doi: 10.3934/dcdss.2020052

European option valuation under the Bates PIDE in finance: A numerical implementation of the Gaussian scheme

1. 

Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137–66731, Iran

2. 

Department of Mathematics, Art and Science Faculty, Siirt University, Siirt, Turkey

* Corresponding author

Received  May 2018 Revised  July 2018 Published  March 2019

Models at which not only the asset price but also the volatility are assumed to be stochastic have received a remarkable attention in financial markets. The objective of the current research is to design a numerical method for solving the stochastic volatility (SV) jump–diffusion model of Bates, at which the presence of a nonlocal integral makes the coding of numerical schemes intensive. A numerical implementation is furnished by gathering several different techniques such as the radial basis function (RBF) generated finite difference (FD) approach, which keeps the sparsity of the FD methods but gives rise to the higher accuracy of the RBF meshless methods. Computational experiments are worked out to reveal the efficacy of the new procedure.

Citation: Fazlollah Soleymani, Ali Akgül. European option valuation under the Bates PIDE in finance: A numerical implementation of the Gaussian scheme. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020052
References:
[1] M. L. Abell and J. P. Braselton, Differential Equations with Mathematica, Fourth Edition, Academic Press, USA, 2016.
[2]

W. AllegrettoY. Lin and N. Yan, A posteriori error analysis for FEM of American options, Discrete Contin. Dyn. Syst., Ser. B, 6 (2006), 957-978. doi: 10.3934/dcdsb.2006.6.957.

[3]

M. Balajewicz and J. Toivanen, Reduced order models for pricing European and American optionsunder stochastic volatility and jump–diffusion models, J. Comput. Sci., 20 (2017), 198-204. doi: 10.1016/j.jocs.2017.01.004.

[4]

L. V. Ballestra and C. Sgarra, The evaluation of American options in a stochastic volatility model with jumps: an efficient finite element approach, Comput. Math. Appl., 60 (2010), 1571-1590. doi: 10.1016/j.camwa.2010.06.040.

[5]

L. V. Ballestra and L. Cecere, A fast numerical method to price American options under the Bates model, Comput. Math. Appl., 72 (2016), 1305-1319. doi: 10.1016/j.camwa.2016.06.041.

[6]

D. Bates, Jumps and stochastic volatility: The exchange rate processes implicit in Deutsche mark options, Rev. Fin. Studies, 9 (1996), 69-107. doi: 10.1093/rfs/9.1.69.

[7]

V. BayonaM. Moscoso and M. Kindelan, Gaussian RBF–FD weights and its corresponding local truncation errors, Eng. Anal. Bound. Elem., 36 (2012), 1361-1369. doi: 10.1016/j.enganabound.2012.03.010.

[8]

F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637-654. doi: 10.1086/260062.

[9]

M. BrianiC. La Chioma and R. Natalini, Convergence of numerical schemes for viscosity solutions to integro–differential degenerate parabolic problems arising in financial theory, Numer. Math., 98 (2004), 607-646. doi: 10.1007/s00211-004-0530-0.

[10]

M. CaliariP. KandolfA. Ostermann and S. Rainer, Comparison of software for computing the action of the matrix exponential, BIT, 54 (2014), 113-128. doi: 10.1007/s10543-013-0446-0.

[11]

S. S. Clift and P. A. Forsyth, Numerical solution of two asset jump diffusion models for option valuation, Appl. Numer. Math., 58 (2008), 743-782. doi: 10.1016/j.apnum.2007.02.005.

[12]

Y. d'HalluinP. A. Forsyth and K. R. Vetzal, Robust numerical methods for contingent claims under jump diffusion processes, IMA J. Numer. Anal., 25 (2005), 87-112. doi: 10.1093/imanum/drh011.

[13]

E. Ekström and J. Tysk, The Black–Scholes equation in stochastic volatility models, J. Math. Anal. Appl., 368 (2010), 498-507. doi: 10.1016/j.jmaa.2010.04.014.

[14]

M. FakharanyV. N. Egorova and R. Company, Numerical valuation of two–asset options under jump diffusion models using Gauss–Hermite quadrature, J. Comput. Appl. Math., 330 (2018), 822-834. doi: 10.1016/j.cam.2017.03.032.

