Sensitivity of robust optimization problems under drift and volatility uncertainty

Paper: arXiv 2311.11248

Authors: Daniel Bartl, Ariel Neufeld, Kyunghyun Park

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.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework using advanced stochastic analysis to address drift and volatility uncertainty via asymptotic expansion. While mathematically rich and highly novel in providing explicit sensitivity terms, it lacks empirical backtesting and real-world data validation.

Research Flowchart

  flowchart TD
    A["Research Goal: Maximize worst-case portfolio gains under drift & volatility uncertainty (ε-neighborhood)"] --> B["Inputs: Baseline drift/volatility processes & uncertainty parameter ε"]
    B --> C["Methodology: Perform asymptotic expansion as ε → 0"]
    C --> D["Computational Process: Evaluate baseline problem & calculate explicit sensitivity correction term"]
    D --> E["Outcome 1: Worst-case value = Baseline value + ε × Correction term"]
    D --> F["Outcome 2: Construct approximately optimal strategies from baseline optimal strategies"]

Cited References (97)

1. First order Martingale model risk and semi-static hedging — → 2410.06906
2. Sensitivity of causal distributionally robust optimization (2024) — link
3. Quadratic expansions in optimal investment with respect to perturbations of the semimartingale model (2024) — link
4. Duality Method for Multidimensional Nonsmooth Constrained Linear Convex Stochastic Control (2023) — link
5. Sensitivity of Multiperiod Optimization Problems with Respect to the Adapted Wasserstein Distance (2023) — link
6. Robust Retirement and Life Insurance with Inflation Risk and Model Ambiguity (2023) — link
7. A Parametric Approach to the Estimation of Convex Risk Functionals Based on Wasserstein Distance (2022) — link
8. Robust Consumption-Investment with Return Ambiguity: A Dual Approach with Volatility Ambiguity (2022) — link

1 of 97 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.