Paper: arXiv 2309.05054

Authors: John Armstrong, Andrei Ionescu

Abstract

We apply rough-path theory to study the discrete-time gamma-hedging strategy. We show that if a trader knows that the market price of a set of European options will be given by a diffusive pricing model, then the discrete-time gamma-hedging strategy will enable them to replicate other European options so long as the underlying pricing signal is sufficiently regular. This is a sure result and does not require that the underlying pricing signal has a quadratic variation corresponding to a probabilisitic pricing model. We show how to generalise this result to exotic derivatives when the gamma is defined to be the Gubinelli derivative of the delta by deriving rough-path versions of the Clark–Ocone formula which hold surely. We illustrate our theory by proving that if the stock price process is sufficiently regular, as is the implied volatility process of a European derivative with maturity $T$ and smooth payoff $f(S_T)$ satisfying $f^{\prime \prime}>0$, one can replicate with certainty any European derivative with smooth payoff and maturity $T$.

Complexity vs Empirical Score

  • Math Complexity: 9.2/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper introduces a highly novel mathematical framework by applying rough-path theory and Gubinelli derivatives to gamma hedging and pathwise replication without probabilistic assumptions. However, it is purely theoretical with no empirical backtesting or real-market data validation included in the summary.

Research Flowchart

  flowchart TD
    A[Research Goal: Sure Replication of Options via Discrete Gamma-Hedging] --> B[Input: Sufficiently Regular Stock & Implied Volatility Paths]
    B --> C[Methodology: Apply Rough-Path Theory & Gubinelli Derivatives]
    C --> D[Computation: Derive Surely-Holding Rough Clark-Ocone Formula]
    D --> E[Outcome 1: Discrete Gamma-Hedging Replicates European Options without Probabilistic Assumptions]
    D --> F[Outcome 2: Generalization to Exotic Derivatives with Convex Payoffs f'' > 0]