[15]

G. E. Fasshauer, Meshfree Approximation Methods with MATLAB, World Scientific Publishing Co., Singapore, 2007. doi: 10.1142/6437.

[16]

R. Feng and J. Duan, High accurate finite differences based on RBF interpolation and its application in solving differential equations, J. Sci. Comput., 76 (2018), 1785-1812. doi: 10.1007/s10915-018-0684-z.

[17]

P. G. Giribone and S. Ligato, Option pricing via radial basis functions: Performance comparison with traditional numerical integration scheme and parameters choice for a reliable pricing, Int. J. Financ. Eng., 2 (2015), 1550018, 30 pp. doi: 10.1142/S2424786315500188.

[18]

W. Hackbusch, Tensor Spaces and Numerical Tensor Calculus, Springer, Berlin, 2012. doi: 10.1007/978-3-642-28027-6.

[19]

R. L. Hardy, Theory and applications of the multiquadric–biharmonic method: 20 years of discovery, Comput. Math. Appl., 19 (1990), 163-208. doi: 10.1016/0898-1221(90)90272-L.

[20]

S. L. Heston, A closed–form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., 6 (1993), 327-343. doi: 10.1093/rfs/6.2.327.

[21]

N. J. Higham, Functions of Matrices: Theory and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2008. doi: 10.1137/1.9780898717778.

[22]

K. J. in 't Hout and S. Foulon, ADI finite difference schemes for option pricing in the Heston model with correlation, Int. J. Numer. Anal. Modeling, 7 (2010), 303-320.

[23]

T. Kluge, Pricing Derivatives in Stochastic Volatility Models Using the Finite Difference Method, Dipl. thesis, TU Chemnitz, 2002.

[24]

T. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), 455-500. doi: 10.1137/07070111X.

[25]

S. Kou, A jump diffusion model for option pricing, Management Sci., 48 (2002), 1086-1101.

[26]

D. C. Lesmana and S. Wang, A numerical scheme for pricing American options with transaction costs under a jump diffusion process, J. Ind. Manag. Optim., 13 (2017), 1793-1813. doi: 10.3934/jimo.2017019.

[27]

J. Loffeld and M. Tokman, Comparative performance of exponential, implicit, and explicit integrators for stiff systems of ODEs, J. Comput. Appl. Math., 241 (2013), 45-67. doi: 10.1016/j.cam.2012.09.038.

[28]

M. D. MarcozziS. Choi and C. S. Chen, On the use of boundary conditions for variational formulations arising in financial mathematics, Appl. Math. Comput., 124 (2001), 197-214. doi: 10.1016/S0096-3003(00)00087-4.

[29]

A. Mayo, Methods for the rapid solution of the pricing PIDEs in exponential and Merton models, J. Comput. Appl. Math., 222 (2008), 128-143. doi: 10.1016/j.cam.2007.10.017.

[30]

R. Merton, Option pricing when underlying stock returns are discontinuous, J. Financ. Ecom., 3 (1976), 125-144. doi: 10.1016/0304-405X(76)90022-2.

[31]

G. H. Meyer, The Time–Discrete Method of Lines for Options and Bonds, A PDE Approach, World Scientific Publishing, USA, 2015.

[32]

C. B. Moler and C. F. Van Loan, Nineteen dubious ways to compute the exponential of a matrix, twenty–five years later, SIAM Rev., 45 (2003), 3-49. doi: 10.1137/S00361445024180.

[33]

M. Mureşan, Introduction to Mathematica with Applications, Springer, Switzerland, 2017.

[34]

H.–K. Pang and H.–W. Sun, Fast exponential time integration for pricing options in stochastic volatility jump diffusion models, East Asian J. Appl. Math., 4 (2014), 52-68. doi: 10.4208/eajam.280313.061013a.

[35]

N. RambeerichD. TangmanA. Gopaul and M. Bhuruth, Exponential time integration for fast finite element solutions of some financial engineering problems, J. Comput. Appl. Math., 224 (2009), 668-678. doi: 10.1016/j.cam.2008.05.047.

