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Solving Black–Scholes equations using fractional generalized homotopy analysis method

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journal contribution
posted on 2020-12-01, 00:00 authored by S R Saratha, G Sai Sundara Krishnan, M Bagyalakshmi, Chee Peng LimChee Peng Lim
This paper aims to solve the Black–Scholes (B–S) model for the European options pricing problem using a hybrid method called fractional generalized homotopy analysis method (FGHAM). The convergence region of the B–S model solutions are clearly identified using h-curve and the closed form series solutions are produced using FGHAM. To verify the convergence of the proposed series solutions, sequence of errors are obtained by estimating the deviation between the exact solution and the series solution, which is increased in number of terms in the series. The convergence of sequence of errors is verified using the convergence criteria and the results are graphically illustrated. Moreover, the FGHAM approach has overcome the difficulties of applying multiple integration and differentiation procedures while obtaining the solution using well-established methods such as homotopy analysis method and homotopy perturbation method. The computational efficiency of the proposed method is analyzed using a comparative study. The advantage of the proposed method is shown with a numerical example using the comparative study between FGHAM and Monte Carlo simulation. Using the numerical example, analytical expression for the implied volatility is derived and the non-local behavior is studied for the various values of the fractional parameter.The results of FGHAM are statistically validated with the exact solution and the other existing computational methods.

History

Journal

Computational and Applied Mathematics

Volume

39

Issue

4

Article number

262

Publisher

Springer

Location

Heidelberg, Germany

ISSN

2238-3603

eISSN

1807-0302

Language

eng

Publication classification

C Journal article; C1 Refereed article in a scholarly journal

Copyright notice

2020, SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional