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On the Quantitative Properties of Some Market Models Involving Fractional Derivatives

Author

Listed:
  • Jean-Philippe Aguilar

    (Covéa Finance, Quantitative Research Department, 8-12 Rue Boissy d’Anglas, FR-75008 Paris, France)

  • Jan Korbel

    (Section for the Science of Complex Systems, Center for Medical Statistics, Informatics, and Intelligent Systems (CeMSIIS), Medical University of Vienna, Spitalgasse 23, 1090 Vienna, Austria
    Complexity Science Hub Vienna, Josefstädterstrasse 39, 1080 Vienna, Austria
    Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, 11519 Prague, Czech Republic
    The Czech Academy of Sciences, Institute of Information Theory and Automation, Pod Vodárenskou Věží 4, 18200 Prague, Czech Republic)

  • Nicolas Pesci

    (Covéa Finance, Quantitative Research Department, 8-12 Rue Boissy d’Anglas, FR-75008 Paris, France)

Abstract

We review and discuss the properties of various models that are used to describe the behavior of stock returns and are related in a way or another to fractional pseudo-differential operators in the space variable; we compare their main features and discuss what behaviors they are able to capture. Then, we extend the discussion by showing how the pricing of contingent claims can be integrated into the framework of a model featuring a fractional derivative in both time and space, recall some recently obtained formulas in this context, and derive new ones for some commonly traded instruments and a model involving a Riesz temporal derivative and a particular case of Riesz–Feller space derivative. Finally, we provide formulas for implied volatility and first- and second-order market sensitivities in this model, discuss hedging and profit and loss policies, and compare with other fractional (Caputo) or non-fractional models.

Suggested Citation

  • Jean-Philippe Aguilar & Jan Korbel & Nicolas Pesci, 2021. "On the Quantitative Properties of Some Market Models Involving Fractional Derivatives," Mathematics, MDPI, vol. 9(24), pages 1-24, December.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:24:p:3198-:d:700072
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    References listed on IDEAS

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    Cited by:

    1. Yuri Luchko, 2023. "Fractional Integrals and Derivatives: “True” versus “False”," Mathematics, MDPI, vol. 11(13), pages 1-2, July.

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