AIMS Mathematics, 2019, 4(4): 1046-1064. doi: 10.3934/math.2019.4.1046

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Multi-dimensional Legendre wavelets approach on the Black-Scholes and Heston Cox Ingersoll Ross equations

Department of Mathematics, Faculty of Mathematical Sciences, Guilan University, P.O. Box 41335-1914, Rasht, Guilan, Iran

The one dimension Legendre Wavelet is a numerical method to solve one dimension equation. In this paper Black-Scholes equation (B-S), that has applied via single asset American option and Heston Cox- Ingersoll- Ross equation (HCIR), as partial differential equations have been studied in the form of stochastic model at first. The Black-Scholes and Heston Cox- Ingersoll- Ross Stochastic differential equations (SDE) models are converted to partial differential equations with a basic lemma in stochastic differential equation which called Ito lemma including derivatives and integration calculus in stochastic differential equations. Multi-dimensional Legendre wavelets method is based upon the expanded properties of Legendre wavelets from high order that is utilized to reduce these equations in to a system of algebraic equations. In fact the properties of Legendre wavelets are leads to reduce the PDEs problems to solution the ODEs systems. To ability and efficiency of the proposed techniques, numerical results and comparison with the other numerical method named Adomian decomposition method (ADM) for different values of parameters are tabulated and plotted.
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1.J. C. Hull, Fundamentals of Futures and Options Markets and Derivagem Package, 6th Edition, Prentice Hall, 2007.

2.S. G. Kou, A Jump-Diffusion Model for Option Pricing, Manage. Sci., 48 (2002), 1086-1101.    

3.S. L. 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.    

4.L. B. G. Andersen, V. Piterbarg, Interest Rate Modeling, 1st ed., Atlantic Financial Press, 2010.

5.E. S. Shreve, Stochastic calculus for finance, Springer, 2000.

6.D. Duffie, J. Pan, K. Singleton, Transform Analysis and Asset Pricing for Affine Jump-Diffusions, Econometrica, 68 (2000), 1343-1376.    

7.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.    

8.M. Razzaghi, S. Yousefi, Legendre wavelets direct method for variational problems, Math. Comput. Simulat., 53 (2000), 185-192.    

9.M. Razzaghi, S. Yousefi, Legendre wavelets method for constrained optimal control problems, Math. Method. Appl. Sci., 25 (2002), 529-539.    

10.M. Razzaghi, S. Yousefi, The Legendre wavelets operational matrix of integration, Int. J. Syst. Sci., 32 (2001), 495-502.    

11.M. Haugh, IEOR E4706 Foundations of financial engineering, 2016.

12.L. A. Grzelak, C. W. Oosterlee, On the Heston model with stochastic interest rates, SIAM J. Financ. Math., 2 (2011), 255-286.    

13.T. Haentjens, ADI FD schemes for the numerical solution of the three-dimensional Heston-Cox-Ingersoll-Ross PDE, AIP Conference Proceedings, 1479 (2012), 2195-2199.

14.T. Haentjens & K. J. In't Hout, Alternating direction implicit finite difference schemes for the Heston-Hull-White PDE, The Journal of Computational Finance, 16 (2012), 83.

15.R. J. Adler, The Geometry of Random Fields, Wiley & Sons, 1981.

16.R. Y. Chang, M. L. Wang, Shifted Legendre directs method for Variational problems, J. Optimiz. Theory App., 39 (1983), 299-307.

17.C. F. Chen, C. H. Hsiao, A Walsh series direct method for solving variational problems, J. Franklin I., 300 (1975), 265-280.    

18.M. Razzaghi, Fourier series direct method for variational problems, Int. J. Control, 48 (1988), 887-895.    

19.N. Ablaoui-Lahmar, O. Belhamitib and S. M. Bahri, A New Legendre Wavelets Decomposition Method for Solving PDEs, Malaya Journal of Matematik (MJM), 2 (2014), 72-81.

20.G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135 (1988), 501-544.    

21.J. Biazar, F. Goldoust, F. Mehrdoust, On the Numerical Solutions of Heston Partial Differential Equation, Mathematical Sciences Letters, 4 (2015), 63.

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