[36]

S. SalmiJ. Toivanen and L. von Sydow, An IMEX–scheme for pricing options under stochastic volatility models with jumps, SIAM J. Sci. Comput., 36 (2014), 817-834. doi: 10.1137/130924905.

[37]

L. O. Scott, Pricing stock options in a jump–diffusion model with stochastic volatility and interest rates: Applications of Fourier inversion methods, Math. Finance, 7 (1997), 413-426. doi: 10.1111/1467-9965.00039.

[38]

M. Sofroniou and R. Knapp, Advanced Numerical Differential Equation Solving in Mathematica, Wolfram Mathematica, Tutorial Collection, USA, 2008.

[39]

V. Stolbunov and P. B. Nair, Sparse radial basis function approximation with spatially variable shape parameters, Appl. Math. Comput., 330 (2018), 170-184. doi: 10.1016/j.amc.2018.02.001.

[40]

D. TangmanA. Gopaul and M. Bhuruth, Exponential time integration and Chebychev discretisation schemes for fast pricing of options, Appl. Numer. Math., 58 (2008), 1309-1319. doi: 10.1016/j.apnum.2007.07.005.

[41]

I. Tolstykh, On using RBF–based differencing formulas for unstructured and mixed structured–unstructured grid calculations, Proc. 16th IMACS World Congress, 228 (2000), 4606-4624.

[42]

M. Trott, The Mathematica Guidebook for Numerics, Springer, New York, NY, USA, 2006. doi: 10.1007/0-387-28814-7.

[43]

C. Van Loan, Computing integrals involving the matrix exponential, IEEE Trans. Autom. Control, 23 (1978), 395-404. doi: 10.1109/TAC.1978.1101743.

[44]

L. von SydowJ. Toivanen and C. Zhang, Adaptive finite differences and IMEX time–stepping to price options under Bates model, Int. J. Comput. Math., 92 (2015), 2515-2529. doi: 10.1080/00207160.2015.1072173.

show all references

References:
[1] M. L. Abell and J. P. Braselton, Differential Equations with Mathematica, Fourth Edition, Academic Press, USA, 2016.
[2]

W. AllegrettoY. Lin and N. Yan, A posteriori error analysis for FEM of American options, Discrete Contin. Dyn. Syst., Ser. B, 6 (2006), 957-978. doi: 10.3934/dcdsb.2006.6.957.

[3]

M. Balajewicz and J. Toivanen, Reduced order models for pricing European and American optionsunder stochastic volatility and jump–diffusion models, J. Comput. Sci., 20 (2017), 198-204. doi: 10.1016/j.jocs.2017.01.004.

[4]

L. V. Ballestra and C. Sgarra, The evaluation of American options in a stochastic volatility model with jumps: an efficient finite element approach, Comput. Math. Appl., 60 (2010), 1571-1590. doi: 10.1016/j.camwa.2010.06.040.

[5]

L. V. Ballestra and L. Cecere, A fast numerical method to price American options under the Bates model, Comput. Math. Appl., 72 (2016), 1305-1319. doi: 10.1016/j.camwa.2016.06.041.

[6]

D. Bates, Jumps and stochastic volatility: The exchange rate processes implicit in Deutsche mark options, Rev. Fin. Studies, 9 (1996), 69-107. doi: 10.1093/rfs/9.1.69.

[7]

V. BayonaM. Moscoso and M. Kindelan, Gaussian RBF–FD weights and its corresponding local truncation errors, Eng. Anal. Bound. Elem., 36 (2012), 1361-1369. doi: 10.1016/j.enganabound.2012.03.010.

[8]

F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Econ., 81 (1973), 637-654. doi: 10.1086/260062.

[9]

M. BrianiC. La Chioma and R. Natalini, Convergence of numerical schemes for viscosity solutions to integro–differential degenerate parabolic problems arising in financial theory, Numer. Math., 98 (2004), 607-646. doi: 10.1007/s00211-004-0530-0.

[10]

M. CaliariP. KandolfA. Ostermann and S. Rainer, Comparison of software for computing the action of the matrix exponential, BIT, 54 (2014), 113-128. doi: 10.1007/s10543-013-0446-0.

