IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i24p3198-d700072.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/24/3198/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/24/3198/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Dilip B. Madan & Peter P. Carr & Eric C. Chang, 1998. "The Variance Gamma Process and Option Pricing," Review of Finance, European Finance Association, vol. 2(1), pages 79-105.
    2. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    3. repec:bla:jfinan:v:58:y:2003:i:2:p:753-778 is not listed on IDEAS
    4. Fang, Fang & Oosterlee, Kees, 2008. "A Novel Pricing Method For European Options Based On Fourier-Cosine Series Expansions," MPRA Paper 9319, University Library of Munich, Germany.
    5. Tarasov, Vasily E. & Tarasova, Valentina V., 2018. "Macroeconomic models with long dynamic memory: Fractional calculus approach," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 466-486.
    6. Peter Carr & Liuren Wu, 2003. "The Finite Moment Log Stable Process and Option Pricing," Journal of Finance, American Finance Association, vol. 58(2), pages 753-777, April.
    7. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    8. Mantegna,Rosario N. & Stanley,H. Eugene, 2007. "Introduction to Econophysics," Cambridge Books, Cambridge University Press, number 9780521039871, November.
    9. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    10. Jean-Philippe Aguilar, 2020. "Explicit option valuation in the exponential NIG model," Papers 2006.04659, arXiv.org, revised Oct 2020.
    11. Hagen Kleinert & Jan Korbel, 2015. "Option Pricing Beyond Black-Scholes Based on Double-Fractional Diffusion," Papers 1503.05655, arXiv.org, revised Mar 2016.
    12. Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, vol. 63(4), pages 511-524, October.
    13. Kleinert, H. & Korbel, J., 2016. "Option pricing beyond Black–Scholes based on double-fractional diffusion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 449(C), pages 200-214.
    14. Cartea, Álvaro & del-Castillo-Negrete, Diego, 2007. "Fractional diffusion models of option prices in markets with jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 374(2), pages 749-763.
    15. Jean-Philippe Aguilar & Cyril Coste & Jan Korbel, 2017. "Series representation of the pricing formula for the European option driven by space-time fractional diffusion," Papers 1712.04990, arXiv.org, revised Oct 2018.
    16. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    17. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    18. Clark, Peter K, 1973. "A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices," Econometrica, Econometric Society, vol. 41(1), pages 135-155, January.
    19. Jean-Philippe Aguilar & Justin Lars Kirkby & Jan Korbel, 2020. "Pricing, Risk and Volatility in Subordinated Market Models," Risks, MDPI, vol. 8(4), pages 1-27, November.
    20. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters, in: Anastasios G Malliaris & William T Ziemba (ed.), THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78, World Scientific Publishing Co. Pte. Ltd..
    21. Laurent E. Calvet & Adlai Fisher, 2008. "Multifractal Volatility: Theory, Forecasting and Pricing," Post-Print hal-00671877, HAL.
    22. Rosinski, Jan, 2007. "Tempering stable processes," Stochastic Processes and their Applications, Elsevier, vol. 117(6), pages 677-707, June.
    23. Anna Maria D’Arcangelis & Giulia Rotundo, 2016. "Complex Networks in Finance," Lecture Notes in Economics and Mathematical Systems, in: Pasquale Commendatore & Mariano Matilla-García & Luis M. Varela & Jose S. Cánovas (ed.), Complex Networks and Dynamics, pages 209-235, Springer.
    24. Peter Carr & Helyette Geman, 2002. "The Fine Structure of Asset Returns: An Empirical Investigation," The Journal of Business, University of Chicago Press, vol. 75(2), pages 305-332, April.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

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

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Dilip B. Madan & Wim Schoutens, 2019. "Arbitrage Free Approximations to Candidate Volatility Surface Quotations," JRFM, MDPI, vol. 12(2), pages 1-21, April.
    2. Chan, Tat Lung (Ron), 2019. "Efficient computation of european option prices and their sensitivities with the complex fourier series method," The North American Journal of Economics and Finance, Elsevier, vol. 50(C).
