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June  2019, 12(3): 625-643. doi: 10.3934/dcdss.2019040

## Barycentric spectral domain decomposition methods for valuing a class of infinite activity Lévy models

 1 Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 002, Republic of South Africa 2 Department of Statistical and Actuarial Sciences, University of Western Ontario, London, Ont., Canada

* Corresponding author: E. Pindza

Received  July 2017 Revised  November 2017 Published  September 2018

A new barycentric spectral domain decomposition methods algorithm for solving partial integro-differential models is described. The method is applied to European and butterfly call option pricing problems under a class of infinite activity Lévy models. It is based on the barycentric spectral domain decomposition methods which allows the implementation of the boundary conditions in an efficient way. After the approximation of the spatial derivatives, we obtained the semi-discrete equations. The computation of these equations is performed by using the barycentric spectral domain decomposition method. This is achieved with the implementation of an exponential time integration scheme. Several numerical tests for the pricing of European and butterfly options are given to illustrate the efficiency and accuracy of this new algorithm. We also show that Greek options, such as Delta and Gamma sensitivity, are computed with no spurious oscillation.

Citation: Edson Pindza, Francis Youbi, Eben Maré, Matt Davison. Barycentric spectral domain decomposition methods for valuing a class of infinite activity Lévy models. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 625-643. doi: 10.3934/dcdss.2019040
##### References:
 [1] A. Almendral and C. W. Oosterlee, Accurate evaluation of European and American options under the CGMY process, SIAM Journal on Scientific Computing, 29 (2007), 93-117.  doi: 10.1137/050637613. [2] R. Baltensperger, J. P. Berrut and B. Noël, Exponential convergence of a linear rational interpolant between transformed Chebyshev points, Mathematics of Computation, 68 (1999), 1109-1120.  doi: 10.1090/S0025-5718-99-01070-4. [3] J. P. Berrut and H. D. Milltelmann, Rational interpolation trough the optimal attachment of poles to the interpolating polynomial, Numerical Algorithms, 23 (2000), 315-328.  doi: 10.1023/A:1019168504808. [4] J. P. Berrut and L. N. Trefethen, Barycentric Lagrange interpolation, SIAM Review, 46 (2004), 501-517.  doi: 10.1137/S0036144502417715. [5] F. Black and M. Scholes, Pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062. [6] M. Briani, C. La Chioma and R. Natalini, Convergence of numerical schemes for viscosity solutions to integro-differential degenerate parabolic problems arising in financial theory, Journal of Numerical Mathematics, 98 (2004), 607-646.  doi: 10.1007/s00211-004-0530-0. [7] P. Carr, H. German, D. B. Madan and M. Yor, The fine structure of asset returns: An empirical investigation, Journal of Business, 75 (2002), 305-332.  doi: 10.1086/338705. [8] T. F. Coleman, Y. Li and A. Verma, Reconstructing the unknown local volatility function, Journal of Computational Finance, 2 (1999), 77-100.  doi: 10.1142/9789812810663_0007. [9] R. Company, L. J$\acute{o}$dar and M. Fakharany, Positive solutions of European option pricing with CGMY process models using double discretization difference schemes, Abstr. Appl. Anal., 2013 (2013), Art. ID 517480, 11 pp. [10] R. Cont and E. Voltchkova, A finite difference scheme for option pricing in jumps diffusion and exponential L$\acute{e}$vy models, SIAM Journal on Numerical Analysis, 43 (2005), 1596-1626.  