IDEAS home Printed from https://ideas.repec.org/a/eee/matcom/v241y2026ipap908-924.html

Improved constrained physics-informed neural networks (ICPINNs) to solve PDE and its application to option pricing

Author

Listed:
  • Tan, Jianguo
  • Zhang, Xingyu

Abstract

Previous deep learning methods, whether using hard or soft constraints, have often exhibited slow convergence when dealing with more than two types of boundary conditions (BCs). In general, option pricing partial differential equations (PDEs) encompass a mix of Dirichlet and non-Dirichlet BCs [1]. Hence, we propose a new method, the ICPINN method, which combines the soft-constraint and hard-constraint BCs to impose constraints based on different types of BCs for PDE. The ICPINN method is highly accurate for solving PDEs with Dirichlet and non-Dirichlet BCs. To verify the capacity of the ICPINN method, we apply it to the Black-Scholes (BS) model, the Heston option pricing PDE, and the three-asset BS model. The results show that the numerical solutions obtained by the ICPINN method can accurately converge to the exact solutions of the three financial option pricing PDEs. At the same time, the ICPINN method is more accurate. It converges faster than Physics-Informed Neural Networks (PINNs), PINNs with hard constraints (hPINNs), and boundary-safe PINNs.

Suggested Citation

  • Tan, Jianguo & Zhang, Xingyu, 2026. "Improved constrained physics-informed neural networks (ICPINNs) to solve PDE and its application to option pricing," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 241(PA), pages 908-924.
    • Handle: RePEc:eee:matcom:v:241:y:2026:i:pa:p:908-924
    DOI: 10.1016/j.matcom.2025.10.005
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378475425004227
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.matcom.2025.10.005?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Kim, Junseok & Kim, Taekkeun & Jo, Jaehyun & Choi, Yongho & Lee, Seunggyu & Hwang, Hyeongseok & Yoo, Minhyun & Jeong, Darae, 2016. "A practical finite difference method for the three-dimensional Black–Scholes equation," European Journal of Operational Research, Elsevier, vol. 252(1), pages 183-190.
    2. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    3. Hainaut, Donatien & Casas, Alex, 2024. "Option pricing in the Heston model with Physics inspired neural networks," LIDAM Discussion Papers ISBA 2024002, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    4. Leif Andersen & Jesper Andreasen, 2000. "Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Option Pricing," Review of Derivatives Research, Springer, vol. 4(3), pages 231-262, October.
    5. Donatien Hainaut & Alex Casas, 2024. "Option pricing in the Heston model with physics inspired neural networks," Annals of Finance, Springer, vol. 20(3), pages 353-376, September.
    6. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    7. Indranil SenGupta, 2014. "Option Pricing with Transaction Costs and Stochastic Interest Rate," Applied Mathematical Finance, Taylor & Francis Journals, vol. 21(5), pages 399-416, November.
    8. Xiang Wang & Jessica Li & Jichun Li, 2023. "A Deep Learning Based Numerical PDE Method for Option Pricing," Computational Economics, Springer;Society for Computational Economics, vol. 62(1), pages 149-164, June.
    9. Rama Cont & Ekaterina Voltchkova, 2005. "A Finite Difference Scheme for Option Pricing in Jump Diffusion and Exponential Lévy Models," Post-Print halshs-00445645, HAL.
    10. Jia, Wantao & Feng, Xiaotong & Hao, Mengli & Ma, Shichao, 2024. "Deep neural network method to predict the dynamical system response under random excitation of combined Gaussian and Poisson white noises," Chaos, Solitons & Fractals, Elsevier, vol. 185(C).
    11. Lishang Jiang, 2005. "Mathematical Modeling and Methods of Option Pricing," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 5855, February.
    12. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    13. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    14. Hainaut, Donatien & Casas, Alex, 2024. "Option pricing in the Heston model with physics inspired neural networks," LIDAM Reprints ISBA 2024043, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    15. Glau, Kathrin & Wunderlich, Linus, 2022. "The deep parametric PDE method and applications to option pricing," Applied Mathematics and Computation, Elsevier, vol. 432(C).
    16. Andrey Itkin & Peter Carr, 2012. "Using Pseudo-Parabolic and Fractional Equations for Option Pricing in Jump Diffusion Models," Computational Economics, Springer;Society for Computational Economics, vol. 40(1), pages 63-104, June.
    17. Margrabe, William, 1978. "The Value of an Option to Exchange One Asset for Another," Journal of Finance, American Finance Association, vol. 33(1), pages 177-186, March.
