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

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

Received  February 2019 Revised  June 2019 Published  October 2020 Early access  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.

Keywords: Space-time numerical scheme, Meshless method, Radial kernels, Black-Scholes equation.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
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
References:
[1]

V. R. Ambati and O. Bokhove, Space-time discontinuous Galerkin finite element method for shallow water flows, J. Comput. Appl. Math., 204 (2007), 452-462.  doi: 10.1016/j.cam.2006.01.047.  Google Scholar

[2]

C. Canuto, M. Y. Hussaini, A. Z. Quarteroni and T. Zang, Option Pricing: Mathematical Models and Computation, Springer, 1993. Google Scholar

[3]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods, Evolution to Complex Geometries and Applications to Fluid Dynamics, Scientific Computation, Springer, Berlin, 2007.  Google Scholar

[4]

C. Chen, A. Karageorghis and Y. Smyrlis, The Method of Fundamental Solutions: A Meshless Method, Dynamic Publishers Atlanta, 2008. Google Scholar

[5]

A. Cohen, Numerical Analysis of Wavelet Methods, Studies in Mathematics and its Applications, 32. North-Holland Publishing Co., Amsterdam, 2003.  Google Scholar

[6]

J. C. Cox, S. A. Ross and M. Rubinstein, Option pricing: A simplified approach, J. Financ. Econ., 7 (1979), 229-263.  doi: 10.1016/0304-405X(79)90015-1.  Google Scholar

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G. E. Fasshauer and M. McCourt, Kernel-Based Approximation Methods Using Matlab, World Scientific pub. Co, 2015. Google Scholar

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G. E. Fasshauer and J. G. Zhang, On choosing optimal shape parameters for RBF approximation, Numerical Algorithms, 45 (2007), 345-368.  doi: 10.1007/s11075-007-9072-8.  Google Scholar

[10]

R. Geske and K. Shastri, Valuation by approximation: A comparison of alternative option valuation techniques, Journal of Financial and Quantitative Analysis, 20 (1985), 45-71.  doi: 10.2307/2330677.  Google Scholar

[11]

M. Hamaidi, A. Naji and A. Charafi, Space-time localized radial basis function collocation method for solving parabolic and hyperbolic equations, Eng. Anal. Bound. Elem., 67 (2016), 152-163.  doi: 10.1016/j.enganabound.2016.03.009.  Google Scholar

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J. Hull and A. White, The use of the control variate technique in option pricing, Journal of Financial and Quantitative analysis, 23 (1988), 237-251.  doi: 10.2307/2331065.  Google Scholar

[13]

M. K. Kadalbajoo, L. P. Tripathi and A. Kumar, A cubic B-spline collocation method for a numerical solution of the generalized Black-Scholes equation, Math. Comput. Modelling, 55 (2012), 1483-1505.  doi: 10.1016/j.mcm.2011.10.040.  Google Scholar

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C. M. Klaija, J. J. W. van der Vegta and H. van der Venb, Space-time discontinuous Galerkin method for the compressible Navier–Stokes equations, J. Comput. Phys., 217 (2006), 589-611.  doi: 10.1016/j.jcp.2006.01.018.  Google Scholar

[15]

M. Li, W. Chen and C. S. Chen, The localized RBFs collocation methods for solving high dimensional PDEs, Eng. Anal. Bound. Elem., 37 (2013), 1300-1304.  doi: 10.1016/j.enganabound.2013.06.001.  Google Scholar

[16]

Z. Li and X. Z. Mao, Global multiquadric collocation method for groundwater contaminant source identification, Environmental modelling & software, 26 (2011), 1611-1621.  doi: 10.1016/j.envsoft.2011.07.010.  Google Scholar

[17]

Z. Li and X. Z. Mao, Global space-time multiquadric method for inverse heat conduction problem, Internat. J. Numer. Methods Engrg., 85 (2011), 355-379.  doi: 10.1002/nme.2975.  Google Scholar

[18]

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[19]

H. Netuzhylov, A Space-Time Meshfree Collocation Method for Coupled Problems on Irregularly-Shaped Domains, Ph.D thesis, Zugl., Braunschweig, Techn. Univ., Diss., 2008. Google Scholar

[20]

R. Mohammadi, Quintic B-spline collocation approach for solving generalized Black–Scholes equation governing option pricing, Comput. Math. Appl., 69 (2015), 777-797.  doi: 10.1016/j.camwa.2015.02.018.  Google Scholar

[21]

S. A. Sauter and C. Schwab, Boundary Element Methods, Springer Series in Computational Mathematics, 39. Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-540-68093-2.  Google Scholar

[22]

J. J.Sudirham, J. J. W. van der Vegt and R. M. J. van Damme, Space-time discontinuous Galerkin method for advection-diffusion problems on time-dependent domains, Appl. Numer. Math., 56 (2006), 1491-1518.  doi: 10.1016/j.apnum.2005.11.003.  Google Scholar

[23]

