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This thesis develops a mathematical framework for the analysis ofcontinuous-time trading strategies which, in contrast to the classical settingof continuous-time finance, does not rely on stochastic integrals or otherprobabilistic notions. Using the recently developed 'non-anticipativefunctional calculus', we first develop a pathwise definition of the gainprocess for a large class of continuous-time trading strategies which includethe important class of delta-hedging strategies, as well as a pathwisedefinition of the self-financing condition. Using these concepts, we propose aframework for analyzing the performance and robustness of delta-hedgingstrategies for path-dependent derivatives across a given set of scenarios. Oursetting allows for general path-dependent payoffs and does not require anyprobabilistic assumption on the dynamics of the underlying asset, therebyextending previous results on robustness of hedging strategies in the settingof diffusion models. We obtain a pathwise formula for the hedging error for ageneral path-dependent derivative and provide sufficient conditions ensuringthe robustness of the delta hedge. We show in particular that robust hedges maybe obtained in a large class of continuous exponential martingale models undera vertical convexity condition on the payoff functional. Under the sameconditions, we show that discontinuities in the underlying asset alwaysdeteriorate the hedging performance. These results are applied to the case ofAsian options and barrier options. The last chapter, independent of the rest ofthe thesis, proposes a novel method, jointly developed with Andrea Pascucci andStefano Pagliarani, for analytical approximations in local volatility modelswith L\'evy jumps. Numerical tests confirm the effectiveness of the method.

Pathwise functional calculus and applications to continuous-time finance (PhD thesis)