Pathwise Analysis and Hedging of Continuous-Time Trading Strategies

Summary

The paper develops an analytic framework for continuous-time trading that does not require stochastic integrals or a probability model. It defines gains and self-financing conditions directly from price paths, while aiming to remain consistent with classical definitions when a probabilistic model is added. The approach uses non-anticipative functional calculus and covers a broad class of path-dependent strategies, including delta hedging, with formulations for both continuous and càdlàg price paths.

A second result gives a pathwise replication theorem extending results from diffusion-model finance and an explicit formula for the hedging error of delta-hedging path-dependent derivatives across specified scenarios. The authors also offer an economic rationale for their assumption about price paths. The supplied description does not state the assumptions in detail or provide empirical validation, so the account chiefly presents a mathematical framework and results rather than evidence of practical trading performance.

Key ideas

• The framework analyzes continuous-time trading directly from price paths without probabilistic tools.
• It defines gain processes and self-financing conditions for a broad class of path-dependent strategies.
• Delta-hedging strategies are included, with versions for continuous and càdlàg price paths.
• A pathwise replication result provides a formula for hedging error across specified scenarios.
• The paper gives an economic justification for its price-path assumption.

Tags

# A pathwise approach to continuous-time trading


# 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 càdlàg 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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