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A pathwise approach to continuous-time trading

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  • Candia Riga

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.

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  • Candia Riga, 2016. "A pathwise approach to continuous-time trading," Papers 1602.04946, arXiv.org.
    • Handle: RePEc:arx:papers:1602.04946
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    References listed on IDEAS

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    1. Bick, Avi & Willinger, Walter, 1994. "Dynamic spanning without probabilities," Stochastic Processes and their Applications, Elsevier, vol. 50(2), pages 349-374, April.
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    Cited by:

    1. Jani Lukkarinen & Mikko S. Pakkanen, 2016. "Arbitrage without borrowing or short selling?," Papers 1604.07690, arXiv.org, revised Oct 2016.
    2. Łochowski, Rafał M. & Perkowski, Nicolas & Prömel, David J., 2018. "A superhedging approach to stochastic integration," Stochastic Processes and their Applications, Elsevier, vol. 128(12), pages 4078-4103.
    3. Terry Lyons & Sina Nejad & Imanol Perez Arribas, 2019. "Numerical method for model-free pricing of exotic derivatives using rough path signatures," Papers 1905.01720, arXiv.org, revised Feb 2020.
    4. Mikko S. Pakkanen & Jani Lukkarinen, 2016. "Arbitrage without borrowing or short selling?," CREATES Research Papers 2016-13, Department of Economics and Business Economics, Aarhus University.
    5. Rafa{l} M. {L}ochowski & Nicolas Perkowski & David J. Promel, 2016. "A superhedging approach to stochastic integration," Papers 1609.02349, arXiv.org, revised Sep 2017.
    6. Terry Lyons & Sina Nejad & Imanol Perez Arribas, 2019. "Nonparametric pricing and hedging of exotic derivatives," Papers 1905.00711, arXiv.org.
    7. Jasdeep Kalsi & Terry Lyons & Imanol Perez Arribas, 2019. "Optimal execution with rough path signatures," Papers 1905.00728, arXiv.org.
    8. Owen Futter & Blanka Horvath & Magnus Wiese, 2023. "Signature Trading: A Path-Dependent Extension of the Mean-Variance Framework with Exogenous Signals," Papers 2308.15135, arXiv.org, revised Aug 2023.
    9. Andrew L. Allan & Chong Liu & David J. Promel, 2021. "A C\`adl\`ag Rough Path Foundation for Robust Finance," Papers 2109.04225, arXiv.org, revised May 2023.

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