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 an existence result for a recursive scheme of parabolic Cauchy problems  . Our main result establishes the nonexistence of pathwise arbitrage opportunities in a class of strategies containing these Delta hedging strategies and under relatively mild conditions on the local-volatility matrix.
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.
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local-volatility
local volatility
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an existence result for a recursive scheme
existence results for recursive schemes
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and via the existence of functional Cauchy problems on path space
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result establishes
results establish
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a class
classes
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local-volatility
local volatility