doi: 10.1137/S0036142903436186. [11] R. Cont and E. Voltchkova, A finite difference scheme for option pricing in jump diffusion and exponential Lvy models, SIAM J. Numer. Anal., 43 (2005), 1596-1626.  doi: 10.1137/S0036142903436186. [12] J. M. Corcuera et al, Completion of Lévy market by power jump assets, Finance Stochastic, 9 (2005), 109–127. doi: 10.1007/s00780-004-0139-2. [13] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Academic Press Inc., Orlando, FL, 2nd edition, 1984. [14] Y. d'Halluin, P. A. Forsyth and K. R. Vetzal, Robust numerical methods for contigent claims under jump-diffusion processs, IMA Journal of Numerical Analysis, 25 (2005), 87-112.  doi: 10.1093/imanum/drh011. [15] B. Dupire, Pricing with a smile, RISK Magazine, 1 (1994), 18-20. [16] M. Fakharany, R. Company and L. Jódar, Solving partial integro-option pricing problems for a wide class of infinite activity Lévy processes, Journal of Computational and applied Mathematics, 296 (2016), 739-752.  doi: 10.1016/j.cam.2015.10.027. [17] R. Frontczak and R. Söchbel, On modified Mellin transforms, Gauss aguerre quadrature, and the valuation of American call options, Journal of Computational and Applied Mathematics, 234 (2010), 1559-1571.  doi: 10.1016/j.cam.2010.02.037. [18] D. Funaro, Polynomial Approximation of Differential Equations, Springer, Berlin, 1992. [19] T. Goll and J. Kallen, Optimal portfolios for logarithmic utility, Stochastic Process. Appl., 89 (2000), 31-48.  doi: 10.1016/S0304-4149(00)00011-9. [20] T. Goll and L. Rüschendorf, Minimax and minimal distance martingale measures and their relationship to portfolio optimization, Finance Stochastic, 5 (2001), 557-581.  doi: 10.1007/s007800100052. [21] D. Gottlieb and C. -W. Shu, On the Gibbs phenomenon Ⅳ: Recovering exponential accuracy in a subinterval from a Gegenbauer partial sum of a piecewise analytic function, Journal of Computational and Applied Mathematics, 64 (1995), 1081-1095.  doi: 10.2307/2153484. [22] S. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327. [23] M. Hochbruck and C. Lubich, On Krylov subspace approximations to the matrix exponential operator, SIAM Journal of Numerical Analysis, 34 (1997), 1911-1925.  doi: 10.1137/S0036142995280572. [24] J. Hull and A. White, The pricing of options with stochastic volatilities, Journal of Finance, 42 (1987), 281-300.  doi: 10.1111/j.1540-6261.1987.tb02568.x. [25] A. K. Kassam and L. N. Trefethen, Fourth-order time stepping for stiff PDEs, SIAM Journal of Scientific Computing, 26 (2005), 1214-1233.  doi: 10.1137/S1064827502410633. [26] G. Klein and J. P. Berrut, Linear barycentric rational quadrature, BIT Numerical Mathematics, 52 (2012), 407-424.  doi: 10.1007/s10543-011-0357-x. [27] P. E. Kloeden, G. J. Neuenkirch and T. Shardlow, The exponential integrator scheme for stochastic partial differential equations: Pathwise error bounds, Journal of Computational and Applied Mathematics, 235 (2011), 1245-1260.  doi: 10.1016/j.cam.2010.08.011. [28] S. G. Kou, A jump-diffusion model for option pricing, Management Science, 8 (2002), 1086-1101. [29] J. L. Lagrange, Leçons élémentaires sur les mathématiques, données à l'Ecole Normal en 1795, in Oeuvres VII, Gauthier-Villars, Paris, 7 (1877), 183–287. [30] D. B. Madan and E. Seneta, The variance gamma model for share market returns, Journal of Business, 63 (1990), 511-524.  doi: 10.1086/296519. [31] D. Madan and M. Yor, Representing the CGMY and Meixner Lvy processes as time changed Brownian motions, Journal of Computational Finance, 12 (2008), 27-47.  