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

Jafar Biazar; Fereshteh Goldoust; Jafar Biazar; Fereshteh Goldoust

doi:10.3934/math.2019.4.1046

AIMS Mathematics

2019, Volume 4, Issue 4: 1046-1064. doi: 10.3934/math.2019.4.1046

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Research article

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

Jafar Biazar ^,,
Fereshteh Goldoust

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

Received: 09 April 2019 Accepted: 12 July 2019 Published: 08 August 2019
MSC : 03F07, 11CXX, 35EXX

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.
- Black-Scholes equation,
- Heston Cox-Ingersoll-Ross equation,
- finance equations,
- Legendre wavelet method,
- stochastic differential equation,
- option pricing
Citation: Jafar Biazar, Fereshteh Goldoust. Multi-dimensional Legendre wavelets approach on the Black-Scholes and Heston Cox Ingersoll Ross equations[J]. AIMS Mathematics, 2019, 4(4): 1046-1064. doi: 10.3934/math.2019.4.1046

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Abstract

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.

References

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