the underlying twodimensional process by time change. The Lagrange formulated optimal stopping problem is transformed to its Mayer formulation through Ito's formula and the optional sampling theorem. We reduce the Lagrange formulated optimal stopping problem to a freeboundary problem where the derivatives are understood in the sense of Schwartz distribution. We then verify that the canonical infinitesimal generator satisfies Hormander's condition which upgrades the weak solution to a strong (smooth/classic) solution. We determine an upper bound on the rates of convergence in the WaldBellman equations depending on the gain function and the transition density function of the underlying Markov process. Making use of theories from parabolic secondorder partial differential equations, we derive an upper bound on the rates of convergence when the transition density function is not known explicitly. We then introduce a technique of constructing the value functions to enable the convergence of WaldBellman equations to continuoustime Mayer formulated optimal stopping problems with finite horizon. The technique is first applied to the detection problem associated with pairs trading strategies, then applied to various Mayer formulated continuoustime optimal stopping problems. Numerical approximations of WaldBellman equations are obtained through Mathematica algorithms. The value functions are then used to generate the corresponding optimal stopping boundaries through numerical calculations in Mathematica.
Date of Award  31 Dec 2023 

Original language  English 

Awarding Institution   The University of Manchester


Supervisor  Goran Peskir (Supervisor) & Ronnie Loeffen (Supervisor) 

OPTIMAL STOPPING FOR PAIRS TRADING STRATEGIES
Bian, Z. (Author). 31 Dec 2023
Student thesis: Phd