Distributionally robust optimization with applications to risk management

Many decision problems can be formulated as mathematical optimization models. While deterministic optimization problems include only known parameters, real-life decision problems almost invariably involve parameters that are subject to uncertainty. Failure to take this uncertainty under consideration may yield decisions which can lead to unexpected or even catastrophic results if certain scenarios are realized. While stochastic programming is a sound approach to decision making under uncertainty, it assumes that the decision maker has complete knowledge about the probability distribution that governs the uncertain parameters. This assumption is usually unjustified as, for most realistic problems, the probability distribution must be estimated from historical data and is therefore itself uncertain. Failure to take this distributional modeling risk into account can result in unduly optimistic risk assessment and suboptimal decisions. Furthermore, for most distributions, stochastic programs involving chance constraints cannot be solved using polynomial-time algorithms. In contrast to stochastic programming, distributionally robust optimization explicitly accounts for distributional uncertainty. In this framework, it is assumed that the decision maker has access to only partial distributional information, such as the firstand second-order moments as well as the support. Subsequently, the problem is solved under the worst-case distribution that complies with this partial information. This worst-case approach effectively immunizes the problem against distributional modeling risk. The objective of this thesis is to investigate how robust optimization techniques can be used for quantitative risk management. In particular, we study how the risk of large-scale derivative portfolios can be computed as well as minimized, while making minimal assumptions about the probability distribution of the underlying asset returns. Our interest in derivative portfolios stems from the fact that careless investment in derivatives can yield large losses or even bankruptcy. We show that by employing robust optimization techniques we are able to capture the substantial risks involved in derivative investments. Furthermore, we investigate how distributionally robust chance constrained programs can be reformulated or approximated as tractable optimization problems. Throughout the thesis, we aim to derive tractable models that are scalable to industrial-size problems.