Data-Driven Robust Optimization Approaches with Applications in Finance

Abstract

Decision making under uncertainty is prevalent in many real-world applications such as stock investing and production planning. The process of determining the optimal decisions relies not only on good quality data but also on the correct modeling assumptions, which are not always possible in reality. This, therefore, introduces instability of the so-called optimal decisions. A traditional method of dealing with decision uncertainty is sensitivity analysis which studies how the uncertainty of the optimal decisions can be attributed to the uncertainty of the model inputs and assumptions. Alternatively, we can use robust optimization. This dissertation explores both approaches in different Finance applications. In particular, with respect to the sensitivity analysis, we study the problem of American option pricing. American options grant the holders the rights to buy/sell a specific underlying asset at a price and time in the future before a pre-determined expiry date. Pricing American options is a continuous-time optimization problem of finding an optimal stopping time that maximizes the expected payoff of the option holders. The optimal stopping time depends on various random parameters such as the interest rate, the underlying asset prices, etc. We are interested in deriving explicit expressions to assess the sensitivity of the options prices as the pricing inputs change. Robust optimization, on the other hand, is a useful tool to ensure that the optimal decisions account for the uncertainty in the data and the modeling process. We propose two new frameworks, the Conditional Value-at-Risk robust optimization and the functionally robust optimization respectively. In Conditional Value-at-Risk robust optimization, we try to find an optimal decision that performs well in a subset of some worst-case estimators of the random parameters of the optimization problem. With functional robust optimization, we construct a decision-making problem using the worst functional fitting of the underlying data. We provide mathematical formulations of the two newly proposed robust optimization frameworks and present algorithms to solve them. Numerical results are presented to showcase the performance of the new problems.