Fractional Black Scholes Option Pricing with Stochastic Arbitrage Return

Bright O. Osu^1, and Chukwunezu A. Ifeoma²

¹Department of Mathematics, Michael Okpara University of Agriculture, Umudike

²Department of Mathematics/Statistics, Federal Polytechnic, Nekede, Owerri

Cite this paper:
Bright O. Osu and Chukwunezu A. Ifeoma. Fractional Black Scholes Option Pricing with Stochastic Arbitrage Return. International Journal of Partial Differential Equations and Applications. 2016; 4(2):20-24. doi: 10.12691/ijpdea-4-2-1

Abstract

Option price and random arbitrage returns change on different time scales allow the development of an asymptotic pricing theory involving the options rather than exact prices. The role that random arbitrage opportunities play in pricing financial derivatives can be determined. In this paper, we construct Green’s functions for terminal boundary value problems of the fractional Black-Scholes equation. We follow further an approach suggested in literature and focus on the pricing bands for options that account for random arbitrage opportunities and got similar result for the fractional Black- Scholes option pricing.

Keywords:
arbitrage returns option pricing green function FBS equation

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References:

[1]	A.N. Adamchuk, S.E. Esipov; Collectively fluctuating assets in the presence of arbitrage opportunities and option pricing, phys. Usp. 40 (12) (1997) 1239-1248.
[2]	Asma, A. E., Adem K. and Bachok, M. T; Homotopy Perturbation Method for Fractional Black-Scholes European Option Pricing Equations Using Sumudu Transform. Hindawi Publishing Corporation Mathematical Problems in EngineeringVolume 2013, Article ID 524852, 7 page.
[3]	F. B. M. Belgacem, A. A. Karaballi, and S. L. Kalla, “Analytical investigations of the Sumudu transform and applications tointegral production equations,” Mathematical Problems in Engineering,no. 3-4, pp. 103-118, 2003.
[4]	L. Borland, Theory of non-Gaussian option pricing, Quant. Finance 2 (2002) 415-431.
[5]	L. Borland, J.-P. Bouchaud, Non-Gaussian option pricing model with skew, cond-mat/0403022, 2004.
[6]	D. Galai; Tests of market efficiency and the Chicago board option exchange, J. Bus. 50 (1997) 167-197.
[7]	K. Ilinski, How to account for the virtual arbitrage in the standard derivative pricing, preprint, condmat/990 2047.
[8]	G. Jumarie. Schrodinger equation for quantum-fractal space-time of order n via the complex-valued fractional Brownian motion, Intern. J. of Modern Physics A 16(31), 5061-5084, 2001.
[9]	Q. D. Katatbeh and F. B. M. Belgacem, “Applications of the Sumudu transform to fractional differential equations,” NonlinearStudies, vol. 18, no. 1, pp. 99-112, 201
[10]	M. Melnikov, Y. A. Melnikov; Construction of Green’s function for the Black-Scholes equation, Electronic Journal of Differential Equations, Vol. 2007(2007), No. 153, Pp. 1-14.
[11]	R.C. Merton, Continuous Time Finance, Blackwell, Oxford, 1992.
[12]	B. O. Osu;An Alternative Approach for the Derivation of the Fractional Black-Scholes Equation for the Pricing of Options and its Solution via the Mellin Transform. International Journal of Applied Science and Mathematics, 2(6), 2394-2894, 2015.
[13]	N.T. Shawagfeh. Analytical approximate solutions for nonlinear fractional differential equations, Appl. Math. And Comp. 131, 517-529, 2002.
[14]	G. Sofianos, Index arbitrage profitability NYSE working paper 90-104, J. Derivatives 1 (N1) (1993).
[15]	Sergei Fedotov, Stephanos Panayides; Stochastic arbitrage return and its implication for option pricing, Physica A 345 (2005) 207-217.
[16]	W. Wyss. The fractional Black-Scholes equation, Fract. Calc. Appl. Anal. 3(1), 51-61, 2000.