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 Ornstein-Uhleneck process based on its mean-reverting property. The mean-reversion rate (and the mean-reverting level) of the underlying Ornstein-Uhleneck 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 Ornstein-Uhleneck process prior to its change in the mean-reversion 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 two-dimensional 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 two-dimensional 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 free-boundary 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 Wald-Bellman equations depending on the gain function and the transition density function of the underlying Markov process. Making use of theories from parabolic second-order 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 Wald-Bellman equations to continuous-time 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 continuous-time optimal stopping problems. Numerical approximations of Wald-Bellman 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|
- The University of Manchester
|Supervisor||Goran Peskir (Supervisor) & Ronnie Loeffen (Supervisor)|