Space-time kernel based numerical method for generalized Black-Scholes equation

Marjan Uddin; Hazrat Ali

doi:10.3934/dcdss.2020221

Article Contents

2020, Volume 13, Issue 10: 2905-2915. Doi: 10.3934/dcdss.2020221

This issue Previous Article Wave-propagation in an incompressible hollow elastic cylinder with residual stress Next Article Smooth and singular traveling wave solutions for the Serre-Green-Naghdi equations

Space-time kernel based numerical method for generalized Black-Scholes equation

Marjan Uddin^, and
Hazrat Ali

Department of Basic Sciences, University of Engineering and Technology Peshawar, Pakistan

^* Corresponding author: Marjan Uddin
^* Corresponding author: Marjan Uddin

Received: January 31, 2019

Revised: May 31, 2019

Early access: December 2019

Published: July 2020

This work is supported by HEC Pakistan

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Abstract

In approximating time-dependent partial differential equations, major error always occurs in the time derivatives as compared to the spatial derivatives. In the present work the time and the spatial derivatives are both approximated using time-space radial kernels. The proposed numerical scheme avoids the time stepping procedures and produced sparse differentiation matrices. The stability and accuracy of the proposed numerical scheme is tested for the generalized Black-Scholes equation.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. A typical centers arrangements in global space-time domain as well as in a local sub-domain, and sparsity of descretized operator of problem 1, where $ m = 100 $, $ n = 10 $

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Figure 2. The exact solution versus the approximate solution in space-time domain corresponding to problem 1, when $ m = 1600 $ and $ n = 10 $ in domain $ (x,t)\in (-2,2)\times(0,1) $

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Figure 3. The numerical solution and error in space-time domain, corresponding to problem 2 when $ m = 400 $ and $ n = 10 $ in the domain $ (x,t)\in (0,1)\times(0,1) $

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Figure 4. Double barrier option prices obtained by space-time local kernel method

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Figure 5. Call option prices obtained by space-time local kernel method

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Figure 6. Put option prices obtained by space-time local kernel method

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Table 1. Sapce-time (ST) solution of problem 1 for different total collocation points $ m $ and stencil size $ n $, and time integration (TI) solution in domain $ (x,t)\in (-2,2)\times(0,1) $

$ m $	$ n $	$ L_{\infty} $	ST method (C.time)	TI method (C.time)
100	10	1.23E-02	0.6783	3.2891
400		9.49E-02	0.7123	6.2821
1600		1.08E-04	0.9129	10.2370
2500		1.26E-04	1.1234	15.2950
100	15	2.37E-02	0.8234	4.3491
400		1.45E-03	10.2356	9.2371
1600		2.45E-03	11.2916	13.9820
2500		3.45E-04	13.1087	20.3582
100	20	2.22E-02	9.1835	10.10491
400		2.89E-03	11.2349	12.7146
1600		9.88E-03	14.8679	21.3812
2500		7.23E-04	16.1955	25.0492

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Table 2. Observed maximum absolute error for example 1 in Reza [20] and Kadabajoo [13] for different $ \theta $ in domain $ (x,t)\in (-2,2)\times(0,1) $

$ M = N $	10	20	40	80	160
[20] for $ \theta=1 $	7.24E-02	3.12E-02	1.39E-02	6.08E-02	2.71E-04
[20] for $ \theta=\frac{1}{2} $	1.12E-03	2.08E-02	3.91E-05	7.19E-06	1.31E-06
[13] for $ \theta=1 $	7.44E-02	3.94E-02	2.02E-02	1.02E-02	5.16E-03
[13] for $ \theta=\frac{1}{2} $	5.89E-03	1.46E-03	3.64E-04	9.10E-05	2.27E-05

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