Barycentric rational collocation-based IMEX-BDF2 scheme for option pricing under jump-diffusion models

Wei Zhang; Hu Li; Wei Zhang; Hu Li

doi:10.3934/era.2026316

Electronic Research Archive

2026, Volume 34, Issue 10: 7299-7329. doi: 10.3934/era.2026316

Previous Article Next Article

Research article

Barycentric rational collocation-based IMEX-BDF2 scheme for option pricing under jump-diffusion models

Wei Zhang ^{1
,
,},
Hu Li ²

1.
School of Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, Sichuan, China
2.
School of Mathematics, Chengdu Normal University, Chengdu 611130, Sichuan, China

Received: 09 May 2026 Revised: 30 July 2026 Accepted: 31 July 2026 Published: 26 August 2026

Jump-diffusion models, such as Merton's and Kou's models, are essential in option pricing for capturing the discontinuous jump characteristics of asset prices. However, the nonlocal integral term in the resulting partial integro-differential equation (PIDE) presents substantial numerical challenges. This paper proposes an efficient and high-order numerical framework for pricing European and American options under these models. We employ the barycentric rational interpolation (BRI) collocation method for spatial discretization to achieve high-order accuracy and ensure numerical stability. For temporal discretization, the second-order implicit–explicit backward differentiation formula (IMEX-BDF2) is utilized, and the nonlocal integral term is evaluated via the Gauss–Legendre quadrature. To handle the early exercise constraint of American options, the operator splitting method is integrated into the framework. A rigorous error estimation and convergence analysis are provided to establish the mathematical reliability of the scheme. Numerical experiments demonstrate that the proposed method is highly efficient, providing accurate estimates for option values and stable numerical approximations for their Greeks (delta and gamma), validating its effectiveness for solving PIDEs in financial engineering.
- jump-diffusion models,
- barycentric rational interpolation,
- IMEX-BDF2 scheme,
- operator splitting method,
- Gauss–Legendre quadrature,
- option pricing
Citation: Wei Zhang, Hu Li. Barycentric rational collocation-based IMEX-BDF2 scheme for option pricing under jump-diffusion models[J]. Electronic Research Archive, 2026, 34(10): 7299-7329. doi: 10.3934/era.2026316

Related Papers:

Abstract

Jump-diffusion models, such as Merton's and Kou's models, are essential in option pricing for capturing the discontinuous jump characteristics of asset prices. However, the nonlocal integral term in the resulting partial integro-differential equation (PIDE) presents substantial numerical challenges. This paper proposes an efficient and high-order numerical framework for pricing European and American options under these models. We employ the barycentric rational interpolation (BRI) collocation method for spatial discretization to achieve high-order accuracy and ensure numerical stability. For temporal discretization, the second-order implicit–explicit backward differentiation formula (IMEX-BDF2) is utilized, and the nonlocal integral term is evaluated via the Gauss–Legendre quadrature. To handle the early exercise constraint of American options, the operator splitting method is integrated into the framework. A rigorous error estimation and convergence analysis are provided to establish the mathematical reliability of the scheme. Numerical experiments demonstrate that the proposed method is highly efficient, providing accurate estimates for option values and stable numerical approximations for their Greeks (delta and gamma), validating its effectiveness for solving PIDEs in financial engineering.

