Optimal Trading Under the American Perpetual Put Option for Geometric Brownian Motion and Mean-reverting Processes

Abstract

This thesis is focused on the perpetual American put option under the geometric Brownian motion and mean-reverting models. Two approaches, which have been applied before to the call option of a mean-reverting process, will be studied in details for the two models. The first approach amounts to solving the associated quasi-variational inequality for the optimal stopping problem. A verification theorem is proved to demonstrate that the solution to the quasi-variational inequality agrees with the value function. The second approach is based on detailed analyses of an auxiliary two-point stopping problem, which leads to an explicit expression for the value function. Both approaches identify an optimal execution rule for the two models.