Nonlinear Optimal Stopping Problems on Finite Horizon

Abstract

This PhD thesis studies the nonlinear optimal stopping problem on the finite horizon which is associated with the optimal mean-variance selling strategy problem. While nonlinear optimal stopping and control problems have been the focus of numerous research studies, this work represents the first investigation of such a problem within a finite horizon setting. Finite horizon problems typically involve a predetermined, limited timeframe for decision-making, which adds a unique dimension to the analysis of the problem and finding its solution. We model the stock price $X$ by geometric Brownian motion with a starting point $x$, a drift $\mu$, and a volatility $\sigma$. The investor intends to find the optimal selling time to maximise the expected return with the lowest risk level. According to this setting, we introduce the Markowitz mean-variance model and construct the optimal stopping problem \begin{equation}\notag \sup_{0 \leq \tau \leq T-t} \left[\mathrm{E}_{t,x}\left(X_{t+\tau} \right) - c \, Var_{t,x}\left(X_{t+\tau} \right) \right], \end{equation} where $t$ runs from $0$ to $T$, the supremum is taken over all stopping times $\tau$ of $X$, and $c > 0$ is a given and fixed constant. We can conclude that it is optimal to stop immediately if $\mu < 0$. Using the Lagrange multipliers approach, we demonstrate that the nonlinear problems for $\mu > 0$ can be simplified to a family of linear optimal stopping problems. After transforming the linear optimal stopping problems to a parabolic free-boundary PDE problem, we employ a change-of-variable formula to get an integral equation for the unknown optimal stopping boundary. To solve the integral equation, we have to determine the solution to the integral equation at $t = T$ first. We recast the optimal stopping problem into a Lagrangian form by Ito's formula to generate the continuation area within which $X$ should not be stopped. This provides us with an idea of how to determine the boundary value for the solution at $t = T$. Then we can also verify the continuous and smooth-fit properties of the optimal stopping boundary. The integral equation is solved using numerical approximation methods, coded in Python. Leveraging the previously derived infinite horizon problem's solution, we estimate the finite horizon case's Lagrange multiplier using its infinite horizon counterpart. We divide the finite time horizon into $n$ subintervals, facilitating the use of a backward induction root-finding algorithm to calculate the optimal stopping boundary values from $T$ down to $0$. The algorithm's precision increases with a longer maturity time T and a greater number of subintervals.

Date of Award	1 Aug 2025
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Goran Peskir (Supervisor) & Denis Denisov (Supervisor)