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Dynamic spanning without probabilities

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  • Bick, Avi
  • Willinger, Walter

Abstract

The paper presents a non-probabilistic approach to continuous-time trading where, in analogy to the binomial option-pricing model, terminal payoffs resulting from a given trading strategy are meaningful 'state-by-state', i.e., path-by-path. In particular, we obtain results of the form: "If a certain trading strategy is applied and if the realized price trajectory satisfies a certain analytical property, then the terminal payoff is...." This way, derivation of the Black and Scholes formula and its extension become an exercise in the analysis of a certain class of real functions. While results of the above forms are of great interest if the analytical property in question is believed to be satisfied for almost all realized price trajectories (for example, if the price is believed to follow a certain stochastic process which has this property with probability 1), they are valid regardless of the stochastic process which presumably generates the possible price trajectories or the probability assigned to the set of all paths having this analytical property.

Suggested Citation

  • Bick, Avi & Willinger, Walter, 1994. "Dynamic spanning without probabilities," Stochastic Processes and their Applications, Elsevier, vol. 50(2), pages 349-374, April.
    • Handle: RePEc:eee:spapps:v:50:y:1994:i:2:p:349-374
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    Citations

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    Cited by:

    1. Alexander Schied & Iryna Voloshchenko, 2015. "Pathwise no-arbitrage in a class of Delta hedging strategies," Papers 1511.00026, arXiv.org, revised Jun 2016.
    2. Alexander Schied & Leo Speiser & Iryna Voloshchenko, 2016. "Model-free portfolio theory and its functional master formula," Papers 1606.03325, arXiv.org, revised May 2018.
    3. 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.
    4. Mark Britten-Jones & Anthony Neuberger, 1996. "Arbitrage pricing with incomplete markets," Applied Mathematical Finance, Taylor & Francis Journals, vol. 3(4), pages 347-363.
    5. Frank Riedel, 2011. "Finance Without Probabilistic Prior Assumptions," Papers 1107.1078, arXiv.org.
    6. 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.
    7. Schied, Alexander, 2014. "Model-free CPPI," Journal of Economic Dynamics and Control, Elsevier, vol. 40(C), pages 84-94.
    8. John Armstrong & Claudio Bellani & Damiano Brigo & Thomas Cass, 2021. "Option pricing models without probability: a rough paths approach," Mathematical Finance, Wiley Blackwell, vol. 31(4), pages 1494-1521, October.
    9. Vladimir Vovk, 2012. "Continuous-time trading and the emergence of probability," Finance and Stochastics, Springer, vol. 16(4), pages 561-609, October.
    10. Alexander Alvarez & Sebastian Ferrando, 2014. "Trajectory Based Models, Arbitrage and Continuity," Papers 1403.5685, arXiv.org, revised Jan 2015.
    11. Frank Riedel, 2015. "Financial economics without probabilistic prior assumptions," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 38(1), pages 75-91, April.
    12. David Hobson & Martin Klimmek, 2011. "Model independent hedging strategies for variance swaps," Papers 1104.4010, arXiv.org, revised May 2011.
    13. Alexander Schied, 2013. "Model-free CPPI," Papers 1305.5915, arXiv.org, revised Jan 2014.
    14. Alexander Alvarez & Sebastian E. Ferrando, 2016. "Trajectory-Based Models, Arbitrage And Continuity," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(03), pages 1-34, May.
    15. David Hobson & Martin Klimmek, 2012. "Model-independent hedging strategies for variance swaps," Finance and Stochastics, Springer, vol. 16(4), pages 611-649, October.
    16. Hans Follmer & Alexander Schied, 2013. "Probabilistic aspects of finance," Papers 1309.7759, arXiv.org.
    17. Candia Riga, 2016. "A pathwise approach to continuous-time trading," Papers 1602.04946, arXiv.org.
    18. Alexander Schied, 2015. "On a class of generalized Takagi functions with linear pathwise quadratic variation," Papers 1501.00837, arXiv.org, revised Aug 2015.
    19. Alexander Alvarez & Sebastian Ferrando & Pablo Olivares, 2011. "Arbitrage and Hedging in a non probabilistic framework," Papers 1103.1006, arXiv.org.
    20. Henry Chiu & Rama Cont, 2023. "A model‐free approach to continuous‐time finance," Mathematical Finance, Wiley Blackwell, vol. 33(2), pages 257-273, April.
    21. Henry Chiu & Rama Cont, 2022. "A model-free approach to continuous-time finance," Papers 2211.15531, arXiv.org.
    22. T. J. Lyons, 1995. "Uncertain volatility and the risk-free synthesis of derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 117-133.
    23. Rimas Norvaiša & Donna Mary Salopek, 2002. "Estimating the p-Variation Index of a Sample Function: An Application to Financial Data Set," Methodology and Computing in Applied Probability, Springer, vol. 4(1), pages 27-53, March.

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