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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• [1] D. S. Bates, Maximum Likelihood Estimation of Latent Affine Processes, Review of Financial Studies 19(2006), 909-965.
• [2] P. Carr and D. Madan, Option valuation using the fast Fourier transform, Journal of Computational Finance 2(4) (1999), 61-73.
• [3] P. Carr and L. Wu, The finite moment log stable process and option pricing, Journal of Finance 58(2) (2003), 597-626.
• [4] A. Cartea, Dynamic hedging of financial instruments when the underlying follows a non-Gaussian process, Working paper, Birkbeck College, University of London (2005).
• [5] Álvaro Cartea, Derivatives pricing with marked point processes using tick-by-tick data, Quant. Finance 13 (2013), no. 1, 111–123. MR 3005353, https://doi.org/10.1080/14697688.2012.661447
• [6] A. Cartea and D. del-Castillo-Negrete, Fractional diffusion models of option prices in markets with jumps, Physica A 374 (2006), 749-763.
• [7] A. Cartea and T. Meyer-Brandis, How duration between trades of underlying securities affects option prices, Review of Finance 14(4) (2010), 749-785.
• [8] S. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies 6 (1993), 327-343.
• [9] F. B. Hildebrand, Introduction to numerical analysis, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1956. MR 0075670
• [10] J. C. Hull, Options, Futures and Other Derivatives, Prentice Hall, 1997.
• [11] M. S. Joshi and C. Yang, Fourier Transforms, Option Pricing and Controls, Working paper, Available at SSRN: http://ssrn.com/abstract=1941464 or https://doi.org/10.2139/ssrn.1941464 (2011).
• [12] W. Schoutens, Lévy process in finance: Pricing financial derivatives, Wiley (2003).
• [13] D. Tavella and C. Randall, Pricing Financial Instruments, The Finite Difference Method, Wiley, New York (2000).
• [14] S. Martin, Option pricing formulae using Fourier transforms: Theory and Application, Working paper, available at: http://pfadintegral.com/articles/option-pricing-formulae-using-fourier-transform (2010).
• [15] K. Matsuda, Introduction to option pricing with Fourier transform: Option pricing with exponential Lévy models, Working paper, available at: www.maxmatsuda.com/Papers/2004/Matsuda%20Intro%20FT%20Pricing.pdf (2004).
• [16] R. Merton, Continuous-Time Finance, Blackwell, 1st edition, 1950.
• [17] Ralf Metzler and Joseph Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (2000), no. 1, 77. MR 1809268, https://doi.org/10.1016/S0370-1573(00)00070-3
• [18] S. J. Press, A compound events model for security prices, Journal of Business 40 (1967), 317-335.
• [19] Walter Wyss, The fractional Black-Scholes equation, Fract. Calc. Appl. Anal. 3 (2000), no. 1, 51–61. MR 1743405

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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