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

Jean-Philippe Aguilar, Jan Korbel and Yuri Luchko
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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
Yuri Luchko: Beuth Technical University of Applied Sciences Berlin, Luxemburger Str. 10, 13353 Berlin, Germany

Mathematics, 2019, vol. 7, issue 9, 1-23

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.

Keywords: fractional diffusion equation; fundamental solution; option pricing; risk sensitivities; portfolio hedging (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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