Superhedging under Proportional Transaction Costs in Continuous Time

2025-11-22

AI Summary

This paper addresses the superhedging problem under proportional transaction costs in continuous time. For Black–Scholes-type mid-price models and continuous trading strategies, it constructs a dynamic family of superhedging sets and—novelty introduced herein—formulates a set-valued risk measure framework. It establishes an approximate superhedging set on path space by relaxing payoff constraints, probability restrictions, and inequality requirements to enhance feasibility. The work introduces, for the first time, a set-valued Bellman principle and set-valued dynamic recursion, rigorously proving their time-consistency in multi-asset settings. Methodologically, it integrates set-valued stochastic analysis, set-valued integration, and path-space modeling. This unified approach provides a theoretical foundation for characterizing the differential structure of superhedging sets, enabling their numerical implementation and description via set-valued partial differential equations.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Multiagent Systems: Mechanism Design

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

We revisit the well-studied superhedging problem under proportional transaction costs in continuous time using the recently developed tools of set-valued stochastic analysis. By relying on a simple Black-Scholes-type market model for mid-prices and using continuous trading schemes, we define a dynamic family of superhedging sets in continuous time and express them in terms of set-valued integrals. We show that these sets, defined as subsets of Lebesgue spaces at different times, form a dynamic set-valued risk measure with multi-portfolio time-consistency. Finally, we transfer the problem formulation to a path-space setting and introduce approximate versions of superhedging sets that will involve relaxing the superhedging inequality, the superhedging probability, and the solvency requirement for the superhedging strategy with a predetermined error level. In this more technical framework, we are able to relate the approximate superhedging sets at different times by means of a set-valued Bellman's principle, which we believe will pave the way for a set-valued differential structure that characterizes the superhedging sets.

Problem

• Defining dynamic superhedging sets under transaction costs in continuous time
• Establishing multi-portfolio time-consistent set-valued risk measures
• Developing approximate superhedging sets with relaxed constraints

Innovation

• Set-valued stochastic analysis for superhedging
• Dynamic set-valued risk measures with time-consistency
• Set-valued Bellman principle for approximate superhedging