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Gamma Hedging without Rough Paths

John Armstrong, Purba Das

TL;DR

The paper addresses the robustness of gamma hedging without rough-path theory by employing a pathwise, partition-based Taylor expansion governed by $p$-th variation and finite quadratic variation. It delivers elementary, model-free proofs showing that the cumulative hedging P&L tends to zero for European options and, in diffusion/Black–Scholes-like settings, enables delta- and gamma-hedging without stochastic integrals or self-financing in continuous time. The framework extends to barrier and Asian options, providing explicit conditions (e.g., gamma-neutrality or variance structure) under which replication holds pathwise. Overall, the work offers a simplified, operator-theoretic, pathwise approach to hedging that sidesteps stochastic calculus while retaining practical replication results and applicability to exotics.

Abstract

We show how the robustness of gamma hedging can be understood without using rough-path theory. Instead, we use the concepts of $p^{th}$ variation along a partition sequence and Taylor's theorem directly, rather than defining an integral and proving a version of Itô's lemma. The same approach allows classical results on delta-hedging to be proved without defining an integral and without the need to define the concept of self-financing in continuous time. We show that the approach can also be applied to barrier options and Asian options

Gamma Hedging without Rough Paths

TL;DR

The paper addresses the robustness of gamma hedging without rough-path theory by employing a pathwise, partition-based Taylor expansion governed by -th variation and finite quadratic variation. It delivers elementary, model-free proofs showing that the cumulative hedging P&L tends to zero for European options and, in diffusion/Black–Scholes-like settings, enables delta- and gamma-hedging without stochastic integrals or self-financing in continuous time. The framework extends to barrier and Asian options, providing explicit conditions (e.g., gamma-neutrality or variance structure) under which replication holds pathwise. Overall, the work offers a simplified, operator-theoretic, pathwise approach to hedging that sidesteps stochastic calculus while retaining practical replication results and applicability to exotics.

Abstract

We show how the robustness of gamma hedging can be understood without using rough-path theory. Instead, we use the concepts of variation along a partition sequence and Taylor's theorem directly, rather than defining an integral and proving a version of Itô's lemma. The same approach allows classical results on delta-hedging to be proved without defining an integral and without the need to define the concept of self-financing in continuous time. We show that the approach can also be applied to barrier options and Asian options
Paper Structure (4 sections, 7 theorems, 37 equations)

This paper contains 4 sections, 7 theorems, 37 equations.

Table of Contents

  1. Pathwise convergence
  2. Integrating against solutions of ODEs
  3. Paths with finite quadratic variation
  4. Asian options

Key Result

Theorem 1.3

Let $\pi$ be a sequence of partitions of $[0,T]$ with mesh tending to zero. For each $\alpha \in \{1,2\}$, let ${\cal V}^\alpha$ be a pair of finite-dimensional normed vector spaces and let $X^\alpha \in C([0,T];U)$ be a path with vanishing $p_\alpha$th variation along $\pi$. Suppose $1 \leq p_1\leq then

Theorems & Definitions (16)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • proof
  • Corollary 1.4
  • proof
  • Corollary 1.5
  • proof
  • Lemma 2.1
  • proof
  • ...and 6 more