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

Abstract

This thesis develops a mathematical framework for the analysis of continuous-time trading strategies which, in contrast to the classical setting of continuous-time finance, does not rely on stochastic integrals or other probabilistic notions. Using the recently developed 'non-anticipative functional calculus', we first develop a pathwise definition of the gain process for a large class of continuous-time trading strategies which include the important class of delta-hedging strategies, as well as a pathwise definition of the self-financing condition. Using these concepts, we propose a framework for analyzing the performance and robustness of delta-hedging strategies for path-dependent derivatives across a given set of scenarios. Our setting allows for general path-dependent payoffs and does not require any probabilistic assumption on the dynamics of the underlying asset, thereby extending previous results on robustness of hedging strategies in the setting of diffusion models. We obtain a pathwise formula for the hedging error for a general path-dependent derivative and provide sufficient conditions ensuring the robustness of the delta hedge. We show in particular that robust hedges may be obtained in a large class of continuous exponential martingale models under a vertical convexity condition on the payoff functional. Under the same conditions, we show that discontinuities in the underlying asset always deteriorate the hedging performance. These results are applied to the case of Asian options and barrier options. The last chapter, independent of the rest of the thesis, proposes a novel method, jointly developed with Andrea Pascucci and Stefano Pagliarani, for analytical approximations in local volatility models with L\'evy jumps. Numerical tests confirm the effectiveness of the method.