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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 and 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.

[2]

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

[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.

[4]

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

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

[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.

[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.

[8]

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

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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.

[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.

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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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

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.

[2]

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

[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.

[4]

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

[5]

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

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

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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