[11]

S. S. Clift and P. A. Forsyth, Numerical solution of two asset jump diffusion models for option valuation, Appl. Numer. Math., 58 (2008), 743-782. doi: 10.1016/j.apnum.2007.02.005.

[12]

Y. d'HalluinP. A. Forsyth and K. R. Vetzal, Robust numerical methods for contingent claims under jump diffusion processes, IMA J. Numer. Anal., 25 (2005), 87-112. doi: 10.1093/imanum/drh011.

[13]

E. Ekström and J. Tysk, The Black–Scholes equation in stochastic volatility models, J. Math. Anal. Appl., 368 (2010), 498-507. doi: 10.1016/j.jmaa.2010.04.014.

[14]

M. FakharanyV. N. Egorova and R. Company, Numerical valuation of two–asset options under jump diffusion models using Gauss–Hermite quadrature, J. Comput. Appl. Math., 330 (2018), 822-834. doi: 10.1016/j.cam.2017.03.032.

[15]

G. E. Fasshauer, Meshfree Approximation Methods with MATLAB, World Scientific Publishing Co., Singapore, 2007. doi: 10.1142/6437.

[16]

R. Feng and J. Duan, High accurate finite differences based on RBF interpolation and its application in solving differential equations, J. Sci. Comput., 76 (2018), 1785-1812. doi: 10.1007/s10915-018-0684-z.

[17]

P. G. Giribone and S. Ligato, Option pricing via radial basis functions: Performance comparison with traditional numerical integration scheme and parameters choice for a reliable pricing, Int. J. Financ. Eng., 2 (2015), 1550018, 30 pp. doi: 10.1142/S2424786315500188.

[18]

W. Hackbusch, Tensor Spaces and Numerical Tensor Calculus, Springer, Berlin, 2012. doi: 10.1007/978-3-642-28027-6.

[19]

R. L. Hardy, Theory and applications of the multiquadric–biharmonic method: 20 years of discovery, Comput. Math. Appl., 19 (1990), 163-208. doi: 10.1016/0898-1221(90)90272-L.

[20]

S. L. Heston, A closed–form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., 6 (1993), 327-343. doi: 10.1093/rfs/6.2.327.

[21]

N. J. Higham, Functions of Matrices: Theory and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2008. doi: 10.1137/1.9780898717778.

[22]

K. J. in 't Hout and S. Foulon, ADI finite difference schemes for option pricing in the Heston model with correlation, Int. J. Numer. Anal. Modeling, 7 (2010), 303-320.

[23]

T. Kluge, Pricing Derivatives in Stochastic Volatility Models Using the Finite Difference Method, Dipl. thesis, TU Chemnitz, 2002.

[24]

T. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), 455-500. doi: 10.1137/07070111X.

[25]

S. Kou, A jump diffusion model for option pricing, Management Sci., 48 (2002), 1086-1101.

[26]

D. C. Lesmana and S. Wang, A numerical scheme for pricing American options with transaction costs under a jump diffusion process, J. Ind. Manag. Optim., 13 (2017), 1793-1813. doi: 10.3934/jimo.2017019.

[27]

J. Loffeld and M. Tokman, Comparative performance of exponential, implicit, and explicit integrators for stiff systems of ODEs, J. Comput. Appl. Math., 241 (2013), 45-67. doi: 10.1016/j.cam.2012.09.038.

[28]

M. D. MarcozziS. Choi and C. S. Chen, On the use of boundary conditions for variational formulations arising in financial mathematics, Appl. Math. Comput., 124 (2001), 197-214. doi: 10.1016/S0096-3003(00)00087-4.

[29]

A. Mayo, Methods for the rapid solution of the pricing PIDEs in exponential and Merton models, J. Comput. Appl. Math., 222 (2008), 128-143. doi: 10.1016/j.cam.2007.10.017.

[30]

R. Merton, Option pricing when underlying stock returns are discontinuous, J. Financ. Ecom., 3 (1976), 125-144. doi: 10.1016/0304-405X(76)90022-2.

[31]

G. H. Meyer, The Time–Discrete Method of Lines for Options and Bonds, A PDE Approach, World Scientific Publishing, USA, 2015.