References

[1]	R. C. Merton, Option pricing when underlying stock returns are discontinuous, J. Financ. Econ., 3 (1976), 125–144. https://doi.org/10.1016/0304-405X(76)90022-2 doi: 10.1016/0304-405X(76)90022-2
[2]	S. G. Kou, A jump-diffusion model for option pricing, Manage. Sci., 48 (2002), 1086–1101.
[3]	K. Zhang, S. Wang, Pricing options under jump diffusion processes with fitted finite volume method, Appl. Math. Comput., 201 (2008), 398–413. https://doi.org/10.1016/j.amc.2007.12.043 doi: 10.1016/j.amc.2007.12.043
[4]	W. S. Wang, M. L. Mao, X. Jiang, L. H. Wang, High order semi-imex BDF schemes for nonlinear partial integro-differential equations arising in finance, ESAIM: M2AN, 59 (2025), 643–670. https://doi.org/10.1051/m2an/2025003 doi: 10.1051/m2an/2025003
[5]	R. Company, V. N. Egorova, L. Jódar, A front-fixing etd numerical method for solving jump-diffusion American option pricing problems, Math. Comput. Simul., 189 (2021), 69–84. https://doi.org/10.1016/j.matcom.2020.07.015 doi: 10.1016/j.matcom.2020.07.015
[6]	W. S. Wang, M. Mao, Z. Wang, An efficient variable step-size method for options pricing under jump-diffusion models with nonsmooth payoff function, ESAIM: M2AN, 55 (2021), 913–938. https://doi.org/10.1051/m2an/2021012 doi: 10.1051/m2an/2021012
[7]	W. S. Wang, Y. Z. Chen, H. Fang, On the variable two-step IMEX BDF method for parabolic integro-differential equations with nonsmooth initial data arising in finance, SIAM J. Numer. Anal., 57 (2019), 1289–1317. https://doi.org/10.1137/18M1194328 doi: 10.1137/18M1194328
[8]	C. C. Christara, N. C. H. Leung, Option pricing in jump diffusion models with quadratic spline collocation, Appl. Math. Comput., 279 (2016), 28–42. https://doi.org/10.1016/j.amc.2015.12.045 doi: 10.1016/j.amc.2015.12.045
[9]	M. Shirzadi, M. Dehghan, A. F. Bastani, Optimal uniform error estimates for moving least-squares collocation with application to option pricing under jump-diffusion processes, Numer. Methods Partial Differ. Equations, 37 (2021), 98–117. https://doi.org/10.1002/num.22520 doi: 10.1002/num.22520
[10]	R. Cont, E. Voltchkova, A finite difference scheme for option pricing in jump diffusion and exponential Lévy models, SIAM J. Numer. Anal., 43 (2005), 1596–1626.
[11]	S. Salmia, J. Toivanen, An iterative method for pricing American options under jump-diffusion models, Appl. Numer. Math., 61 (2011), 821–831. https://doi.org/10.1016/j.apnum.2011.02.002 doi: 10.1016/j.apnum.2011.02.002
[12]	A. Ebrahimijahan, M. Dehghan, M. Abbaszadeh, A reduced-order model based on integrated radial basis functions with partition of unity method for option pricing under jump–diffusion models, Eng. Anal. Boundary Elem., 155 (2023), 48–61. https://doi.org/10.1016/j.enganabound.2023.05.032 doi: 10.1016/j.enganabound.2023.05.032
[13]	M. Haghi, R. Mollapourasl, M. Vanmaele, An RBF-FD method for pricing American options under jump-diffusion models, Comput. Math. Appl., 76 (2018), 2434–2459. https://doi.org/10.1016/j.camwa.2018.08.040 doi: 10.1016/j.camwa.2018.08.040
[14]	M. K. Kadalbajoo, L. P. Tripathi, A. Kumar, Second order accurate IMEX methods for option pricing under Merton and Kou jump-diffusion models, J. Sci. Comput., 65 (2015), 979–1024. https://doi.org/10.1007/s10915-015-0001-z doi: 10.1007/s10915-015-0001-z
[15]	Y. Z. Chen, W. S. Wang, A. G. Xiao, An efficient algorithm for options under Merton's jump-diffusion model on nonuniform grids, Comput. Econ., 53 (2019), 1565–1591. https://doi.org/10.1007/s10614-018-9823-8 doi: 10.1007/s10614-018-9823-8
[16]	J. Toivanen, Numerical valuation of European and American options under Kou's jump-diffusion model, SIAM J. Sci. Comput., 30 (2008), 1949–1970. https://doi.org/10.1137/060674697 doi: 10.1137/060674697
[17]	L. Boen, K. J. In't Hout, Operator splitting schemes for American options under the two-asset Merton jump-diffusion model, Appl. Numer. Math., 153 (2020), 114–131. https://doi.org/10.1016/j.apnum.2020.02.004 doi: 10.1016/j.apnum.2020.02.004
[18]	S. Ikonen, J. Toivanen, Operator splitting methods for pricing American put options under stochastic volatility, Numer. Math., 113 (2009), 299–324. https://doi.org/10.1007/s00211-009-0227-5 doi: 10.1007/s00211-009-0227-5
[19]	V. Maurya, A. Singh, V. S. Yadav, M. K. Rajpoot, Efficient pricing of options in jump-diffusion models: Novel implicit-explicit methods for numerical valuation, Math. Comput. Simul., 217 (2024), 202–225. https://doi.org/10.1016/j.matcom.2023.10.025 doi: 10.1016/j.matcom.2023.10.025
[20]	P. A. Forsyth, K. R. Vetzal, Quadratic convergence for valuing American options using a penalty method, SIAM J. Sci. Comput., 23 (2002), 2095–2122. https://doi.org/10.1137/S1064827500382324 doi: 10.1137/S1064827500382324
[21]	R. Zvan, P. A. Forsyth, K. R. Vetzal, Penalty methods for American options with stochastic volatility, J. Comput. Appl. Math., 91 (1998), 199–218. https://doi.org/10.1016/S0377-0427(98)00037-5 doi: 10.1016/S0377-0427(98)00037-5
