Applied and Computational Mathematics

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The Mellin Transform Method as an Alternative Analytic Solution for the Valuation of Geometric Asian Option

Received: 19 July 2014    Accepted: 5 August 2014    Published: 5 August 2014
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Abstract

This paper presents the Mellin transform method as an alternative analytic solution for the valuation of geometric Asian option. Asian options are options in which the variable is the average price over a period of time. The analytical solution of the Black-Scholes partial differential equation for Asian option is known as an explicit formula, this is due to the fact that the geometric average of a set of lognormal random variables is lognormally distributed. We derive a closed form solution for a continuous geometric Asian option by means of the partial differential equations. We also provide an alternative method for solving geometric Asian options partial differential equations using the Mellin transform method.

DOI 10.11648/j.acm.s.2014030601.11
Published in Applied and Computational Mathematics (Volume 3, Issue 6-1, December 2014)

This article belongs to the Special Issue Computational Finance

Page(s) 1-7
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Analytic Solution, Asian Option, Black-Scholes Model, Mellin Transform Method, Partial Differential Equation

References
[1] Fadugba S.E., Adefolaju F.H. and Akindutire O.H. (2012), On the stability and accuracy of finite difference method for options pricing, Mathematical Theory and Modeling, Vol. 2, No. 6, pp. 101-108.
[2] Fadugba S.E., Okunlola J.T. and Adeyemo A.O. (2013), On the robustness of binomial model and finite difference method for pricing European options, International Journal of IT, Engineering and Applied Sciences Research, Vol. 2, No. 2, pp. 5-11.
[3] Fadugba S.E. and Edogbanya O.H. (2014), On the valuation of credit risk via reduced-form approach, Global Journal of Science Frontier Research, Vol. 14, No. 1, Version 1, pp. 49-61.
[4] Geman H. and Yor M. (1993), Bessel processes, Asian options and perpetuities, Mathematical Financial, Vol. 3, 349 -375, doi:10.1111/j.1467-9965.1993.tb00092.x.
[5] Lapeyre B. and Temam E. (2001), Competitive Monte Carlo methods for the pricing of Asian options, Journal of computational finance, Vol. 5, pp. 39-57.
[6] Nwozo C.R. and Fadugba S.E. (2012), Monte Carlo method for pricing some path dependent options, International Journal of Applied Mathematics, Vol. 25, No. 6, pp. 763-778.
[7] Nwozo C. R. and Fadugba S. E. (2012), Some numerical methods for options valuation, Communications in Mathematical Finance, Vol. 1, No. 1, pp. 51-74, United Kingdom
[8] Nwozo C.R. and Fadugba S.E. (2014), Performance measure of Laplace transforms for pricing path dependent options, International Journal of Pure and Applied Mathematics, Vol. 94, No. 2, pp. 175-197, dx.doi.org/10.12732/ijpam.v94i2.5.
[9] Panini R. and Srivastav R.P. (2004), Option Pricing with Mellin Transforms, Mathematical and Computer Modelling, Vol. 40, pp. 43-56, doi:10.1016/j.mcm.2004.07.008.
[10] Rogers L.C.G. and Shi Z. (1992), The value of an Asian option, Journal of Applied Probability, Vol. 32 pp. 1077-1088, doi:10.2307/3215221.
[11] Temam E. (2001), Couverture Approche des options exotiques. Pricing des options Asiatiques Ph.D. Thesis}, University of Paris VI.
[12] Turnbull S.M. and Wakeman L.M.(1991), A quick algorithm for pricing European average options, Journal of Financial and Quantitative Analysis, Vol. 26, pp. 377-389, doi:10.2307/2331213.
[13] Vasilieva O. (2009), A New Method of Pricing Multi-Options using Mellin Transforms and Integral Equations, Master's Thesis in Financial Mathematics, School of Information Science, Computer and Electrical Engineering, Halmsta University.
[14] Yakubovich S.B. and Nguyen T.H. (1991), The Double Mellin-Barnes Type Integrals and their Applications to Convolution Theory, Series on Soviet Mathematics, World Scientific, 199.
[15] Yang Z., Oliver C.O. and Menkens O. (2011), Pricing and hedging of Asian options: quasi-explicit solutions via Malliavin calculus, Math Meth Oper Res, 74 pp. 93-120, doi:10.1007/s00186-011-0352-7.
[16] Zieneb A.E. and Rokiah R.A. (2011), Analytical Solution for an Arithmetic Asian Option using Mellin Transforms, Vol. 5, pp. 1259-1265
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    Fadugba Sunday Emmanuel. (2014). The Mellin Transform Method as an Alternative Analytic Solution for the Valuation of Geometric Asian Option. Applied and Computational Mathematics, 3(6-1), 1-7. https://doi.org/10.11648/j.acm.s.2014030601.11

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    ACS Style

    Fadugba Sunday Emmanuel. The Mellin Transform Method as an Alternative Analytic Solution for the Valuation of Geometric Asian Option. Appl. Comput. Math. 2014, 3(6-1), 1-7. doi: 10.11648/j.acm.s.2014030601.11

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    AMA Style

    Fadugba Sunday Emmanuel. The Mellin Transform Method as an Alternative Analytic Solution for the Valuation of Geometric Asian Option. Appl Comput Math. 2014;3(6-1):1-7. doi: 10.11648/j.acm.s.2014030601.11

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  • @article{10.11648/j.acm.s.2014030601.11,
      author = {Fadugba Sunday Emmanuel},
      title = {The Mellin Transform Method as an Alternative Analytic Solution for the Valuation of Geometric Asian Option},
      journal = {Applied and Computational Mathematics},
      volume = {3},
      number = {6-1},
      pages = {1-7},
      doi = {10.11648/j.acm.s.2014030601.11},
      url = {https://doi.org/10.11648/j.acm.s.2014030601.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.s.2014030601.11},
      abstract = {This paper presents the Mellin transform method as an alternative analytic solution for the valuation of geometric Asian option. Asian options are options in which the variable is the average price over a period of time. The analytical solution of the Black-Scholes partial differential equation for Asian option is known as an explicit formula, this is due to the fact that the geometric average of a set of lognormal random variables is lognormally distributed. We derive a closed form solution for a continuous geometric Asian option by means of the partial differential equations. We also provide an alternative method for solving geometric Asian options partial differential equations using the Mellin transform method.},
     year = {2014}
    }
    

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    AB  - This paper presents the Mellin transform method as an alternative analytic solution for the valuation of geometric Asian option. Asian options are options in which the variable is the average price over a period of time. The analytical solution of the Black-Scholes partial differential equation for Asian option is known as an explicit formula, this is due to the fact that the geometric average of a set of lognormal random variables is lognormally distributed. We derive a closed form solution for a continuous geometric Asian option by means of the partial differential equations. We also provide an alternative method for solving geometric Asian options partial differential equations using the Mellin transform method.
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Author Information
  • Department of Mathematical Sciences, Ekiti State University, Ado Ekiti, Nigeria

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