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Article

On the Quantitative Properties of Some Market Models Involving Fractional Derivatives

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Covéa Finance, Quantitative Research Department, 8-12 Rue Boissy d’Anglas, FR-75008 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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The Czech Academy of Sciences, Institute of Information Theory and Automation, Pod Vodárenskou Věží 4, 18200 Prague, Czech Republic
*
Author to whom correspondence should be addressed.
Academic Editor: Manuel Alberto M. Ferreira
Mathematics 2021, 9(24), 3198; https://doi.org/10.3390/math9243198
Received: 23 November 2021 / Revised: 7 December 2021 / Accepted: 9 December 2021 / Published: 11 December 2021
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. View Full-Text
Keywords: option pricing; fractional calculus; stable laws; lévy processes; Riemann–Liouville derivative; Riesz–Feller derivative; fractional partial differential equation; fractional diffusion equation option pricing; fractional calculus; stable laws; lévy processes; Riemann–Liouville derivative; Riesz–Feller derivative; fractional partial differential equation; fractional diffusion equation
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MDPI and ACS Style

Aguilar, J.-P.; Korbel, J.; Pesci, N. On the Quantitative Properties of Some Market Models Involving Fractional Derivatives. Mathematics 2021, 9, 3198. https://doi.org/10.3390/math9243198

AMA Style

Aguilar J-P, Korbel J, Pesci N. On the Quantitative Properties of Some Market Models Involving Fractional Derivatives. Mathematics. 2021; 9(24):3198. https://doi.org/10.3390/math9243198

Chicago/Turabian Style

Aguilar, Jean-Philippe, Jan Korbel, and Nicolas Pesci. 2021. "On the Quantitative Properties of Some Market Models Involving Fractional Derivatives" Mathematics 9, no. 24: 3198. https://doi.org/10.3390/math9243198

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