doi: 10.21314/JCF.2008.181. [32] C. Markakis, and L. Barack, High-order difference and pseudospectral methods for discontinuous problems, arXiv: 1406.4865v1, [maths. NA], (2014) 1-9. [33] R. C. Merton, Option pricing when the underlying stocks are discontinuous, Journal of Financial Economics, 5 (1976), 125-144.  doi: 10.1016/0304-405X(76)90022-2. [34] J. Niesen and W. M. Wright, A Krylov subspace method for option pricing, Technical report SSRN 1799124, 2011. doi: 10.2139/ssrn.1799124. [35] E. Ngounda, K. C. Patidar and E. Pindza, Contour Integral Method for European Options with Jumps, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 478-492.  doi: 10.1016/j.cnsns.2012.08.003. [36] S. A. Orszag, Spectral methods for problems in complex geometries, Journal of Computational Physics, 37 (1980), 70-92.  doi: 10.1016/0021-9991(80)90005-4. [37] H. K. Pang and H. W. Sun, Fast exponential time integration for pricing options in stochastic volatility jump diffusion models, East Asian Journal on Applied Mathematics, 4 (2014), 52-68.  doi: 10.4208/eajam.280313.061013a. [38] A. Papapantoleon, An Introduction to Lévy Processes with Applications in Finance, Lecture Notes, University of Freiburg, 2008. [39] H. P. Pfeiffer, L. E. Kidder, M. A. Scheel and S. A. Teukolsky, A multi-domain spectral method for solving elliptic equations, Computer Physics Communications, 152 (2003), 253-273.  doi: 10.1016/S0010-4655(02)00847-0. [40] E. Pindza, K. C. Patidar and E. Ngounda, Robust Spectral Method for Numerical Valuation of European Options under Merton's Jump-Diffusion Model, Numerical Methods for Partial Differential Equations, 30 (2014), 1169-1188.  doi: 10.1002/num.21864. [41] N. Rambeerich, D. Y. Tangaman and M. Bhuruth, Numerical Pricing Of American Option Under Infinite Activity Lévy Processes, Journal of Futures Markets, 31 (2011), 809-829.  doi: 10.1002/fut.20497. [42] Y. Saad, Analysis of some Krylov subspace approximations to the matrix exponential operator, SIAM Journal of Numerical Analysis, 29 (1992), 209-228.  doi: 10.1137/0729014. [43] T. Schmelzer and L. N. Trefethen, Evaluating matrix functions for exponential integrators via Carathéodory-Fejér approximation and contour integrals, Electronic Transactions on Numerical Analysis, 29 (2007), 1-18. [44] W. Schoutens, J. L. Teugels and L. processes, Polynomials and martingales, Communications in Statistics. Stochastic Models, 14 (1998), 335-349.  doi: 10.1080/15326349808807475. [45] W. Schoutens, Lévy Processes in Finance: Pricing Financial Derivatives, Wiley, New York, 2003. doi: 10.1002/0470870230. [46] D. Y. Tangman, A. Gopaul and M. Bhuruth, Exponential time integration and Chebyshev discretisation schemes for fast pricing options, Applied Numerical Mathematics, 58 (2008), 1309-1319.  doi: 10.1016/j.apnum.2007.07.005. [47] D. Tavella and C. Randall, Pricing Financial Instruments: The Finite Difference Method, Wiley, New York, 2000. [48] L. N. Trefethen, Approximation Theory and Approximation Practice, SIAM, Philadelphia, PA, 2013. [49] L. N. Trefethen, Is Gauss quadrature better than Clenshaw urtis?, SIAM Review, 50 (2008), 67-87.  doi: 10.1137/060659831. [50] L. N. Trefethen, Spectral Methods in Matlab, SIAM, Philadelphia, PA, 2000. doi: 10.1137/1.9780898719598. [51] L. N. Trefethen and H. M. Gutknecht, The Carathéodory-Fejér method for real rational approximation, SIAM Journal on Numerical Analysis, 20 (1983), 420-436.  doi: 10.1137/0720030. [52] I. R. Wang, J. W. Wan and P. A. Forsyth, Robust numerical valuation of European and American options under the CGMY process, Journal of Computational Finance, 10 (2007), 31-69.  doi: 10.21314/JCF.2007.169. [53] B. D. Welfert, Generation of pseudospectral differentiation matrices Ⅰ, SIAM Journal on Numerical Analysis, 34 (1997), 1640-1657.  doi: 10.1137/S0036142993295545.

