A pathwise approach to continuous-time trading

TL;DR

This paper introduces a purely analytic, pathwise framework for continuous-time trading strategies that does not depend on probabilistic models, enabling explicit hedging and replication results for path-dependent derivatives.

Contribution

It develops a novel non-probabilistic, pathwise approach to continuous-time trading, including definitions of gain processes and hedging error formulas, extending classical results.

Findings

Pathwise definition of gain process for self-financing strategies

Explicit formula for hedging error in delta-hedging

Generalization of replication results beyond diffusion models

Abstract

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.