A pathwise approach to continuous-time trading
This paper develops a mathematical framework for the analysis of
continuous-time trading strategies which, in contrast to the classical setting
of continuous-time mathematical finance, does not rely on stochastic integrals
or other probabilistic notions. Our purely analytic framework allows for the
derivation of a pathwise self-financial condition for continuous-time trading
strategies, which is consistent with the classical definition in case a
probability model is introduced. Our first proposition provides us with a
pathwise definition of the gain process for a large class of continuous-time,
path-dependent, self-finacing trading strategies, including the important class
of 'delta-hedging' strategies, and is based on the recently developed
'non-anticipative functional calculus'. Two versions of the statement involve
respectively continuous and c\adl\ag price paths. The second proposition is a
pathwise replication result that generalizes the ones obtained in the classical
framework of diffusion models. Moreover, it gives an explicit and purely
pathwise formula for the hedging error of delta-hedging strategies for
path-dependent derivatives across a given set of scenarios. We also provide an
economic justification of our main assumption on price paths.
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