show all references

##### References:
 [1] A. Almendral and C. W. Oosterlee, Accurate evaluation of European and American options under the CGMY process, SIAM Journal on Scientific Computing, 29 (2007), 93-117.  doi: 10.1137/050637613. [2] R. Baltensperger, J. P. Berrut and B. Noël, Exponential convergence of a linear rational interpolant between transformed Chebyshev points, Mathematics of Computation, 68 (1999), 1109-1120.  doi: 10.1090/S0025-5718-99-01070-4. [3] J. P. Berrut and H. D. Milltelmann, Rational interpolation trough the optimal attachment of poles to the interpolating polynomial, Numerical Algorithms, 23 (2000), 315-328.  doi: 10.1023/A:1019168504808. [4] J. P. Berrut and L. N. Trefethen, Barycentric Lagrange interpolation, SIAM Review, 46 (2004), 501-517.  doi: 10.1137/S0036144502417715. [5] F. Black and M. Scholes, Pricing of options and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062. [6] M. Briani, C. La Chioma and R. Natalini, Convergence of numerical schemes for viscosity solutions to integro-differential degenerate parabolic problems arising in financial theory, Journal of Numerical Mathematics, 98 (2004), 607-646.  doi: 10.1007/s00211-004-0530-0. [7] P. Carr, H. German, D. B. Madan and M. Yor, The fine structure of asset returns: An empirical investigation, Journal of Business, 75 (2002), 305-332.  doi: 10.1086/338705. [8] T. F. Coleman, Y. Li and A. Verma, Reconstructing the unknown local volatility function, Journal of Computational Finance, 2 (1999), 77-100.  doi: 10.1142/9789812810663_0007. [9] R. Company, L. J$\acute{o}$dar and M. Fakharany, Positive solutions of European option pricing with CGMY process models using double discretization difference schemes, Abstr. Appl. Anal., 2013 (2013), Art. ID 517480, 11 pp. [10] R. Cont and E. Voltchkova, A finite difference scheme for option pricing in jumps diffusion and exponential L$\acute{e}$vy models, SIAM Journal on Numerical Analysis, 43 (2005), 1596-1626.  doi: 10.1137/S0036142903436186. [11] R. Cont and E. Voltchkova, A finite difference scheme for option pricing in jump diffusion and exponential Lvy models, SIAM J. Numer. Anal., 43 (2005), 1596-1626.  doi: 10.1137/S0036142903436186. [12] J. M. Corcuera et al, Completion of Lévy market by power jump assets, Finance Stochastic, 9 (2005), 109–127. doi: 10.1007/s00780-004-0139-2. [13] P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Academic Press Inc., Orlando, FL, 2nd edition, 1984. [14] Y. d'Halluin, P. A. Forsyth and K. R. Vetzal, Robust numerical methods for contigent claims under jump-diffusion processs, IMA Journal of Numerical Analysis, 25 (2005), 87-112.  doi: 10.1093/imanum/drh011. [15] B. Dupire, Pricing with a smile, RISK Magazine, 1 (1994), 18-20. [16] M. Fakharany, R. Company and L. Jódar, Solving partial integro-option pricing problems for a wide class of infinite activity Lévy processes, Journal of Computational and applied Mathematics, 296 (2016), 739-752.  doi: 10.1016/j.cam.2015.10.027. [17] R. Frontczak and R. Söchbel, On modified Mellin transforms, Gauss aguerre quadrature, and the valuation of American call options, Journal of Computational and Applied Mathematics, 234 (2010), 1559-1571.  doi: 10.1016/j.cam.2010.02.037. [18] D. Funaro, Polynomial Approximation of Differential Equations, Springer, Berlin, 1992. [19] T. Goll and J. Kallen, Optimal portfolios for logarithmic utility, Stochastic Process. Appl., 89 (2000), 31-48.  