Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Analytically pricing European-style options under the modified Black-Scholes equation with a spatial-fractional derivative

Authors: Wenting Chen, Xiang Xu and Song-ping Zhu
Journal: Quart. Appl. Math. 72 (2014), 597-611
MSC (2010): Primary 97M30, 35R11
DOI: https://doi.org/10.1090/S0033-569X-2014-01373-2
Published electronically: June 10, 2014
MathSciNet review: 3237565
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Abstract: This paper investigates the option pricing under the FMLS (finite moment log stable) model, which can effectively capture the leptokurtic feature observed in many financial markets. However, under the FMLS model, the option price is governed by a modified Black-Scholes equation with a spatial-fractional derivative. In comparison with standard derivatives of integer order, the fractional-order derivatives are characterized by their ``globalness'', i.e., the rate of change of a function near a point is affected by the property of the function defined in the entire domain of definition rather than just near the point itself. This has added an additional degree of difficulty not only when a purely numerical solution is sought but also when an analytical method is attempted. Despite this difficulty, we have managed to find an explicit closed-form analytical solution for European-style options after successfully solving the FPDE (fractional partial differential equation) derived from the FMLS model. After the validity of the put-call parity under the FMLS model is verified both financially and mathematically, we have also proposed an efficient numerical evaluation technique to facilitate the implementation of our formula so that it can be easily used in trading practice.

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Additional Information

Wenting Chen
Affiliation: School of Mathematics and Applied Statistics, University of Wollongong NSW 2522, Australia
Email: wtchen@uow.edu.au

Xiang Xu
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, 310007, China
Email: xxu@zju.edu.cn

Song-ping Zhu
Affiliation: School of Mathematics and Applied Statistics, University of Wollongong NSW 2522, Australia
Email: spz@uow.edu.au

DOI: https://doi.org/10.1090/S0033-569X-2014-01373-2
Keywords: Fractional partial differential equation, closed-form analytical solution, put-call parity
Received by editor(s): August 2, 2012
Received by editor(s) in revised form: June 24, 2013
Published electronically: June 10, 2014
Additional Notes: The third author is the corresponding author and also a Tang-Au-Qing Chair Professor (adjunct) of Jilin University, Jilin, 130012, China
Article copyright: © Copyright 2014 Brown University

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