Essays in financial engineering

Abstract

This thesis consists of three essays that apply techniques of operations research to problems in financial engineering. In particular, we study problems in portfolio optimization and options pricing. The first essay is motivated by the fact that derivative securities are equivalent to specific dynamic trading strategies in complete markets. This suggests the possibility of constructing buy-and-hold portfolios of options that mimic certain dynamic investment policies, e.g., asset-allocation rules. We explore this possibility by solving the following problem: given an optimal dynamic investment policy, find a set of options at the start of the investment horizon which will come closest to the optimal dynamic investment policy. We solve this problem for several combinations of preferences, return dynamics, and optimality criteria, and show that under certain conditions, a portfolio consisting of just a few european options is an excellent substitute for considerably more complex dynamic investment policies. In the second essay, we develop a method for pricing and exercising high-dimensional American options. The approach is based on approximate dynamic programming using nonlinear regression to approximate the value function. Using the approximate dynamic programming solutions, we construct upper and lower bounds on the option prices. These bounds can be evaluated by Monte Carlo simulation, and they are general enough to be used in conjunction with other approximate methods for pricing American options.

(cont.) We characterize the theoretical worst-case performance of the pricing bounds and examine how they may be used for hedging and exercising the option. We also discuss the implications for the design of the approximate pricing algorithm and illustrate its performance on a set of sample problems where we price call options on the maximum and the geometric mean of a collection of stocks. The third essay explores the possibility of solving high-dimensional portfolio optimization problems using approximate dynamic programming. In particular, we employ approximate value iteration where the portfolio strategy at each time period is obtained using quadratic approximations to the approximate value function. We then compare the resulting solution to the best heuristic strategies available. Though the approximate dynamic programming solutions are often competitive, they are sometimes dominated by the best heuristic strategy. On such occasions we conclude that inaccuracies in the quadratic approximations are responsible for the poor performance. Finally, we compare our results to other recent work in this area and suggest possible methods for improving these algorithms.