doi: 10.1016/S0304-4149(00)00011-9. [20] T. Goll and L. Rüschendorf, Minimax and minimal distance martingale measures and their relationship to portfolio optimization, Finance Stochastic, 5 (2001), 557-581.  doi: 10.1007/s007800100052. [21] D. Gottlieb and C. -W. Shu, On the Gibbs phenomenon Ⅳ: Recovering exponential accuracy in a subinterval from a Gegenbauer partial sum of a piecewise analytic function, Journal of Computational and Applied Mathematics, 64 (1995), 1081-1095.  doi: 10.2307/2153484. [22] S. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies, 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327. [23] M. Hochbruck and C. Lubich, On Krylov subspace approximations to the matrix exponential operator, SIAM Journal of Numerical Analysis, 34 (1997), 1911-1925.  doi: 10.1137/S0036142995280572. [24] J. Hull and A. White, The pricing of options with stochastic volatilities, Journal of Finance, 42 (1987), 281-300.  doi: 10.1111/j.1540-6261.1987.tb02568.x. [25] A. K. Kassam and L. N. Trefethen, Fourth-order time stepping for stiff PDEs, SIAM Journal of Scientific Computing, 26 (2005), 1214-1233.  doi: 10.1137/S1064827502410633. [26] G. Klein and J. P. Berrut, Linear barycentric rational quadrature, BIT Numerical Mathematics, 52 (2012), 407-424.  doi: 10.1007/s10543-011-0357-x. [27] P. E. Kloeden, G. J. Neuenkirch and T. Shardlow, The exponential integrator scheme for stochastic partial differential equations: Pathwise error bounds, Journal of Computational and Applied Mathematics, 235 (2011), 1245-1260.  doi: 10.1016/j.cam.2010.08.011. [28] S. G. Kou, A jump-diffusion model for option pricing, Management Science, 8 (2002), 1086-1101. [29] J. L. Lagrange, Leçons élémentaires sur les mathématiques, données à l'Ecole Normal en 1795, in Oeuvres VII, Gauthier-Villars, Paris, 7 (1877), 183–287. [30] D. B. Madan and E. Seneta, The variance gamma model for share market returns, Journal of Business, 63 (1990), 511-524.  doi: 10.1086/296519. [31] D. Madan and M. Yor, Representing the CGMY and Meixner Lvy processes as time changed Brownian motions, Journal of Computational Finance, 12 (2008), 27-47.  doi: 10.21314/JCF.2008.181. [32] C. Markakis, and L. Barack, High-order difference and pseudospectral methods for discontinuous problems, arXiv: 1406.4865v1, [maths. NA], (2014) 1-9. [33] R. C. Merton, Option pricing when the underlying stocks are discontinuous, Journal of Financial Economics, 5 (1976), 125-144.  doi: 10.1016/0304-405X(76)90022-2. [34] J. Niesen and W. M. Wright, A Krylov subspace method for option pricing, Technical report SSRN 1799124, 2011. doi: 10.2139/ssrn.1799124. [35] E. Ngounda, K. C. Patidar and E. Pindza, Contour Integral Method for European Options with Jumps, Communications in Nonlinear Science and Numerical Simulation, 18 (2013), 478-492.  doi: 10.1016/j.cnsns.2012.08.003. [36] S. A. Orszag, Spectral methods for problems in complex geometries, Journal of Computational Physics, 37 (1980), 70-92.  doi: 10.1016/0021-9991(80)90005-4. [37] H. K. Pang and H. W. Sun, Fast exponential time integration for pricing options in stochastic volatility jump diffusion models, East Asian Journal on Applied Mathematics, 4 (2014), 52-68.  doi: 10.4208/eajam.280313.061013a. [38] A. Papapantoleon, An Introduction to Lévy Processes with Applications in Finance, Lecture Notes, University of Freiburg, 2008. [39] H. P. Pfeiffer, L. E. Kidder, M. A. Scheel and S. A. Teukolsky, A multi-domain spectral method for solving elliptic equations, Computer Physics Communications, 152 (2003), 253-273.  