Abstract
We consider a strictly pathwise setting for Delta hedging exotic options, based on Follmer’s pathwise Ito 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.
Highlights
The price evolution of a risky asset is usually modeled as a stochastic process defined on some probability space and is subject to model uncertainty
A theory of hedging European options of the form H = h(S(T )) for one-dimensional asset price trajectories S = (S(t))0≤t≤T was developed by Bick and Willinger (1994) by using Follmer’s (1981) approach to pathwise Itocalculus
Using ideas from (Schied and Stadje 2007), we show that such options can be hedged in a strictly pathwise sense if a certain recursive scheme of terminal-value problems (1.1) can be solved, and we provide sufficient conditions for the existence and uniqueness of the corresponding solutions
Summary
The price evolution of a risky asset is usually modeled as a stochastic process defined on some probability space and is subject to model uncertainty. Suppose that f : Rd(+) → R is a continuous function for which there exists a solution v to the following terminal-value problem,
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