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Superhedging under Proportional Transaction Costs in Continuous Time

Atiqah Almuzaini, Çağın Ararat, Jin Ma

TL;DR

The paper addresses dynamic superhedging under proportional transaction costs in continuous time within a set-valued stochastic analysis framework. It constructs a tractable multi-asset Black-Scholes-type market with a solvency cone and derives a dynamic family of superhedging sets in $\mathbb{L}^2$ via a functional set-valued integral, establishing multi-portfolio time-consistency. It then develops both a functional (Lebesgue-integral) formulation and a path-space (process- and pathwise) formulation, proving a set-valued dynamic programming principle and a pathwise DP principle, and introduces approximate notions to connect the two viewpoints and to motivate potential set-valued differential structures. The results advance the theory and computation of time-consistent, multivariate superhedging under transaction costs, and lay groundwork for a set-valued stochastic calculus and Bellman-type formalism in conic market models.

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.

Superhedging under Proportional Transaction Costs in Continuous Time

TL;DR

The paper addresses dynamic superhedging under proportional transaction costs in continuous time within a set-valued stochastic analysis framework. It constructs a tractable multi-asset Black-Scholes-type market with a solvency cone and derives a dynamic family of superhedging sets in via a functional set-valued integral, establishing multi-portfolio time-consistency. It then develops both a functional (Lebesgue-integral) formulation and a path-space (process- and pathwise) formulation, proving a set-valued dynamic programming principle and a pathwise DP principle, and introduces approximate notions to connect the two viewpoints and to motivate potential set-valued differential structures. The results advance the theory and computation of time-consistent, multivariate superhedging under transaction costs, and lay groundwork for a set-valued stochastic calculus and Bellman-type formalism in conic market models.

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.

Paper Structure

This paper contains 5 sections, 17 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Solvency Cone and Consistent Prices
  4. Functional Formulation of the Superhedging Problem
  5. Pathwise Formulation of the Superhedging Problem

Key Result

Theorem 2.1

Let $M\in \mathscr{C}(\mathbb{L}^p_\mathscr{E}(\Omega,\mathbb{R}^d))$, where $p\in [1,+\infty)$. Then, $M$ is $\mathscr{H}$-decomposable if and only if there exists a unique $F\in \mathscr{A}^p_\mathscr{H}(\Omega,\mathscr{C} (\mathbb{R}^d))$ such that $M=\mathbb{S}^p_\mathscr{H}(F)$.

Theorems & Definitions (25)

  • Theorem 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4
  • Proposition 3.1
  • Remark 3.2
  • Theorem 3.3
  • Proposition 3.4
  • Corollary 3.5
  • Definition 3.6
  • ...and 15 more