    3. Jean-Philippe Aguilar & Jan Korbel & Yuri Luchko, 2019. "Applications of the Fractional Diffusion Equation to Option Pricing and Risk Calculations," Mathematics, MDPI, vol. 7(9), pages 1-23, September.
    4. Carr, Peter & Wu, Liuren, 2007. "Stochastic skew in currency options," Journal of Financial Economics, Elsevier, vol. 86(1), pages 213-247, October.
    5. Liuren Wu, 2006. "Dampened Power Law: Reconciling the Tail Behavior of Financial Security Returns," The Journal of Business, University of Chicago Press, vol. 79(3), pages 1445-1474, May.
    6. Xu Guo & Yutian Li, 2016. "Valuation of American options under the CGMY model," Quantitative Finance, Taylor & Francis Journals, vol. 16(10), pages 1529-1539, October.
    7. Bates, David S., 2012. "U.S. stock market crash risk, 1926–2010," Journal of Financial Economics, Elsevier, vol. 105(2), pages 229-259.
    8. David S. Bates, 2009. "U.S. Stock Market Crash Risk, 1926-2006," NBER Working Papers 14913, National Bureau of Economic Research, Inc.
    9. Pedro Febrer & João Guerra, 2021. "Residue Sum Formula for Pricing Options under the Variance Gamma Model," Mathematics, MDPI, vol. 9(10), pages 1-29, May.
    10. Henri Bertholon & Alain Monfort & Fulvio Pegoraro, 2006. "Pricing and Inference with Mixtures of Conditionally Normal Processes," Working Papers 2006-28, Center for Research in Economics and Statistics.
    11. repec:dau:papers:123456789/1392 is not listed on IDEAS
    12. Kao, Lie-Jane & Wu, Po-Cheng & Lee, Cheng-Few, 2012. "Time-changed GARCH versus the GARJI model for prediction of extreme news events: An empirical study," International Review of Economics & Finance, Elsevier, vol. 21(1), pages 115-129.
    13. Aldrich, Eric M. & Heckenbach, Indra & Laughlin, Gregory, 2016. "A compound duration model for high-frequency asset returns," Journal of Empirical Finance, Elsevier, vol. 39(PA), pages 105-128.
    14. Carr, Peter & Wu, Liuren, 2004. "Time-changed Levy processes and option pricing," Journal of Financial Economics, Elsevier, vol. 71(1), pages 113-141, January.
    15. Svetlana Boyarchenko & Sergei Levendorskiĭ, 2019. "Sinh-Acceleration: Efficient Evaluation Of Probability Distributions, Option Pricing, And Monte Carlo Simulations," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(03), pages 1-49, May.
    16. Ballotta, Laura, 2005. "A Lévy process-based framework for the fair valuation of participating life insurance contracts," Insurance: Mathematics and Economics, Elsevier, vol. 37(2), pages 173-196, October.
    17. Lasko Basnarkov & Viktor Stojkoski & Zoran Utkovski & Ljupco Kocarev, 2019. "Option Pricing With Heavy-Tailed Distributions Of Logarithmic Returns," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(07), pages 1-35, November.
    18. Jingzhi Huang & Liuren Wu, 2004. "Specification Analysis of Option Pricing Models Based on Time- Changed Levy Processes," Finance 0401002, University Library of Munich, Germany.
    19. Khaled Salhi, 2017. "Pricing European options and risk measurement under exponential Lévy models — a practical guide," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 4(02n03), pages 1-36, June.
    20. Yanhui Mi, 2016. "A modified stochastic volatility model based on Gamma Ornstein–Uhlenbeck process and option pricing," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 3(02), pages 1-16, June.
    21. Jean-Philippe Aguilar, 2021. "The value of power-related options under spectrally negative Lévy processes," Review of Derivatives Research, Springer, vol. 24(2), pages 173-196, July.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:9:y:2021:i:24:p:3198-:d:700072. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.