doi: 10.1016/S0010-4655(02)00847-0. [40] E. Pindza, K. C. Patidar and E. Ngounda, Robust Spectral Method for Numerical Valuation of European Options under Merton's Jump-Diffusion Model, Numerical Methods for Partial Differential Equations, 30 (2014), 1169-1188.  doi: 10.1002/num.21864. [41] N. Rambeerich, D. Y. Tangaman and M. Bhuruth, Numerical Pricing Of American Option Under Infinite Activity Lévy Processes, Journal of Futures Markets, 31 (2011), 809-829.  doi: 10.1002/fut.20497. [42] Y. Saad, Analysis of some Krylov subspace approximations to the matrix exponential operator, SIAM Journal of Numerical Analysis, 29 (1992), 209-228.  doi: 10.1137/0729014. [43] T. Schmelzer and L. N. Trefethen, Evaluating matrix functions for exponential integrators via Carathéodory-Fejér approximation and contour integrals, Electronic Transactions on Numerical Analysis, 29 (2007), 1-18. [44] W. Schoutens, J. L. Teugels and L. processes, Polynomials and martingales, Communications in Statistics. Stochastic Models, 14 (1998), 335-349.  doi: 10.1080/15326349808807475. [45] W. Schoutens, Lévy Processes in Finance: Pricing Financial Derivatives, Wiley, New York, 2003. doi: 10.1002/0470870230. [46] D. Y. Tangman, A. Gopaul and M. Bhuruth, Exponential time integration and Chebyshev discretisation schemes for fast pricing options, Applied Numerical Mathematics, 58 (2008), 1309-1319.  doi: 10.1016/j.apnum.2007.07.005. [47] D. Tavella and C. Randall, Pricing Financial Instruments: The Finite Difference Method, Wiley, New York, 2000. [48] L. N. Trefethen, Approximation Theory and Approximation Practice, SIAM, Philadelphia, PA, 2013. [49] L. N. Trefethen, Is Gauss quadrature better than Clenshaw urtis?, SIAM Review, 50 (2008), 67-87.  doi: 10.1137/060659831. [50] L. N. Trefethen, Spectral Methods in Matlab, SIAM, Philadelphia, PA, 2000. doi: 10.1137/1.9780898719598. [51] L. N. Trefethen and H. M. Gutknecht, The Carathéodory-Fejér method for real rational approximation, SIAM Journal on Numerical Analysis, 20 (1983), 420-436.  doi: 10.1137/0720030. [52] I. R. Wang, J. W. Wan and P. A. Forsyth, Robust numerical valuation of European and American options under the CGMY process, Journal of Computational Finance, 10 (2007), 31-69.  doi: 10.21314/JCF.2007.169. [53] B. D. Welfert, Generation of pseudospectral differentiation matrices Ⅰ, SIAM Journal on Numerical Analysis, 34 (1997), 1640-1657.  doi: 10.1137/S0036142993295545.
Spectral domain decomposition method matrix structures
Numerical valuation of European call options for the CGMY, Meixner and GH model with their Greeks for the parameters in Table 2
Convergence of the SDDM and FDM for European vanilla call options for the parameters in Table 2
Numerical valuation of European butterfly call options for the CGMY, Meixner, and GH model with $N = 16, K_{1} = 40, K_{2} = 50, K_{3} = 60$ for the parameters in Table 2
Convergence of the SDDM and FDM for European vanilla butterfly call options for the parameters in Table 2
Density functions for Lévy Processes
 Model Lévy density function CGMY $f(y)=\frac{C_{-}e^{-G|y|}}{|y|^{1+Y}}{\bf{1}}_{y<0}+ \frac{C_{+}e^{-M|y|}}{|y|^{1+Y}}{\bf{1}}_{y>0}$ Meixner $f(y)=\frac{Ae^{-ay}}{y\sinh(by)}$ GH process $f(y)=\frac{e^{\beta y}}{|y|}\left(\int_{0}^{\infty}\frac{e^{-\sqrt{2\zeta+\alpha^{2}}|y|}}{\pi^{2}\zeta \left(J^{2}_{|\lambda|}\left(\delta\sqrt{2\zeta}\right)+Y^{2}_{|\lambda|}\left(\delta\sqrt{2\zeta}\right)\right)}d\zeta+\max(0,\lambda)e^{-\alpha|y|}\right)$
 Model Lévy density function CGMY $f(y)=\frac{C_{-}e^{-G|y|}}{|y|^{1+Y}}{\bf{1}}_{y<0}+ \frac{C_{+}e^{-M|y|}}{|y|^{1+Y}}{\bf{1}}_{y>0}$ Meixner $f(y)=\frac{Ae^{-ay}}{y\sinh(by)}$ GH process $f(y)=\frac{e^{\beta y}}{|y|}\left(\int_{0}^{\infty}\frac{e^{-\sqrt{2\zeta+\alpha^{2}}|y|}}{\pi^{2}\zeta \left(J^{2}_{|\lambda|}\left(\delta\sqrt{2\zeta}\right)+Y^{2}_{|\lambda|}\left(\delta\sqrt{2\zeta}\right)\right)}d\zeta+\max(0,\lambda)e^{-\alpha|y|}\right)$
