We develop a non-anticipative calculus for path-dependent functionals of a semimartingale, using a notion of pathwise functional derivative proposed by B. Dupire. The key ingredient is a functional extension of the Ito formula, which is used to derive a martingale representation formula for square integrable martingales. Regular functionals of a semimartingale S which have the local martingale property are characterized as solutions of a functional dierential equation, for which a uniqueness result is given.

This result is used to derive a universal pricing equation for the price of path-dependent derivatives with underlying asset S: this pricing equation is shown to be a functional equation whose coefficients involve the local characteristics of S. Using these results we derive a general formula for the hedging strategy of a path-dependent contingent

claim and present a numerical method for computing this hedging strategy. By contrast with methods based on Malliavin calculus, this representation is based on non-anticipative quantities which may be computed pathwise and leads to simple simulation-based estimators for computing hedging strategies for path-dependent options.