Pathwise no-arbitrage in a class of Delta hedging strategies

TL;DR

This paper develops a pathwise framework for Delta hedging exotic options using Föllmer's calculus, proving the absence of arbitrage in certain strategy classes under mild volatility conditions.

Contribution

It introduces a pathwise approach to Delta hedging with existence results for strategies and demonstrates no arbitrage opportunities in this setting.

Findings

Existence of Delta hedging strategies in a pathwise setting.

Nonexistence of arbitrage opportunities under mild conditions.

Framework based on Föllmer's pathwise Itô calculus.

Abstract

We consider a strictly pathwise setting for Delta hedging exotic options, based on F\"ollmer's pathwise It\=o calculus. Price trajectories are $d$-dimensional continuous functions whose pathwise quadratic variations and covariations are determined by a given local volatility matrix. The existence of Delta hedging strategies in this pathwise setting is established via existence results for recursive schemes of parabolic Cauchy problems and via the existence of functional Cauchy problems on path space. Our main results establish the nonexistence of pathwise arbitrage opportunities in classes of strategies containing these Delta hedging strategies and under relatively mild conditions on the local volatility matrix.