[32]

C. B. Moler and C. F. Van Loan, Nineteen dubious ways to compute the exponential of a matrix, twenty–five years later, SIAM Rev., 45 (2003), 3-49. doi: 10.1137/S00361445024180.

[33]

M. Mureşan, Introduction to Mathematica with Applications, Springer, Switzerland, 2017.

[34]

H.–K. Pang and H.–W. Sun, Fast exponential time integration for pricing options in stochastic volatility jump diffusion models, East Asian J. Appl. Math., 4 (2014), 52-68. doi: 10.4208/eajam.280313.061013a.

[35]

N. RambeerichD. TangmanA. Gopaul and M. Bhuruth, Exponential time integration for fast finite element solutions of some financial engineering problems, J. Comput. Appl. Math., 224 (2009), 668-678. doi: 10.1016/j.cam.2008.05.047.

[36]

S. SalmiJ. Toivanen and L. von Sydow, An IMEX–scheme for pricing options under stochastic volatility models with jumps, SIAM J. Sci. Comput., 36 (2014), 817-834. doi: 10.1137/130924905.

[37]

L. O. Scott, Pricing stock options in a jump–diffusion model with stochastic volatility and interest rates: Applications of Fourier inversion methods, Math. Finance, 7 (1997), 413-426. doi: 10.1111/1467-9965.00039.

[38]

M. Sofroniou and R. Knapp, Advanced Numerical Differential Equation Solving in Mathematica, Wolfram Mathematica, Tutorial Collection, USA, 2008.

[39]

V. Stolbunov and P. B. Nair, Sparse radial basis function approximation with spatially variable shape parameters, Appl. Math. Comput., 330 (2018), 170-184. doi: 10.1016/j.amc.2018.02.001.

[40]

D. TangmanA. Gopaul and M. Bhuruth, Exponential time integration and Chebychev discretisation schemes for fast pricing of options, Appl. Numer. Math., 58 (2008), 1309-1319. doi: 10.1016/j.apnum.2007.07.005.

[41]

I. Tolstykh, On using RBF–based differencing formulas for unstructured and mixed structured–unstructured grid calculations, Proc. 16th IMACS World Congress, 228 (2000), 4606-4624.

[42]

M. Trott, The Mathematica Guidebook for Numerics, Springer, New York, NY, USA, 2006. doi: 10.1007/0-387-28814-7.

[43]

C. Van Loan, Computing integrals involving the matrix exponential, IEEE Trans. Autom. Control, 23 (1978), 395-404. doi: 10.1109/TAC.1978.1101743.

[44]

L. von SydowJ. Toivanen and C. Zhang, Adaptive finite differences and IMEX time–stepping to price options under Bates model, Int. J. Comput. Math., 92 (2015), 2515-2529. doi: 10.1080/00207160.2015.1072173.