The parameters for Lévy models used in both examples
 Model Parameters GBM (Black-Scholes) $K =50, \ r = 0.05, \sigma = 0.2, q=0$ and $T =0.5.$ CGMY $C_{-} = 0.3, \ C_{+} = 0.1, \ \ G = 15,\ M = 25$ and $Y = 20.$ Meixner $A=15, \ a=-1.5$ and $b=50$ GH process $\alpha=4, \ \beta=-3.2, \ \delta=1.4775$ and $\lambda=-3$
 Model Parameters GBM (Black-Scholes) $K =50, \ r = 0.05, \sigma = 0.2, q=0$ and $T =0.5.$ CGMY $C_{-} = 0.3, \ C_{+} = 0.1, \ \ G = 15,\ M = 25$ and $Y = 20.$ Meixner $A=15, \ a=-1.5$ and $b=50$ GH process $\alpha=4, \ \beta=-3.2, \ \delta=1.4775$ and $\lambda=-3$
The benchmark European call option values under Lévy processes with different values of S and $N = 150$ for the parameters in Table 2
 Model $S$ 40 50 60 CGMY 0.2210443864 3.3785900783 11.3681462140 Meixner 1.3420365535 5.4934725848 12.7780678851 GH processes 0.3237597911 3.8485639686 11.9164490861
 Model $S$ 40 50 60 CGMY 0.2210443864 3.3785900783 11.3681462140 Meixner 1.3420365535 5.4934725848 12.7780678851 GH processes 0.3237597911 3.8485639686 11.9164490861
Absolute errors (${\bf{AE}}$) of the benchmark and the European call option apply to the CGMY, Meixner and GH processes models with different values of $N$ and $S$ for the parameters in Table 2
 $SDDM$ $FDM$ $S$ $40$ 50 60 $40$ 50 60 $N$ ${\bf{AE}}$ ${\bf{AE}}$ ${\bf{AE}}$ CPU ${\bf{AE}}$ ${\bf{AE}}$ ${\bf{AE}}$ CPU CGMY 10 $1.15e^{-4}$ $1.28e^{-4}$ $1.35e^{-4}$ $0.30$ $1.14e^{-2}$ $5.72e^{-2}$ $9.55e^{-3}$ 0.6 15 $1.23e^{-5}$ $1.73e^{-5}$ $1.45e^{-5}$ $0.34$ $2.03e^{-3}$ $1.55e^{-2}$ $1.75e^{-3}$ 0.72 20 $2.75e^{-7}$ $2.13e^{-7}$ $2.34e^{-7}$ $0.40$ $6.25e^{-4}$ $4.09e^{-3}$ $6.06e^{-4}$ 0.87 25 $3.33e^{-10}$ $3.15e^{-10}$ $3.24e^{-10}$ $0.53$ $2.75e^{-4}$ $1.91e^{-3}$ $2.37e^{-4}$ 1.32 Meixner 10 $2.12e^{-4}$ $2.45e^{-4}$ $2.35e^{-4}$ $0.32$ $1.46e^{-2}$ $4.35e^{-2}$ $8.51e^{-3}$ 0.65 15 $2.78e^{-5}$ $2.65e^{-5}$ $2.67e^{-5}$ $0.35$ $2.66e^{-3}$ $1.16e^{-2}$ $2.13e^{-3}$ 0.78 20 $3.40e^{-7}$ $3.23e^{-7}$ $3.14e^{-7}$ 0.41 $6.45e^{-4}$ $3.95e^{-3}$ $5.45e^{-4}$ 0.86 25 $4.77e^{-10}$ $4.65e^{-10}$ $4.33e^{-10}$ $0.55$ $2.35e^{-4}$ $1.02e^{-3}$ $2.14e^{-4}$ 1.41 GH processes 10 $3.33e^{-4}$ $3.29e^{-4}$ $3.17e^{-4}$ $0.65$ $1.45e^{-2}$ $5.33e^{-2}$ $7.13e^{-3}$ 1.33 15 $4.55e^{-5}$ $4.370e^{-5}$ $4.14e^{-5}$ $0.82$ $2.15e^{-3}$ $1.04e^{-2}$ $5.72e^{-2}$ 1.61 20 $5.14e^{-7}$ $5.21e^{-7}$ $5.14e^{-7}$ $1.41$ $5.61e^{-4}$ $4.02e^{-3}$ $6.12e^{-4}$ 2.94 25 $7.11e^{-10}$ $7.25e^{-10}$ $7.33e^{-10}$ $1.82$ $2.36e^{-4}$ $1.21e^{-3}$ $3.01e^{-4}$ 3.51
 $SDDM$ $FDM$ $S$ $40$ 50 60 $40$ 50 60 $N$ ${\bf{AE}}$ ${\bf{AE}}$ ${\bf{AE}}$ CPU ${\bf{AE}}$ ${\bf{AE}}$ ${\bf{AE}}$ CPU CGMY 10 $1.15e^{-4}$ $1.28e^{-4}$ $1.35e^{-4}$ $0.30$ $1.14e^{-2}$ $5.72e^{-2}$ $9.55e^{-3}$ 0.6 15 $1.23e^{-5}$ $1.73e^{-5}$ $1.45e^{-5}$ $0.34$ $2.03e^{-3}$ $1.55e^{-2}$ $1.75e^{-3}$ 0.72 20 $2.75e^{-7}$ $2.13e^{-7}$ $2.34e^{-7}$ $0.40$ $6.25e^{-4}$ $4.09e^{-3}$ $6.06e^{-4}$ 0.87 25 $3.33e^{-10}$ $3.15e^{-10}$ $3.24e^{-10}$ $0.53$ $2.75e^{-4}$ $1.91e^{-3}$ $2.37e^{-4}$ 1.32 Meixner 10 $2.12e^{-4}$ $2.45e^{-4}$ $2.35e^{-4}$ $0.32$ $1.46e^{-2}$ $4.35e^{-2}$ $8.51e^{-3}$ 0.65 15 $2.78e^{-5}$ $2.65e^{-5}$ $2.67e^{-5}$ $0.35$ $2.66e^{-3}$ $1.16e^{-2}$ $2.13e^{-3}$ 0.78 20 $3.40e^{-7}$ $3.23e^{-7}$ $3.14e^{-7}$ 0.41 $6.45e^{-4}$ $3.95e^{-3}$ $5.45e^{-4}$ 0.86 25 $4.77e^{-10}$ $4.65e^{-10}$ $4.33e^{-10}$ $0.55$ $2.35e^{-4}$ $1.02e^{-3}$ $2.14e^{-4}$ 1.41 GH processes 10 $3.33e^{-4}$ $3.29e^{-4}$ $3.17e^{-4}$ $0.65$ $1.45e^{-2}$ $5.33e^{-2}$ $7.13e^{-3}$ 1.33 15 $4.55e^{-5}$ $4.370e^{-5}$ $4.14e^{-5}$ $0.82$ $2.15e^{-3}$ $1.04e^{-2}$ $5.72e^{-2}$ 1.61 20 $5.14e^{-7}$ $5.21e^{-7}$ $5.14e^{-7}$ $1.41$ $5.61e^{-4}$ $4.02e^{-3}$ $6.12e^{-4}$ 2.94 25 $7.11e^{-10}$ $7.25e^{-10}$ $7.33e^{-10}$ $1.82$ $2.36e^{-4}$ $1.21e^{-3}$ $3.01e^{-4}$ 3.51
