# American Institute of Mathematical Sciences

October  2020, 13(10): 2905-2915. doi: 10.3934/dcdss.2020221

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

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

* Corresponding author: Marjan Uddin

Received  February 2019 Revised  June 2019 Published  December 2019

Fund Project: This work is supported by HEC Pakistan

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.

Citation: Marjan Uddin, Hazrat Ali. Space-time kernel based numerical method for generalized Black-Scholes equation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2905-2915. doi: 10.3934/dcdss.2020221
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##### References:
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 $m = 100$, $n = 10$
The exact solution versus the approximate solution in space-time domain corresponding to problem 1, when $m = 1600$ and $n = 10$ in domain $(x,t)\in (-2,2)\times(0,1)$
The numerical solution and error in space-time domain, corresponding to problem 2 when $m = 400$ and $n = 10$ in the domain $(x,t)\in (0,1)\times(0,1)$
Double barrier option prices obtained by space-time local kernel method
Call option prices obtained by space-time local kernel method
Put option prices obtained by space-time local kernel method
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
 $m$ $n$ $L_{\infty}$ 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
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 $\theta=1$ 0.0724 0.0312 0.0139 0.0608 0.000271 [20] for $\theta=\frac{1}{2}$ 0.00112 0.0208 3.91e-05 7.19e-06 1.31e-06 [13] for $\theta=1$ 0.0744 0.0394 0.0202 0.0102 0.00516 [13] for $\theta=\frac{1}{2}$ 0.00589 0.00146 0.000364 9.1e-05 2.27e-05
 $M = N$ 10 20 40 80 160 [20] for $\theta=1$ 0.0724 0.0312 0.0139 0.0608 0.000271 [20] for $\theta=\frac{1}{2}$ 0.00112 0.0208 3.91e-05 7.19e-06 1.31e-06 [13] for $\theta=1$ 0.0744 0.0394 0.0202 0.0102 0.00516 [13] for $\theta=\frac{1}{2}$ 0.00589 0.00146 0.000364 9.1e-05 2.27e-05
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