Some Mean-Variance Related Optimal Stopping Problems

Specialeforsvar: Johan Viderø Gunnarsson

Titel: Some Mean-Variance Related Optimal Stopping Problems

Resume: Assuming that the stock price process is a geometric Brownian motion, we study the optimal stopping problems arising from trying to maximize the mean of the stock price while at the same time wanting to minimize a certain measure of risk. By considering the standard deviation and the lower semivariance, we avoid some of the drawbacks of using the variance as a measure of risk. We also consider the closely related problems of maximizing the mean with a constraint on the measure of risk, and of minimizing the measure of risk with a constraint on the mean. First we outline useful results from  standard optimal stopping theory and give a detailed example of a standard optimal stopping problem. We introduce the different forms of optimality including our modified definition of dynamic optimality. Then we solve the cases with non-negative drift and relatively large drift, respectively, that are identical for both measures of risk. Using a Lagrangian approach, we solve the standard deviation problems, and finally, using a more
direct approach, we solve the lower semivariance problems.

Vejleder: Jesper Lund Pedersen
Censor: Kennth Bruhn