  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.
