Deep learning provides surrogates for computationally intensive interest rate option pricing, but practical deployment requires not only accurate prices but also reliable sensitivities and robustness to regime shifts. We presented a unified, task-driven comparison of a deep operator network (DeepONet), physics-informed neural networks (PINNs), and the deep backward stochastic differential equation (DeepBSDE) for European bond call options under the one-factor Hull–White and two-factor G2++ Gaussian affine term structure models. Using historical US Treasury term structures, we constructed smooth yield curve inputs and sampled broad option/model parameter configurations; closed-form formulas supply reference prices and analytical volatility sensitivities (Vegas). DeepONet was trained by supervised operator learning on price labels, whereas the PINN and DeepBSDE relied solely on partial differential equation (PDE) and backward stochastic differential equation (BSDE) constraints without price supervision. Although training is primarily aligned with pricing, we evaluated out-of-sample price accuracy together with automatic differentiation-based Vega accuracy and price robustness under an out-of-distribution volatility stress test. Across both models, supervised operator learning via DeepONet achieved the highest pricing accuracy on the held-out test set and the highest Vega accuracy, and exhibited the smallest degradation in pricing accuracy under the out-of-distribution volatility stress test.
Citation: Sanghyun Lee, Jeonggyu Huh, Seungwon Jeong. DeepONet-based surrogate modeling for bond option pricing[J]. AIMS Mathematics, 2026, 11(3): 5853-5896. doi: 10.3934/math.2026242
Deep learning provides surrogates for computationally intensive interest rate option pricing, but practical deployment requires not only accurate prices but also reliable sensitivities and robustness to regime shifts. We presented a unified, task-driven comparison of a deep operator network (DeepONet), physics-informed neural networks (PINNs), and the deep backward stochastic differential equation (DeepBSDE) for European bond call options under the one-factor Hull–White and two-factor G2++ Gaussian affine term structure models. Using historical US Treasury term structures, we constructed smooth yield curve inputs and sampled broad option/model parameter configurations; closed-form formulas supply reference prices and analytical volatility sensitivities (Vegas). DeepONet was trained by supervised operator learning on price labels, whereas the PINN and DeepBSDE relied solely on partial differential equation (PDE) and backward stochastic differential equation (BSDE) constraints without price supervision. Although training is primarily aligned with pricing, we evaluated out-of-sample price accuracy together with automatic differentiation-based Vega accuracy and price robustness under an out-of-distribution volatility stress test. Across both models, supervised operator learning via DeepONet achieved the highest pricing accuracy on the held-out test set and the highest Vega accuracy, and exhibited the smallest degradation in pricing accuracy under the out-of-distribution volatility stress test.
| [1] | R. Assabumrungrat, K. Minami, M. Hirano, Error analysis of option pricing via deep PDE solvers: Empirical study, In 2024 16th IIAI International Congress on Advanced Applied Informatics (IIAI-AAI), 329–336. IEEE, 2024. http://dx.doi.org/10.1109/IIAI-AAI63651.2024.00068 |
| [2] | E. Bayraktar, Q. Feng, Z. Zhang, Z. Zhang, Deep neural operator learning for probabilistic models, arXiv preprint, 2025. http://dx.doi.org/10.48550/arXiv.2511.07235 |
| [3] | D. Brigo, F. Mercurio, Interest rate models–-Theory and practice: With smile, inflation and credit, Springer, 2006. http://dx.doi.org/10.1007/978-3-540-34604-3 |
| [4] |
T. Chen, H. Chen, Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems, IEEE T. Neural Networ., 6 (1995), 911–917. http://dx.doi.org/10.1109/72.392253 doi: 10.1109/72.392253
|
| [5] |
Y. Chen, X. Lü, H. Tian, R. H. Li, Physics-informed neural network for barrier option pricing in coupled financial quantitative system with varying interest rate and volatility, Eng. Anal. Bound. Elem., 180 (2025), 106457. http://dx.doi.org/10.1016/j.enganabound.2025.106457 doi: 10.1016/j.enganabound.2025.106457
|
| [6] | Y. Cui, L. Li, G. Zhang, W. Zhang, Learning pricing maps for financial derivatives with deep operator networks, Available at SSRN 4670716, 2023. http://dx.doi.org/10.2139/ssrn.4670716 |
