A robust optimization approach to finance

Abstract

An important issue in real-world optimization problems is how to treat uncertain coefficients. Robust optimization is a modeling methodology that takes a deterministic view: the optimal solution is required to remain feasible for any realization of the uncertain coefficients within prescribed uncertainty sets. The focus of this thesis is on robust linear programming problems in which the uncertainty sets are polytopes. The assumption of polyhedral uncertainty leads to compact, efficiently solvable linear formulations. In the first part of the thesis, we study special types of polyhedral uncertainty sets that allow for incorporating moment information about the distribution of the uncertain coefficients, and for controlling the tradeoff between robustness and optimality. We provide probabilistic guarantees on the feasibility of optimal solutions obtained with such uncertainty sets for any realization of the uncertain coefficients. We then illustrate the versatility of robust polyhedral formulations by studying three financial applications: single period portfolio optimization, multiperiod portfolio management, and credit risk estimation. In the area of single period portfolio optimization, we propose ways of modeling inaccuracy in parameter estimates, and explore the benefits of robust optimal strategies through computational experiments with the statistical estimation of a particular measure of portfolio risk - sample shortfall. We emphasize the advantages of linear, as opposed to nonlinear, robust formulations in large portfolio problems with integrality constraints.

(cont.) In the area of multiperiod portfolio management, we propose robust polyhedral formulations that use some minimal information about long-term direction of movement of asset returns to make informed decisions about portfolio rebalancing over the short term. The suggested formulations allow for including considerations of transaction costs and taxes while keeping the dimension of the problem low. In the area of credit risk estimation, we propose a model for estimating the survival probability distribution and the fair prices of credit risky bonds from market prices of similar credit risky securities. We address the issue of uncertainty in key parameters of the model, such as discount factors, by using robust optimization modeling. We also suggest a method for classification of credit risky bonds based on integer programming techniques.