Sensitivity of robust optimization problems under drift and volatility uncertainty
Daniel Bartl, Ariel Neufeld, Kyunghyun Park
TL;DR
The paper analyzes robust optimization under drift and volatility uncertainty in continuous-time trading. It establishes a precise first-order sensitivity expansion V(ε) = V(0) + ε V'(0) + O(ε^2) when the uncertainty set is a small ε-neighborhood around baseline parameters, with V'(0) given explicitly via BSDE-derived terms Y^*, Z^*, \\mathcal{Y}^*, \\mathcal{Z}^*. An envelope theorem shows that strategies optimal for the baseline problem perform nearly as well as those optimized for the robust problem, up to second-order terms. The results are illustrated with a robust mean-variance hedging example, and the proofs rely on BSDE representations, GKW decompositions, and stability analyses of the forward-backward system. This provides a practical pathway to approximate robust policies using baseline solutions and explicit sensitivity corrections.
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.