Pathwise no-arbitrage in a class of Delta hedging strategies

Alexander Schied; Iryna Voloshchenko

doi:10.1186/s41546-016-0003-2

Article Contents

2016, Volume 1: 3. Doi: 10.1186/s41546-016-0003-2

This volume Previous Article On approximation of BSDE and multi-step MLE-processes Next Article Mean-field stochastic linear quadratic optimal control problems: closed-loop solvability

Pathwise no-arbitrage in a class of Delta hedging strategies

Alexander Schied^, and
Iryna Voloshchenko

Department of Mathematics, University of Mannheim, 68131 Mannheim, Germany

Alexander Schied, schied@uni-mannheim.de

Received: January 06, 2016

Revised: June 09, 2016

Published: September 2020

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Abstract

We consider a strictly pathwise setting for Delta hedging exotic options, based on Föllmer's pathwise Itô 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.

Keywords:

Pathwise hedging,
Exotic options,
Pathwise arbitrage,
Pathwise Itô calculus,
Föllmer integral,
Local volatility,
Functional Itô formula,
Functional Cauchy problem on path space.

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