On the Quantitative Properties of Some Market Models Involving Fractional Derivatives

Jean-Philippe Aguilar, Jan Korbel and Nicolas Pesci
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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
Nicolas Pesci: Covéa Finance, Quantitative Research Department, 8-12 Rue Boissy d’Anglas, FR-75008 Paris, France

Mathematics, 2021, vol. 9, issue 24, 1-24

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.

Keywords: option pricing; fractional calculus; stable laws; lévy processes; Riemann–Liouville derivative; Riesz–Feller derivative; fractional partial differential equation; fractional diffusion equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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