Figure 1.  Results based on GRBF–FDI in Problem 4.1. Top left: Numerical solution. Top right: List plot of the numerical solution indicating the non–uniform nodes' distribution. Bottom left: Contour plot of the solution. Bottom right: The sparsity pattern of the system of ODEs' coefficient matrix
Figure 2.  Results based on GRBF–FDI in Problem 4.2. Top left: Numerical solution. Top right: List plot of the numerical solution indicating the non–uniform nodes' distribution. Bottom left: Contour plot of the solution. Bottom right: The sparsity pattern of the system of ODEs' coefficient matrix
Figure 3.  Results based on GRBF–FDI in Problem 4.3. Top left: Numerical solution. Top right: List plot of the numerical solution indicating the non–uniform nodes' distribution. Bottom left: Contour plot of the solution. Bottom right: The sparsity pattern of the system of ODEs' coefficient matrix
Table 1.  Numerical reports of call vanilla option pricing for Problem 4.1
Scheme $ m $ $ n $ $ N $ $ k $ $ u $ $ \varepsilon $ Time (s)
SFD–EM
20 20 400 400 8.7001 $ 1.947\times10^{-1} $ 0.91
40 25 1000 2000 8.5974 $ 2.973\times10^{-1} $ 2.67
40 40 1600 2000 8.6739 $ 2.209\times10^{-1} $ 5.39
65 45 2925 4000 8.8609 $ 3.397\times10^{-2} $ 15.77
80 50 4000 10000 8.8745 $ 2.036\times10^{-2} $ 33.95
HFM–DM
10 10 100 250 8.3465 $ 5.483\times10^{-1} $ 0.45
15 15 225 250 8.6980 $ 1.968\times10^{-1} $ 0.64
25 20 500 400 8.8601 $ 3.473\times10^{-2} $ 1.12
30 30 900 600 8.8705 $ 2.431\times10^{-2} $ 2.03
50 30 1500 2000 8.8852 $ 9.624\times10^{-3} $ 5.41
80 30 2400 5000 8.8905 $ 4.320\times10^{-3} $ 13.10
GRBF–FDI
10 10 100 NR 8.4182 $ 4.766\times10^{-1} $ 0.08
15 15 225 NR 8.7578 $ 1.370\times10^{-1} $ 0.13
25 20 500 NR 8.8513 $ 4.355\times10^{-2} $ 0.14
30 30 900 NR 8.8659 $ 2.891\times10^{-2} $ 0.18
60 30 900 NR 8.8905 $ 4.321\times10^{-3} $ 0.58
80 30 2400 NR 8.8952 $ \bf{3.305\times10^{-4}} $ 1.39
Scheme $ m $ $ n $ $ N $ $ k $ $ u $ $ \varepsilon $ Time (s)
SFD–EM
20 20 400 400 8.7001 $ 1.947\times10^{-1} $ 0.91
40 25 1000 2000 8.5974 $ 2.973\times10^{-1} $ 2.67
40 40 1600 2000 8.6739 $ 2.209\times10^{-1} $ 5.39
65 45 2925 4000 8.8609 $ 3.397\times10^{-2} $ 15.77
80 50 4000 10000 8.8745 $ 2.036\times10^{-2} $ 33.95
HFM–DM
10 10 100 250 8.3465 $ 5.483\times10^{-1} $ 0.45
15 15 225 250 8.6980 $ 1.968\times10^{-1} $ 0.64
25 20 500 400 8.8601 $ 3.473\times10^{-2} $ 1.12
30 30 900 600 8.8705 $ 2.431\times10^{-2} $ 2.03
50 30 1500 2000 8.8852 $ 9.624\times10^{-3} $ 5.41
80 30 2400 5000 8.8905 $ 4.320\times10^{-3} $ 13.10
GRBF–FDI
10 10 100 NR 8.4182 $ 4.766\times10^{-1} $ 0.08
15 15 225 NR 8.7578 $ 1.370\times10^{-1} $ 0.13
25 20 500 NR 8.8513 $ 4.355\times10^{-2} $ 0.14
30 30 900 NR 8.8659 $ 2.891\times10^{-2} $ 0.18
60 30 900 NR 8.8905 $ 4.321\times10^{-3} $ 0.58
80 30 2400 NR 8.8952 $ \bf{3.305\times10^{-4}} $ 1.39
