AN ANALYTICAL SOLUTION FOR THE BLACK-SCHOLES EQUATION USING FUNCTIONAL PERTURBATION METHOD

Document Type : Original Article

Authors
Somayeh Pourghanbar 1
Mojtaba Ranjbar 1
Ebrahim Nasrabadi 2
1 Department of Mathematics, Azarbaijan shahid madani University, Tabriz, Iran.
2 Department of Mathematics, University of Birjand, Birjand, Iran
10.29252/maco.1.1.8
Abstract
One of the greatest accomplishments in modern financial theory, in terms of both approach and applicability has been the Black-Scholes option pricing model. It is widely recognized that the value of a European option can be obtained by solving the BlackScholes equation. In this paper we use functional perturbation method (FPM) for solving Black-Scholes equation to price a European call option. The FPM is a tool based on considering the differential operator as a functional. The equation is expanded functionally by Frechet series. Then a number of successive partial differential equations (PDEs) are obtained that have constant coefficients and differ only in their right hand side part. Therefore we do not need to resolve the different equations for each step. In contrast to methods that
have implicit solutions, the FPM yields a closed form explicit solution.

Keywords

[1] E. Altus. Analysis of Bernoulli beams with 3D stochastic hetrogeneity Probabilist. Eng. Mech., 18: 301–314 ,2003.
[2] E. Altus. Microstress estimate of stochastically hetrogeneous structures by the functional perturbation method: A one dimentional example, Probabilist. Eng. Mech., 21: 434–441, 2006.
[3] E. Altus, Statistical modeling of heterogeneous microbeams. Int. J. Solids. Struct., 38: 5915–5034, 2001.
[4] E. Altus and E. Totry, Buckling of stochastically heterogeneous beams using a functional perturbation method, Int. J. Solids. Struct., 40: 6547–6565, 2003.
[5] E. Altus. A. Proskura and S. Givli, A new functional perturbation method for linear non-homogeneous materials. Int. J. Solids. Struct., 42: 1577–1595, 2005.
[6] E. Altus. E. Totry and S. Givli, Optimized functional perturbation method and morphology based effective properties of randomly heterogeneous beams, Int. J. Solids. Struct.,42: 2345–2359, 2005.
[7] M. Avellaneda and A. Paras, Dynamic hedging portfolios for derivative securities in the presence of large transaction costs, Appl. Math. Finance., 1: 165–193, 1994.
[8] B. Bensaid. J. Lesne. H. Pages and J. Scheinkman, Derivative asset pricing with transaction costs, Math. Finance., 2: 63–82, 1992.
[9] M. Beran, Statistical continuum mechanics, Inter-Science Publishers, 1968.
[10] F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81: 637–54,1973.
[11] P. Boyle and T. Vorst, Option replication in discrete time with transaction costs, J. Finance., 47: 271–293, 1973.
[12] M. Dehghan and S. Pourghanbar, Solution of the Black-Scholes Equation for Pricing of Barrier Option, Z Naturforsch., 66a: 289–296, 2011.
[13] S. Givli and E. Altus, Improved functional perturbation method for reliability analysis of randomly heterogeneous beams, Struct. Saf., 28: 378–391, 2006.
[14] H.C. Hull, Fundamentals of Futures and Options Markets, Fourth edition, Prentice Hall, 2002.
[15] R.P. Kanwal, Generalized functions: Theary and techniques, Birkhauser, Basel, 1998.
[16] S. Nachum and E. Altus. Natural frequenvies and mode shapes of deterministic and stochastic non homogeneous rods and beams, J. Sound. Vib., 302: 903–924, 2007.
[17] B. Oksendal, Stochastic differential Equations: An Introduction with Applications, Sixth Edition, Springer, 2000.
[18] S. Poon, A Practical Guide to Forecasting financial market volatility, John Wiley & Sons Ltd. 2005.
[19] S.A. Ramu and R. Ganesan, Response and stability of stochastic beam column using stochastic FEM, Comp. Struct., 54: 207–221, 1994.
[20] E.M. Totry. E. Altus. and A. Proskura, A novel application of the FPM to the buckling differential equation of non uniform beams, Probabilist. Eng. Mech., bf 23: 339–346, 2008.
[21] E. Totry. E. Altus. and A. Proskura, Buckling of non uniform beams by a direct functional perturbation method, Probabilist. Eng. Mech., 22: 88–99, 2007.
[22] P. Wilmott. S. Howison and J. Dewynne, The Mathematics of Financial Derivatives, Cambridge Univ Press, Cambridge, 1995.
[23] P. Wilmott, Introduces quantitative finance, Second Edition, John Wiley & sons, 2007.
[24] P. Wilmott. J. Dewynne and S. Howison, Option pricing mathematical models and computation, Oxford Financial Press, Oxford, 1993.
Mathematical Analysis and Convex Optimization
Volume 1, Issue 1
June 2020
Pages 65-74