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Open AccessFeature PaperEditor’s ChoiceArticle

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

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BRED Banque Populaire, Modeling Department, 18 quai de la Râpée, 75012 Paris, France
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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
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Complexity Science Hub Vienna, Josefstädterstrasse 39, 1080 Vienna, Austria
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Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, 11519 Prague, Czech Republic
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Beuth Technical University of Applied Sciences Berlin, Luxemburger Str. 10, 13353 Berlin, Germany
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Author to whom correspondence should be addressed.
Mathematics 2019, 7(9), 796; https://doi.org/10.3390/math7090796
Received: 15 August 2019 / Revised: 26 August 2019 / Accepted: 26 August 2019 / Published: 1 September 2019
(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)
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. View Full-Text
Keywords: fractional diffusion equation; fundamental solution; option pricing; risk sensitivities; portfolio hedging fractional diffusion equation; fundamental solution; option pricing; risk sensitivities; portfolio hedging
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Aguilar, J.-P.; Korbel, J.; Luchko, Y. Applications of the Fractional Diffusion Equation to Option Pricing and Risk Calculations. Mathematics 2019, 7, 796.

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