T. E. Tezduyar, S. Sathe, R. Keedy and K. Stein, Space-time finite element techniques for computation of fluid-structure interactions, Comput. Methods Appl. Mech. Engrg., 195 (2006), 2002-2027.  doi: 10.1016/j.cma.2004.09.014.  Google Scholar

[24]

C. Turchetti, M. Conti, P. Crippa and S. Orcioni, On the approximation of stochastic processes by approximate identity neural networks, IEEE Transactions on Neural Networks, 9 (1998), 1069-1085.  doi: 10.1109/72.728353.  Google Scholar

[25]

C. Turchetti, P. Crippa, M. Pirani and G. Biagetti, Representation of nonlinear random transformations by non-Gaussian stochastic neural networks, IEEE transactions on neural networks, 19 (2008), 1033-1060.  doi: 10.1109/TNN.2007.2000055.  Google Scholar

[26]

M. Uddin, H. Ali and A. Ali, Kernel-based local meshless method for solving multi-dimensional wave equations in irregular domain, CMES-Computer Modeling In Engineering & Sciences, 107 (2015), 463-479.   Google Scholar

[27]

M. Uddin and H. Ali, The space time kernel based numerical method for Burgers equations, Mathematics, 6 (2018), 212-222.  doi: 10.3390/math6100212.  Google Scholar

[28]

M. Uddin, K. Kamran, M. Usman and A. Ali, On the Laplace-transformed-based local meshless method for fractional-order diffusion equation, Int. J. Comput. Methods Eng. Sci. Mech., 19 (2018), 221-225.  doi: 10.1080/15502287.2018.1472150.  Google Scholar

[29]

B. M. Vaganan and E. E. Priya, Generalized Cole-Hopf transformations for generalized Burgers equations, Pramana, 85 (2015), 861-867.  doi: 10.1007/s12043-015-1107-4.  Google Scholar

[30]

D. L. Young, C. C. Tsai, K. Murugesana, C. M. Fan and C. W. Chen, Time-dependent fundamental solutions for homogeneous diffusion problems, Engineering Analysis with Boundary Elements, 28 (2004), 1463-1473.  doi: 10.1016/j.enganabound.2004.07.003.  Google Scholar

[31]

H. Zhang, F. Liu, I. Turner and Q. Yang, Numerical solution of the time fractional Black-Scholes model governing European options, Comput. Math. Appl., 71 (2016), 1772-1783.  doi: 10.1016/j.camwa.2016.02.007.  Google Scholar

show all references

References:
[1]

V. R. Ambati and O. Bokhove, Space-time discontinuous Galerkin finite element method for shallow water flows, J. Comput. Appl. Math., 204 (2007), 452-462.  doi: 10.1016/j.cam.2006.01.047.  Google Scholar

[2]

C. Canuto, M. Y. Hussaini, A. Z. Quarteroni and T. Zang, Option Pricing: Mathematical Models and Computation, Springer, 1993. Google Scholar

[3]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods, Evolution to Complex Geometries and Applications to Fluid Dynamics, Scientific Computation, Springer, Berlin, 2007.  Google Scholar

[4]

C. Chen, A. Karageorghis and Y. Smyrlis, The Method of Fundamental Solutions: A Meshless Method, Dynamic Publishers Atlanta, 2008. Google Scholar

[5]

A. Cohen, Numerical Analysis of Wavelet Methods, Studies in Mathematics and its Applications, 32. North-Holland Publishing Co., Amsterdam, 2003.  Google Scholar

[6]

J. C. Cox, S. A. Ross and M. Rubinstein, Option pricing: A simplified approach, J. Financ. Econ., 7 (1979), 229-263.  doi: 10.1016/0304-405X(79)90015-1.  Google Scholar

[7]

G. E. Fasshauer, Meshfree Approximation Methods with MATLAB, With 1 CD-ROM (Windows, Macintosh and UNIX), Interdisciplinary Mathematical Sciences, 6. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. doi: 10.1142/6437.  Google Scholar

[8]

G. E. Fasshauer and M. McCourt, Kernel-Based Approximation Methods Using Matlab, World Scientific pub. Co, 2015. Google Scholar

[9]

G. E. Fasshauer and J. G. Zhang, On choosing optimal shape parameters for RBF approximation, Numerical Algorithms, 45 (2007), 345-368.  doi: 10.1007/s11075-007-9072-8.  Google Scholar

[10]

R. Geske and K. Shastri, Valuation by approximation: A comparison of alternative option valuation techniques, Journal of Financial and Quantitative Analysis, 20 (1985), 45-71.  doi: 10.2307/2330677.  Google Scholar

[11]

M. Hamaidi, A. Naji and A. Charafi, Space-time localized radial basis function collocation method for solving parabolic and hyperbolic equations, Eng. Anal. Bound. Elem., 67 (2016), 152-163.  doi: 10.1016/j.enganabound.2016.03.009.  Google Scholar

[12]

J. Hull and A. White, The use of the control variate technique in option pricing, Journal of Financial and Quantitative analysis, 23 (1988), 237-251.  doi: 10.2307/2331065.  Google Scholar

[13]

