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Non-concave stochastic optimal control in finite discrete time under model uncertainty

Ariel Neufeld, Julian Sester

TL;DR

This work develops a general, time-consistent framework for non-concave robust stochastic control in finite discrete time under model uncertainty, allowing non-dominated priors and path-dependent ambiguity. A dynamic programming principle is established, with one-step optimization and measurable selectors yielding both an optimal control and a worst-case measure, under Hölder regularity and compactness assumptions. Ambiguity sets are instantiated via Wasserstein balls around path-dependent references and via parametric families, with precise stability guarantees and bounds comparing robust versus non-robust values. The framework is applied to data-driven hedging with asymmetric loss, demonstrating improved hedging performance during crises due to robustness against model misspecification, and it is backed by a thorough numerical method using neural networks and deep DP. Overall, the paper provides a versatile, theoretically grounded approach to robust planning under non-convex utilities with practical finance applications.

Abstract

In this article we present a general framework for non-concave robust stochastic control problems under model uncertainty in a discrete time finite horizon setting. Our framework allows to consider a variety of different path-dependent ambiguity sets of probability measures comprising, as a natural example, the ambiguity set defined via Wasserstein-balls around path-dependent reference measures with path-dependent radii, as well as parametric classes of probability distributions. We establish a dynamic programming principle which allows to derive both optimal control and worst-case measure by solving recursively a sequence of one-step optimization problems. Moreover, we derive upper bounds for the difference of the values of the robust and non-robust stochastic control problem in the Wasserstein uncertainty and parameter uncertainty case. As a concrete application, we study the robust hedging problem of financial derivatives under an asymmetric (and non-convex) loss function accounting for different preferences of sell- and buy side when it comes to the hedging of financial derivatives. As our entirely data-driven ambiguity set of probability measures, we consider Wasserstein-balls around the empirical measure derived from real financial data. We demonstrate that during adverse scenarios such as a financial crisis, our robust approach outperforms typical model-based hedging strategies such as the classical Delta-hedging strategy as well as the hedging strategy obtained in the non-robust setting with respect to the empirical measure and therefore overcomes the problem of model misspecification in such critical periods.

Non-concave stochastic optimal control in finite discrete time under model uncertainty

TL;DR

This work develops a general, time-consistent framework for non-concave robust stochastic control in finite discrete time under model uncertainty, allowing non-dominated priors and path-dependent ambiguity. A dynamic programming principle is established, with one-step optimization and measurable selectors yielding both an optimal control and a worst-case measure, under Hölder regularity and compactness assumptions. Ambiguity sets are instantiated via Wasserstein balls around path-dependent references and via parametric families, with precise stability guarantees and bounds comparing robust versus non-robust values. The framework is applied to data-driven hedging with asymmetric loss, demonstrating improved hedging performance during crises due to robustness against model misspecification, and it is backed by a thorough numerical method using neural networks and deep DP. Overall, the paper provides a versatile, theoretically grounded approach to robust planning under non-convex utilities with practical finance applications.

Abstract

In this article we present a general framework for non-concave robust stochastic control problems under model uncertainty in a discrete time finite horizon setting. Our framework allows to consider a variety of different path-dependent ambiguity sets of probability measures comprising, as a natural example, the ambiguity set defined via Wasserstein-balls around path-dependent reference measures with path-dependent radii, as well as parametric classes of probability distributions. We establish a dynamic programming principle which allows to derive both optimal control and worst-case measure by solving recursively a sequence of one-step optimization problems. Moreover, we derive upper bounds for the difference of the values of the robust and non-robust stochastic control problem in the Wasserstein uncertainty and parameter uncertainty case. As a concrete application, we study the robust hedging problem of financial derivatives under an asymmetric (and non-convex) loss function accounting for different preferences of sell- and buy side when it comes to the hedging of financial derivatives. As our entirely data-driven ambiguity set of probability measures, we consider Wasserstein-balls around the empirical measure derived from real financial data. We demonstrate that during adverse scenarios such as a financial crisis, our robust approach outperforms typical model-based hedging strategies such as the classical Delta-hedging strategy as well as the hedging strategy obtained in the non-robust setting with respect to the empirical measure and therefore overcomes the problem of model misspecification in such critical periods.
Paper Structure (24 sections, 18 theorems, 146 equations, 6 figures, 2 tables, 2 algorithms)

This paper contains 24 sections, 18 theorems, 146 equations, 6 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Setting
  3. Dynamic Programming Principle
  4. Modeling Ambiguity and Controls
  5. Modeling ambiguity with the Wasserstein distance
  6. Modeling ambiguity in parametric models
  7. Modeling of control sets
  8. Bounding the difference of value functions with and without model uncertainty
  9. Comparison to existing literature and our contributions
  10. Application: Data-driven hedging with asymmetric loss function
  11. Setting to data-driven hedging with asymmetric loss function
  12. Numerical method
  13. Empirical Experiments
  14. Data
  15. Hedging at-the-money call options
  16. ...and 9 more sections

Key Result

Theorem 3.1

Suppose that Assumption asu_A, Assumption asu_psi, and Assumption asu_P are fulfilled. Then the following holds.

Figures (6)

  • Figure 1: The three plots show the loss function described in \ref{['eq_u_prospect']} for different choices of the parameters $\mathfrak{a}$ and $\mathfrak{b}$.
  • Figure 2: The graph depicts the normalized evolution of the price of the stocks of Google, Ebay, Amazon, Microsoft, and Apple and the separation of the data into a training period (beginning of January $2010$ until beginning of February $2020$) and the two test periods: beginning of February $2020$ until mid of April $2020$, and mid of April $2020$ until end of June $2020$.
  • Figure 3: Cumulated hedging errors and error distributions for different strategies during Test Period 1. The top panels show the cumulative hedging errors over time, while the bottom panels display the corresponding histograms of hedging error distributions obtained over all $40$ hedges.
  • Figure 4: Cumulated hedging errors and error distributions for different strategies during Test Period 2. The top panels show the cumulative hedging errors over time, while the bottom panels display the corresponding histograms of hedging error distributions obtained over all $40$ hedges.
  • Figure 5: Cumulated hedging errors and error distributions for different strategies when hedging a basket option during Test Period 1. The left panel shows the cumulative hedging errors over time, while the right panel displays the corresponding histograms of hedging error distributions obtained over all $40$ hedges.
  • ...and 1 more figures

Theorems & Definitions (44)

  • Remark 2.2
  • Remark 2.5
  • Remark 2.6
  • Theorem 3.1
  • Theorem 4.1
  • Remark 4.2
  • Theorem 4.3
  • Remark 4.4
  • Proposition 4.5: Normal distributions
  • Remark 4.6
  • ...and 34 more