    18. Saadet Eskiizmirliler & Korhan Günel & Refet Polat, 2021. "On the Solution of the Black–Scholes Equation Using Feed-Forward Neural Networks," Computational Economics, Springer;Society for Computational Economics, vol. 58(3), pages 915-941, October.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Guo, Jingjun & Kang, Weiyi & Wang, Yubing, 2024. "Multi-perspective option price forecasting combining parametric and non-parametric pricing models with a new dynamic ensemble framework," Technological Forecasting and Social Change, Elsevier, vol. 204(C).
    2. Xun Li & Ping Lin & Xue-Cheng Tai & Jinghui Zhou, 2015. "Pricing Two-asset Options under Exponential L\'evy Model Using a Finite Element Method," Papers 1511.04950, arXiv.org.
    3. Kuldip Singh Patel & Mani Mehra, 2018. "Fourth-Order Compact Scheme For Option Pricing Under The Merton’S And Kou’S Jump-Diffusion Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(04), pages 1-26, June.
    4. Chan, Tat Lung (Ron), 2019. "Efficient computation of european option prices and their sensitivities with the complex fourier series method," The North American Journal of Economics and Finance, Elsevier, vol. 50(C).
    5. Sina Kazemian & Ghazal Farhani & Amirhessam Yazdi, 2025. "An uncertainty-aware physics-informed neural network solution for the Black-Scholes equation: a novel framework for option pricing," Papers 2511.05519, arXiv.org.
    6. Mark Broadie & Jerome B. Detemple, 2004. "ANNIVERSARY ARTICLE: Option Pricing: Valuation Models and Applications," Management Science, INFORMS, vol. 50(9), pages 1145-1177, September.
    7. Kuldip Singh Patel & Mani Mehra, 2018. "Fourth order compact scheme for option pricing under Merton and Kou jump-diffusion models," Papers 1804.07534, arXiv.org.
    8. Karel in 't Hout & Jari Toivanen, 2015. "Application of Operator Splitting Methods in Finance," Papers 1504.01022, arXiv.org.
    9. Kozarski, R., 2013. "Pricing and hedging in the VIX derivative market," Other publications TiSEM 221fefe0-241e-4914-b6bd-c, Tilburg University, School of Economics and Management.
    10. Carr, Peter & Wu, Liuren, 2004. "Time-changed Levy processes and option pricing," Journal of Financial Economics, Elsevier, vol. 71(1), pages 113-141, January.
    11. Zafar Ahmad & Reilly Browne & Rezaul Chowdhury & Rathish Das & Yushen Huang & Yimin Zhu, 2023. "Fast American Option Pricing using Nonlinear Stencils," Papers 2303.02317, arXiv.org, revised Oct 2023.
    12. Jamal Amani Rad & Kourosh Parand, 2014. "Numerical pricing of American options under two stochastic factor models with jumps using a meshless local Petrov-Galerkin method," Papers 1412.6064, arXiv.org.
    13. Wang, Libin & Liu, Lixia, 2025. "Some bivariate options pricing in a regime-switching stochastic volatility jump-diffusion model with stochastic intensity, stochastic interest and dependent jump," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 229(C), pages 468-490.
    14. Yingzi Chen & Wansheng Wang & Aiguo Xiao, 2019. "An Efficient Algorithm for Options Under Merton’s Jump-Diffusion Model on Nonuniform Grids," Computational Economics, Springer;Society for Computational Economics, vol. 53(4), pages 1565-1591, April.
    15. Liming Feng & Vadim Linetsky, 2008. "Pricing Options in Jump-Diffusion Models: An Extrapolation Approach," Operations Research, INFORMS, vol. 56(2), pages 304-325, April.
    16. Maya Briani & Lucia Caramellino & Giulia Terenzi & Antonino Zanette, 2016. "Numerical stability of a hybrid method for pricing options," Papers 1603.07225, arXiv.org, revised Dec 2019.
    17. Leif Andersen & Alexander Lipton, 2012. "Asymptotics for Exponential Levy Processes and their Volatility Smile: Survey and New Results," Papers 1206.6787, arXiv.org.
    18. Kuldip Singh Patel & Mani Mehra, 2018. "Compact finite difference method for pricing European and American options under jump-diffusion models," Papers 1804.09043, arXiv.org.
    19. Ravi Kashyap, 2022. "Options as Silver Bullets: Valuation of Term Loans, Inventory Management, Emissions Trading and Insurance Risk Mitigation using Option Theory," Annals of Operations Research, Springer, vol. 315(2), pages 1175-1215, August.
    20. Bilel Jarraya & Abdelfettah Bouri, 2013. "A Theoretical Assessment on Optimal Asset Allocations in Insurance Industry," International Journal of Finance & Banking Studies, Center for the Strategic Studies in Business and Finance, vol. 2(4), pages 30-44, October.

    More about this item

    Keywords

    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:matcom:v:241:y:2026:i:pa:p:908-924. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/mathematics-and-computers-in-simulation/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.