The benchmark values of the European butterfly call option values under Lévy processes with different values of S and $N = 100$ for the parameters in Table 2
 Model $S$ 40 50 60 CGMY 2.2845953002 4.6814621409 2.1592689295 Meixner 2.2689295039 3.7101827676 2.3159268929 GH processes 2.3942558746 4.2898172323 1.7989556135
 Model $S$ 40 50 60 CGMY 2.2845953002 4.6814621409 2.1592689295 Meixner 2.2689295039 3.7101827676 2.3159268929 GH processes 2.3942558746 4.2898172323 1.7989556135
Absolute errors (${\bf{AE}}$) of the benchmark and the European call option apply to the CGMY, Meixner, and GH processes models with different values of $N$ and $S$ for the parameters in Table 2
 $SDDM$ $FDM$ $S$ $40$ 50 60 $40$ 50 60 $N$ ${\bf{AE}}$ ${\bf{AE}}$ ${\bf{AE}}$ CPU ${\bf{AE}}$ ${\bf{AE}}$ ${\bf{AE}}$ CPU CGMY 07 $1.88e^{-4}$ $1.76e^{-4}$ $1.76e^{-4}$ $0.20$ $1.24e^{-2}$ $5.81e^{-2}$ $8.98e^{-3}$ 0.62 10 $1.45e^{-5}$ $1.81e^{-5}$ $1.71e^{-5}$ $0.28$ $1.98e^{-3}$ $1.63e^{-2}$ $2.05e^{-3}$ 0.73 13 $2.91e^{-7}$ $2.33e^{-7}$ $2.25e^{-7}$ $0.33$ $5.99e^{-4}$ $4.29e^{-3}$ $5.66e^{-4}$ 0.87 16 $3.72e^{-10}$ $3.34e^{-10}$ $3.81e^{-10}$ $0.44$ $2.83e^{-4}$ $2.11e^{-3}$ $2.48e^{-4}$ 1.41 Meixner 07 $2.45e^{-4}$ $3.32e^{-4}$ $3.22e^{-4}$ $0.24$ $1.56e^{-2}$ $4.28e^{-2}$ $8.88e^{-3}$ 0.63 10 $2.72e^{-5}$ $2.75e^{-5}$ $2.46e^{-5}$ $0.24$ $1.99e^{-3}$ $1.23e^{-2}$ $2.34e^{-3}$ 0.75 13 $3.54e^{-7}$ $3.28e^{-7}$ $3.69e^{-7}$ 0.34 $6.25e^{-4}$ $4.05e^{-3}$ $5.32e^{-4}$ 0.85 16 $5.68e^{-10}$ $5.55e^{-10}$ $5.88e^{-10}$ $0.42$ $2.44e^{-4}$ $1.24e^{-3}$ $2.51e^{-4}$ 1.42 GH processes 07 $3.12e^{-4}$ $3.02e^{-4}$ $3.45e^{-4}$ $0.51$ $1.25e^{-2}$ $5.71e^{-2}$ $6.97e^{-3}$ 1.32 10 $5.51e^{-5}$ $6.1e^{-5}$ $5.92e^{-5}$ $0.68$ $2.13e^{-3}$ $1.21e^{-2}$ $5.87e^{-2}$ 1.63 13 $7.11e^{-7}$ $7.25e^{-7}$ $7.33e^{-7}$ $0.82$ $5.11e^{-4}$ $4.14e^{-3}$ $6.22e^{-4}$ 2.91 16 $8.25e^{-10}$ $9.12e^{-10}$ $9.23e^{-10}$ $1.32$ $2.53e^{-4}$ $1.31e^{-3}$ $3.12e^{-4}$ 3.48
 $SDDM$ $FDM$ $S$ $40$ 50 60 $40$ 50 60 $N$ ${\bf{AE}}$ ${\bf{AE}}$ ${\bf{AE}}$ CPU ${\bf{AE}}$ ${\bf{AE}}$ ${\bf{AE}}$ CPU CGMY 07 $1.88e^{-4}$ $1.76e^{-4}$ $1.76e^{-4}$ $0.20$ $1.24e^{-2}$ $5.81e^{-2}$ $8.98e^{-3}$ 0.62 10 $1.45e^{-5}$ $1.81e^{-5}$ $1.71e^{-5}$ $0.28$ $1.98e^{-3}$ $1.63e^{-2}$ $2.05e^{-3}$ 0.73 13 $2.91e^{-7}$ $2.33e^{-7}$ $2.25e^{-7}$ $0.33$ $5.99e^{-4}$ $4.29e^{-3}$ $5.66e^{-4}$ 0.87 16 $3.72e^{-10}$ $3.34e^{-10}$ $3.81e^{-10}$ $0.44$ $2.83e^{-4}$ $2.11e^{-3}$ $2.48e^{-4}$ 1.41 Meixner 07 $2.45e^{-4}$ $3.32e^{-4}$ $3.22e^{-4}$ $0.24$ $1.56e^{-2}$ $4.28e^{-2}$ $8.88e^{-3}$ 0.63 10 $2.72e^{-5}$ $2.75e^{-5}$ $2.46e^{-5}$ $0.24$ $1.99e^{-3}$ $1.23e^{-2}$ $2.34e^{-3}$ 0.75 13 $3.54e^{-7}$ $3.28e^{-7}$ $3.69e^{-7}$ 0.34 $6.25e^{-4}$ $4.05e^{-3}$ $5.32e^{-4}$ 0.85 16 $5.68e^{-10}$ $5.55e^{-10}$ $5.88e^{-10}$ $0.42$ $2.44e^{-4}$ $1.24e^{-3}$ $2.51e^{-4}$ 1.42 GH processes 07 $3.12e^{-4}$ $3.02e^{-4}$ $3.45e^{-4}$ $0.51$ $1.25e^{-2}$ $5.71e^{-2}$ $6.97e^{-3}$ 1.32 10 $5.51e^{-5}$ $6.1e^{-5}$ $5.92e^{-5}$ $0.68$ $2.13e^{-3}$ $1.21e^{-2}$ $5.87e^{-2}$ 1.63 13 $7.11e^{-7}$ $7.25e^{-7}$ $7.33e^{-7}$ $0.82$ $5.11e^{-4}$ $4.14e^{-3}$ $6.22e^{-4}$ 2.91 16 $8.25e^{-10}$ $9.12e^{-10}$ $9.23e^{-10}$ $1.32$ $2.53e^{-4}$ $1.31e^{-3}$ $3.12e^{-4}$ 3.48
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