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Applications of the Fractional Diffusion Equation to Option Pricing and Risk Calculations

Author

Listed:
  • Jean-Philippe Aguilar

    (BRED Banque Populaire, Modeling Department, 18 quai de la Râpée, 75012 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)

  • Yuri Luchko

    (Beuth Technical University of Applied Sciences Berlin, Luxemburger Str. 10, 13353 Berlin, Germany)

Abstract

In this article, we first provide a survey of the exponential option pricing models and show that in the framework of the risk-neutral approach, they are governed by the space-fractional diffusion equation. Then, we introduce a more general class of models based on the space-time-fractional diffusion equation and recall some recent results in this field concerning the European option pricing and the risk-neutral parameter. We proceed with an extension of these results to the class of exotic options. In particular, we show that the call and put prices can be expressed in the form of simple power series in terms of the log-forward moneyness and the risk-neutral parameter. Finally, we provide the closed-form formulas for the first and second order risk sensitivities and study the dependencies of the portfolio hedging and profit-and-loss calculations upon the model parameters.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:9:p:796-:d:262856
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    References listed on IDEAS

    as
    1. Robert J. Elliott & John Van Der Hoek, 2003. "A General Fractional White Noise Theory And Applications To Finance," Mathematical Finance, Wiley Blackwell, vol. 13(2), pages 301-330, April.
    2. 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.
    3. Matjaž Perc & Mahmut Ozer & Janja Hojnik, 2019. "Correction: Social and juristic challenges of artificial intelligence," Palgrave Communications, Palgrave Macmillan, vol. 5(1), pages 1-1, December.
    4. Jean-Philippe Aguilar & Jan Korbel, 2019. "Simple Formulas for Pricing and Hedging European Options in the Finite Moment Log-Stable Model," Risks, MDPI, vol. 7(2), pages 1-14, April.
    5. Scalas, Enrico & Gorenflo, Rudolf & Mainardi, Francesco, 2000. "Fractional calculus and continuous-time finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 284(1), pages 376-384.
    6. Wang, Xiao-Tian, 2010. "Scaling and long range dependence in option pricing, IV: Pricing European options with transaction costs under the multifractional Black–Scholes model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(4), pages 789-796.
    7. Xiao, Wei-Lin & Zhang, Wei-Guo & Zhang, Xi-Li & Wang, Ying-Luo, 2010. "Pricing currency options in a fractional Brownian motion with jumps," Economic Modelling, Elsevier, vol. 27(5), pages 935-942, September.
    8. 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.
    9. 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.
    10. Vasily E. Tarasov, 2019. "On History of Mathematical Economics: Application of Fractional Calculus," Mathematics, MDPI, vol. 7(6), pages 1-28, June.
    11. Mainardi, Francesco & Raberto, Marco & Gorenflo, Rudolf & Scalas, Enrico, 2000. "Fractional calculus and continuous-time finance II: the waiting-time distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 468-481.
    12. Lina Song & Weiguo Wang, 2013. "Solution of the Fractional Black-Scholes Option Pricing Model by Finite Difference Method," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-10, June.
    13. Peter Carr & Liuren Wu, 2003. "The Finite Moment Log Stable Process and Option Pricing," Journal of Finance, American Finance Association, vol. 58(2), pages 753-778, April.
    14. 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.
    15. Wang, Xiao-Tian, 2010. "Scaling and long-range dependence in option pricing I: Pricing European option with transaction costs under the fractional Black–Scholes model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(3), pages 438-444.
    16. Jin-Chuan Duan & Ivilina Popova & Peter Ritchken, 2002. "Option pricing under regime switching," Quantitative Finance, Taylor & Francis Journals, vol. 2(2), pages 116-132.
    17. 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.
    18. Wang, Xiao-Tian & Zhu, En-Hui & Tang, Ming-Ming & Yan, Hai-Gang, 2010. "Scaling and long-range dependence in option pricing II: Pricing European option with transaction costs under the mixed Brownian–fractional Brownian model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(3), pages 445-451.
    19. Jean-Philippe Aguilar & Jan Korbel, 2018. "Option Pricing Models Driven by the Space-Time Fractional Diffusion: Series Representation and Applications," Papers 1802.09864, arXiv.org.
    20. Matjaž Perc & Mahmut Ozer & Janja Hojnik, 2019. "Social and juristic challenges of artificial intelligence," Palgrave Communications, Palgrave Macmillan, vol. 5(1), pages 1-7, December.
    21. Hagen Kleinert & Jan Korbel, 2015. "Option Pricing Beyond Black-Scholes Based on Double-Fractional Diffusion," Papers 1503.05655, arXiv.org, revised Mar 2016.
    22. 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.
    23. Cipian Necula, 2008. "Option Pricing in a Fractional Brownian Motion Environment," Advances in Economic and Financial Research - DOFIN Working Paper Series 2, Bucharest University of Economics, Center for Advanced Research in Finance and Banking - CARFIB.
    24. Longjin, Lv & Ren, Fu-Yao & Qiu, Wei-Yuan, 2010. "The application of fractional derivatives in stochastic models driven by fractional Brownian motion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(21), pages 4809-4818.
    25. 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.
    26. Jean-Philippe Aguilar, 2018. "On expansions for the Black-Scholes prices and hedge parameters," Papers 1809.06736, arXiv.org, revised Jun 2019.
    27. 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..
    28. Gong, Xiaoli & Zhuang, Xintian, 2017. "American option valuation under time changed tempered stable Lévy processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 466(C), pages 57-68.
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