[22]	J. P. Berrut, R. Baltensperger, The linear rational pseudospectral method for boundary value problems, BIT Numer. Math., 41 (2001), 868–879. https://doi.org/10.1023/A:1021916623407 doi: 10.1023/A:1021916623407
[23]	R. Baltensperger, J. P. Berrut, The linear rational collocation method, J. Comput. Appl. Math., 134 (2001), 243–258. https://doi.org/10.1016/S0377-0427(00)00552-5 doi: 10.1016/S0377-0427(00)00552-5
[24]	J. Li, J. Z. Qu, Barycentric lagrange interpolation collocation method for solving the Sine-Gordon equation, Wave Motion, 120 (2023), 103159. https://doi.org/10.1016/j.wavemoti.2023.103159 doi: 10.1016/j.wavemoti.2023.103159
[25]	R. Jiwari, Barycentric rational interpolation and local radial basis functions based numerical algorithms for multidimensional Sine-Gordon equation, Numer. Methods Partial Differ. Equations, 37 (2021), 1965–1992. https://doi.org/10.1002/num.22636 doi: 10.1002/num.22636
[26]	Y. F. Deng, Z. F. Weng, Barycentric interpolation collocation method based on Crank-Nicolson scheme for the Allen-Cahn equation, AIMS Math., 6 (2021), 3857–3873. https://doi.org/10.3934/math.2021229 doi: 10.3934/math.2021229
[27]	O. Oruç, Two meshless methods based on local radial basis function and barycentric rational interpolation for solving 2D viscoelastic wave equation, Comput. Math. Appl., 79 (2020), 3272–3288. https://doi.org/10.1016/j.camwa.2020.01.025 doi: 10.1016/j.camwa.2020.01.025
[28]	T. Liu, H. Y. Liu, Y. Y. Ma, Approximate solution of a kind of time-fractional evolution equations based on fast L1 formula and barycentric Lagrange interpolation, Fractal Fract., 8 (2024), 48. https://doi.org/10.3390/fractalfract8110675 doi: 10.3390/fractalfract8110675
[29]	A. Almendral, C. W. Oosterlee, Numerical valuation of options with jumps in the underlying, Appl. Numer. Math., 53 (2005), 1–18. https://doi.org/10.1016/j.apnum.2004.08.037 doi: 10.1016/j.apnum.2004.08.037
[30]	M. S. Floater, K. Hormann, Barycentric rational interpolation with no poles and high rates of approximation, Numer. Math., 107 (2007), 315–331. https://doi.org/10.1007/s00211-007-0093-y doi: 10.1007/s00211-007-0093-y
[31]	G. Klein, J. P. Berrut, Linear rational finite differences from derivatives of barycentric rational interpolants, SIAM J. Numer. Anal., 50 (2012), 643–656. https://www.jstor.org/stable/41582753
[32]	J. P. Berrut, M. S. Floater, G. Klein, Convergence rates of derivatives of a family of barycentric rational interpolants, Appl. Numer. Math., 61 (2011), 989–1000. https://doi.org/10.1016/j.apnum.2011.05.001 doi: 10.1016/j.apnum.2011.05.001
[33]	M. Costabile, A. Leccadito, I. Massabó, E. Russo, Option pricing under regime-switching jump-diffusion models, J. Comput. Appl. Math., 256 (2014), 152–167. https://doi.org/10.1016/j.cam.2013.07.046 doi: 10.1016/j.cam.2013.07.046
[34]	R. Yadav, D. K. Yadav, A. Kumar, RBF based some implicit-explicit finite difference schemes for pricing option under extended jump-diffusion model, Eng. Anal. Boundary Elem., 156 (2023), 392–406. https://doi.org/10.1016/j.enganabound.2023.08.021 doi: 10.1016/j.enganabound.2023.08.021
[35]	M. K. Kadalbajoo, A. Kumar, L. P. Tripathi, A radial basis function based implicit-explicit method for option pricing under jump-diffusion models, Appl. Numer. Math., 110 (2016), 159–173. https://doi.org/10.1016/j.apnum.2016.08.006 doi: 10.1016/j.apnum.2016.08.006
[36]	J. G. Heywood, R. Rannacher, Finite-element approximation of the nonstationary Navier–Stokes problem part Ⅳ: error analysis for second-order time discretization, SIAM J. Numer. Anal., 27 (1990), 353–384.
[37]	J. Shen, On error estimates of the projection methods for the Navier-Stokes equations: second-order schemes, Math. Comput., 65 (1996), 1039–1066.
[38]	C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Berlin, Germany: Springer, 2006. https://doi.org/10.1007/978-3-540-30726-6
[39]	S. Ikonen, J. Toivanen, Operator splitting methods for American option pricing, Appl. Math. Lett., 17 (2004), 809–814. https://doi.org/10.1016/j.aml.2004.06.010 doi: 10.1016/j.aml.2004.06.010
[40]	Y. Kwon, Y. Lee, A second-order finite difference method for option pricing under jump-diffusion models, SIAM J. Numer. Anal., 49 (2011), 2598–2617. https://doi.org/10.1137/090777529 doi: 10.1137/090777529
[41]	Y. Kwon, Y. Lee, A second-order tridiagonal method for American options under jump-diffusion models, SIAM J. Sci. Comput., 33 (2011), 1860–1872. https://doi.org/10.1137/100806552 doi: 10.1137/100806552
[42]	J. Toivanen, A high-order front-tracking finite difference method for pricing American options under jump-diffusion models, J. Comput. Finance, 13 (2010), 61–79. https://doi.org/10.21314/JCF.2010.222 doi: 10.21314/JCF.2010.222

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)