Sensitivity of robust optimization problems under drift and volatility uncertainty

Abstract

We examine optimization problems in which an investor has the opportunity to trade in $d$ stocks with the goal of maximizing her worst-case cost of cumulative gains and losses. Here, worst-case refers to taking into account all possible drift and volatility processes for the stocks that fall within a $\varepsilon$-neighborhood of predefined fixed baseline processes. Although solving the worst-case problem for a fixed $\varepsilon>0$ is known to be very challenging in general, we show that it can be approximated as $\varepsilon\to 0$ by the baseline problem (computed using the baseline processes) in the following sense: Firstly, the value of the worst-case problem is equal to the value of the baseline problem plus $\varepsilon$ times a correction term. This correction term can be computed explicitly and quantifies how sensitive a given optimization problem is to model uncertainty. Moreover, approximately optimal trading strategies for the worst-case problem can be obtained using optimal strategies from the corresponding baseline problem.