Table 2.  Numerical reports of put vanilla option pricing for Problem 4.2
Scheme $ m $ $ n $ $ N $ $ k $ $ Re(\lambda_{\max}) $ $ \epsilon $ Time (s)
SFD–EM
20 20 400 400 -516.30 $ 2.582\times10^{-2} $ 0.15
40 25 1000 2000 -2438.80 $ 2.368\times10^{-2} $ 0.99
40 40 1600 2000 -2512.03 $ 2.109\times10^{-2} $ 1.55
65 45 2925 4000 -7041.97 $ 4.009\times10^{-3} $ 5.01
80 50 4000 10000 -10932.60 $ 2.972\times10^{-3} $ 19.43
HFM–DM
10 10 100 250 -109.47 $ 5.948\times10^{-2} $ 0.10
15 15 225 250 -298.15 $ 2.459\times10^{-2} $ 0.12
25 20 500 500 -951.58 $ 3.982\times10^{-3} $ 0.22
30 30 900 1000 -1418.65 $ 3.015\times10^{-3} $ 0.51
80 30 2400 10000 -11134.30 $ 4.722\times10^{-4} $ 11.12
GRBF–FDI
10 10 100 NR -238.16 $ 8.173\times10^{-2} $ 0.09
15 15 225 NR -639.62 $ 3.579\times10^{-2} $ 0.10
25 20 500 NR -2032.56 $ 1.210\times10^{-2} $ 0.15
30 30 900 NR -3030.1 $ 8.521\times10^{-3} $ 0.19
60 30 1800 NR -13211.4 $ 1.746\times10^{-3} $ 0.57
80 30 2400 NR -24023.6 $ \bf{5.879\times10^{-4}} $ 1.28
Scheme $ m $ $ n $ $ N $ $ k $ $ Re(\lambda_{\max}) $ $ \epsilon $ Time (s)
SFD–EM
20 20 400 400 -516.30 $ 2.582\times10^{-2} $ 0.15
40 25 1000 2000 -2438.80 $ 2.368\times10^{-2} $ 0.99
40 40 1600 2000 -2512.03 $ 2.109\times10^{-2} $ 1.55
65 45 2925 4000 -7041.97 $ 4.009\times10^{-3} $ 5.01
80 50 4000 10000 -10932.60 $ 2.972\times10^{-3} $ 19.43
HFM–DM
10 10 100 250 -109.47 $ 5.948\times10^{-2} $ 0.10
15 15 225 250 -298.15 $ 2.459\times10^{-2} $ 0.12
25 20 500 500 -951.58 $ 3.982\times10^{-3} $ 0.22
30 30 900 1000 -1418.65 $ 3.015\times10^{-3} $ 0.51
80 30 2400 10000 -11134.30 $ 4.722\times10^{-4} $ 11.12
GRBF–FDI
10 10 100 NR -238.16 $ 8.173\times10^{-2} $ 0.09
15 15 225 NR -639.62 $ 3.579\times10^{-2} $ 0.10
25 20 500 NR -2032.56 $ 1.210\times10^{-2} $ 0.15
30 30 900 NR -3030.1 $ 8.521\times10^{-3} $ 0.19
60 30 1800 NR -13211.4 $ 1.746\times10^{-3} $ 0.57
80 30 2400 NR -24023.6 $ \bf{5.879\times10^{-4}} $ 1.28
Table 3.  Parameter settings for the Bates model
Descriptions Parameters Values
Correlation between the Brownian motions $ \rho $ -0.5
Rate of interest $ r $ 0.03
Dividend yield $ q $ 0
Variance volatility $ \sigma $ 0.25
Mean reversal rate $ \kappa $ 2
Mean level of variance $ \theta $ 0.04
Price of strike $ E $ 100
Rate of jump arrival $ \lambda $ 0.2
Time to expiry $ T $ 0.5
Jump size log–variance $ \hat{\sigma} $ 0.4
Jump size log–mean $ \gamma $ -0.5
Descriptions Parameters Values
Correlation between the Brownian motions $ \rho $ -0.5
Rate of interest $ r $ 0.03
Dividend yield $ q $ 0
Variance volatility $ \sigma $ 0.25
Mean reversal rate $ \kappa $ 2
Mean level of variance $ \theta $ 0.04
Price of strike $ E $ 100
Rate of jump arrival $ \lambda $ 0.2
Time to expiry $ T $ 0.5
Jump size log–variance $ \hat{\sigma} $ 0.4
Jump size log–mean $ \gamma $ -0.5
Table 4.  Numerical reports of put option pricing in Problem 4.3