M. K. Kadalbajoo, L. P. Tripathi and A. Kumar, A cubic B-spline collocation method for a numerical solution of the generalized Black-Scholes equation, Math. Comput. Modelling, 55 (2012), 1483-1505.  doi: 10.1016/j.mcm.2011.10.040.  Google Scholar

[14]

C. M. Klaija, J. J. W. van der Vegta and H. van der Venb, Space-time discontinuous Galerkin method for the compressible Navier–Stokes equations, J. Comput. Phys., 217 (2006), 589-611.  doi: 10.1016/j.jcp.2006.01.018.  Google Scholar

[15]

M. Li, W. Chen and C. S. Chen, The localized RBFs collocation methods for solving high dimensional PDEs, Eng. Anal. Bound. Elem., 37 (2013), 1300-1304.  doi: 10.1016/j.enganabound.2013.06.001.  Google Scholar

[16]

Z. Li and X. Z. Mao, Global multiquadric collocation method for groundwater contaminant source identification, Environmental modelling & software, 26 (2011), 1611-1621.  doi: 10.1016/j.envsoft.2011.07.010.  Google Scholar

[17]

Z. Li and X. Z. Mao, Global space-time multiquadric method for inverse heat conduction problem, Internat. J. Numer. Methods Engrg., 85 (2011), 355-379.  doi: 10.1002/nme.2975.  Google Scholar

[18]

F. Moukalled, L. Mangani and M. Darwish, The Finite Volume Method in Computational Fluid Dynamics, Fluid Mechanics and its Applications, 113. Springer, Cham, 2016. doi: 10.1007/978-3-319-16874-6.  Google Scholar

[19]

H. Netuzhylov, A Space-Time Meshfree Collocation Method for Coupled Problems on Irregularly-Shaped Domains, Ph.D thesis, Zugl., Braunschweig, Techn. Univ., Diss., 2008. Google Scholar

[20]

R. Mohammadi, Quintic B-spline collocation approach for solving generalized Black–Scholes equation governing option pricing, Comput. Math. Appl., 69 (2015), 777-797.  doi: 10.1016/j.camwa.2015.02.018.  Google Scholar

[21]

S. A. Sauter and C. Schwab, Boundary Element Methods, Springer Series in Computational Mathematics, 39. Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-540-68093-2.  Google Scholar

[22]

J. J.Sudirham, J. J. W. van der Vegt and R. M. J. van Damme, Space-time discontinuous Galerkin method for advection-diffusion problems on time-dependent domains, Appl. Numer. Math., 56 (2006), 1491-1518.  doi: 10.1016/j.apnum.2005.11.003.  Google Scholar

[23]

T. E. Tezduyar, S. Sathe, R. Keedy and K. Stein, Space-time finite element techniques for computation of fluid-structure interactions, Comput. Methods Appl. Mech. Engrg., 195 (2006), 2002-2027.  doi: 10.1016/j.cma.2004.09.014.  Google Scholar

[24]

C. Turchetti, M. Conti, P. Crippa and S. Orcioni, On the approximation of stochastic processes by approximate identity neural networks, IEEE Transactions on Neural Networks, 9 (1998), 1069-1085.  doi: 10.1109/72.728353.  Google Scholar

[25]

C. Turchetti, P. Crippa, M. Pirani and G. Biagetti, Representation of nonlinear random transformations by non-Gaussian stochastic neural networks, IEEE transactions on neural networks, 19 (2008), 1033-1060.  doi: 10.1109/TNN.2007.2000055.  Google Scholar

[26]

M. Uddin, H. Ali and A. Ali, Kernel-based local meshless method for solving multi-dimensional wave equations in irregular domain, CMES-Computer Modeling In Engineering & Sciences, 107 (2015), 463-479.   Google Scholar

[27]

M. Uddin and H. Ali, The space time kernel based numerical method for Burgers equations, Mathematics, 6 (2018), 212-222.  doi: 10.3390/math6100212.  Google Scholar

[28]

M. Uddin, K. Kamran, M. Usman and A. Ali, On the Laplace-transformed-based local meshless method for fractional-order diffusion equation, Int. J. Comput. Methods Eng. Sci. Mech., 19 (2018), 221-225.  doi: 10.1080/15502287.2018.1472150.  Google Scholar

[29]

B. M. Vaganan and E. E. Priya, Generalized Cole-Hopf transformations for generalized Burgers equations, Pramana, 85 (2015), 861-867.  doi: 10.1007/s12043-015-1107-4.  Google Scholar

[30]

D. L. Young, C. C. Tsai, K. Murugesana, C. M. Fan and C. W. Chen, Time-dependent fundamental solutions for homogeneous diffusion problems, Engineering Analysis with Boundary Elements, 28 (2004), 1463-1473.  doi: 10.1016/j.enganabound.2004.07.003.  Google Scholar

[31]

H. Zhang, F. Liu, I. Turner and Q. Yang, Numerical solution of the time fractional Black-Scholes model governing European options, Comput. Math. Appl., 71 (2016), 1772-1783.  doi: 10.1016/j.camwa.2016.02.007.  Google Scholar

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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$ 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
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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$ 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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