Scheme $ m $ $ n $ $ N $ $ k $ $ Re(\lambda_{\max}) $ $ \epsilon $ Time (s)
SFD–EM
10 10 100 250 -104.46 $ 4.321\times10^{-1} $ 0.10
15 15 225 500 -276.31 $ 8.815\times10^{-2} $ 0.23
25 20 500 1000 -884.68 $ 8.121\times10^{-2} $ 0.74
30 30 900 2000 -1316.01 $ 1.498\times10^{-2} $ 1.78
45 30 1350 2000 -3209.72 $ 4.817\times10^{-3} $ 3.77
60 30 1800 5000 -1316.01 $ 1.834\times10^{-3} $ 7.50
80 30 2400 5000 -10982.70 $ 1.434\times10^{-3} $ 13.92
GRBF–FDI
10 10 100 NR -158.68 $ 5.584\times10^{-2} $ 0.16
15 15 225 NR -435.54 $ 2.018\times10^{-2} $ 0.30
25 20 500 NR -1381.41 $ 6.922\times10^{-3} $ 0.89
30 30 900 NR -2054.73 $ 5.998\times10^{-3} $ 1.77
45 30 1350 NR -4874.35 $ 2.042\times10^{-3} $ 4.26
60 30 1800 NR -8904.53 $ 1.055\times10^{-3} $ 8.26
80 30 2400 NR -16165.60 $ \bf{5.754\times10^{-4}} $ 15.68
Scheme $ m $ $ n $ $ N $ $ k $ $ Re(\lambda_{\max}) $ $ \epsilon $ Time (s)
SFD–EM
10 10 100 250 -104.46 $ 4.321\times10^{-1} $ 0.10
15 15 225 500 -276.31 $ 8.815\times10^{-2} $ 0.23
25 20 500 1000 -884.68 $ 8.121\times10^{-2} $ 0.74
30 30 900 2000 -1316.01 $ 1.498\times10^{-2} $ 1.78
45 30 1350 2000 -3209.72 $ 4.817\times10^{-3} $ 3.77
60 30 1800 5000 -1316.01 $ 1.834\times10^{-3} $ 7.50
80 30 2400 5000 -10982.70 $ 1.434\times10^{-3} $ 13.92
GRBF–FDI
10 10 100 NR -158.68 $ 5.584\times10^{-2} $ 0.16
15 15 225 NR -435.54 $ 2.018\times10^{-2} $ 0.30
25 20 500 NR -1381.41 $ 6.922\times10^{-3} $ 0.89
30 30 900 NR -2054.73 $ 5.998\times10^{-3} $ 1.77
45 30 1350 NR -4874.35 $ 2.042\times10^{-3} $ 4.26
60 30 1800 NR -8904.53 $ 1.055\times10^{-3} $ 8.26
80 30 2400 NR -16165.60 $ \bf{5.754\times10^{-4}} $ 15.68
Table 5.  Numerical results of the AMG–STS method for Problem 4.3
$m$ $n$ $\varepsilon$ at $s=90$ $\varepsilon$ at $s=100$ $\varepsilon$ at $s=110$ $\epsilon$
17 9 $1.081\times10^{0}$ $1.577\times10^{0}$ $1.968\times10^{-1}$ $1.512\times10^{-1}$
33 17 $2.808\times10^{-2}$ $5.205\times10^{-1}$ $1.389\times10^{-1}$ $4.948\times10^{-2}$
65 33 $4.783\times10^{-3}$ $1.253\times10^{-1}$ $2.845\times10^{-2}$ $1.166\times10^{-2}$
129 65 $7.383\times10^{-3}$ $3.098\times10^{-2}$ $5.255\times10^{-3}$ $2.834\times10^{-3}$
257 129 $1.700\times10^{-5}$ $7.781\times10^{-3}$ $3.455\times10^{-3}$ $8.313\times10^{-4}$
$m$ $n$ $\varepsilon$ at $s=90$ $\varepsilon$ at $s=100$ $\varepsilon$ at $s=110$ $\epsilon$
17 9 $1.081\times10^{0}$ $1.577\times10^{0}$ $1.968\times10^{-1}$ $1.512\times10^{-1}$
33 17 $2.808\times10^{-2}$ $5.205\times10^{-1}$ $1.389\times10^{-1}$ $4.948\times10^{-2}$
65 33 $4.783\times10^{-3}$ $1.253\times10^{-1}$ $2.845\times10^{-2}$ $1.166\times10^{-2}$
129 65 $7.383\times10^{-3}$ $3.098\times10^{-2}$ $5.255\times10^{-3}$ $2.834\times10^{-3}$
257 129 $1.700\times10^{-5}$ $7.781\times10^{-3}$ $3.455\times10^{-3}$ $8.313\times10^{-4}$
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