| [7] | R. Culkin, S. R. Das, Machine learning in finance: The case of deep learning for option pricing, J. Invest. Manag., 15 (2017), 92–100. |
| [8] |
B. Deng, Y. Shin, L. Lu, Z. Zhang, G. E. Karniadakis, Approximation rates of DeepONets for learning operators arising from advection-diffusion equations, Neural Networks, 153 (2022), 411–426. http://dx.doi.org/10.1016/j.neunet.2022.06.019 doi: 10.1016/j.neunet.2022.06.019
|
| [9] | A. Dhiman, Y. Hu, Physics informed neural network for option pricing, arXiv preprint, 2023. http://dx.doi.org/10.48550/arXiv.2312.06711 |
| [10] |
W. E, J. Han, A. Jentzen, Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, Commun. Math. Stat., 5 (2017), 349–380. http://dx.doi.org/10.1007/s40304-017-0117-6 doi: 10.1007/s40304-017-0117-6
|
| [11] | R. Ferguson, A. Green, Deeply learning derivatives, arXiv preprint, 2018. http://dx.doi.org/10.48550/arXiv.1809.02233 |
| [12] | N. Ganesan, Y. Yu, B. Hientzsch, Pricing barrier options with deepBSDEs, arXiv preprint, 2020. http://dx.doi.org/10.48550/arXiv.2005.10966 |
| [13] |
M. Germain, H. Pham, X. Warin, Approximation error analysis of some deep backward schemes for nonlinear PDEs, SIAM J. Sci. Comput., 44 (2022), A28–A56. http://dx.doi.org/10.1137/20M1355355 doi: 10.1137/20M1355355
|
| [14] |
S. Ghadimi, G. Lan, Stochastic first-and zeroth-order methods for nonconvex stochastic programming, SIAM J. Optimization, 23 (2013), 2341–2368. http://dx.doi.org/10.1137/120880811 doi: 10.1137/120880811
|
| [15] |
K. Glau, L. Wunderlich, The deep parametric PDE method and applications to option pricing, Appl. Math. Comput., 432 (2022), 127355. http://dx.doi.org/10.1016/j.amc.2022.127355 doi: 10.1016/j.amc.2022.127355
|
| [16] |
D. Hainaut, A. Casas, Option pricing in the Heston model with physics inspired neural networks, Ann. Financ., 20 (2024), 353–376. http://dx.doi.org/10.1007/s10436-024-00452-7 doi: 10.1007/s10436-024-00452-7
|
| [17] |
J. Han, A. Jentzen, W. E, Solving high-dimensional partial differential equations using deep learning, P. Natl. Acad. Sci., 115 (2018), 8505–8510. http://dx.doi.org/10.1073/pnas.1718942115 doi: 10.1073/pnas.1718942115
|
| [18] | A. Hirsa, T. Karatas, A. Oskoui, Supervised deep neural networks (DNNs) for pricing/calibration of vanilla/exotic options under various different processes, arXiv preprint, 2019. http://dx.doi.org/10.48550/arXiv.1902.05810 |
| [19] |
J. Hull, A. White, Pricing interest-rate-derivative securities, Rev. Financ. Stud., 3 (1990), 573–592. http://dx.doi.org/10.1093/rfs/3.4.573 doi: 10.1093/rfs/3.4.573
|
| [20] | J. C. Hull, S. Basu, Options, futures, and other derivatives, Pearson Education India, 2016. |
| [21] |
C. Huré, H. Pham, X. Warin, Deep backward schemes for high-dimensional nonlinear PDEs, Math. Comput., 89 (2020), 1547–1579. http://dx.doi.org/10.1090/mcom/3514 doi: 10.1090/mcom/3514
|
| [22] |
J. M. Hutchinson, A. W. Lo, T. Poggio, A nonparametric approach to pricing and hedging derivative securities via learning networks, J. Financ., 49 (1994), 851–889. http://dx.doi.org/10.1111/j.1540-6261.1994.tb00081.x doi: 10.1111/j.1540-6261.1994.tb00081.x
|
| [23] | J. Kienitz, S. K. Acar, Q. Liang, N. Nowaczyk, Deep option pricing-term structure models, Available at SSRN 3498398, 2019. http://dx.doi.org/10.2139/ssrn.3498398 |
| [24] |
S. Lanthaler, S. Mishra, G. E. Karniadakis, Error estimates for deeponets: A deep learning framework in infinite dimensions, T. Math. Appl., 6 (2022), tnac001. http://dx.doi.org/10.1093/imatrm/tnac001 doi: 10.1093/imatrm/tnac001
|
| [25] |
W. Lefebvre, G. Loeper, H. Pham, Differential learning methods for solving fully nonlinear PDEs, Digit. Financ., 5 (2023), 183–229. http://dx.doi.org/10.1007/s42521-023-00077-x doi: 10.1007/s42521-023-00077-x
|
| [26] |
R. B. Litterman, J. Scheinkman, Common factors affecting bond returns, J. Fix. Income Summer, 1 (1991), 54–61. http://dx.doi.org/10.3905/jfi.1991.692347 doi: 10.3905/jfi.1991.692347
|
| [27] |
L. Lu, P. Jin, G. Pang, Z. Zhang, G. E. Karniadakis, Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators, Nat. Mach. Intell., 3 (2021), 218–229. http://dx.doi.org/10.1038/s42256-021-00302-5 doi: 10.1038/s42256-021-00302-5
|
| [28] | K. A. Malandain, S. Kalici, H. Chakhoyan, DeepSVM: Learning stochastic volatility models with physics-informed deep operator networks, arXiv preprint, 2025. http://dx.doi.org/10.48550/arXiv.2512.07162 |
| [29] |
M. Malliaris, L. Salchenberger, A neural network model for estimating option prices, Appl. Intell., 3 (1993), 193–206. http://dx.doi.org/10.1007/BF00871937 doi: 10.1007/BF00871937
|
| [30] | B. Negyesi, C. W. Oosterlee, A deep BSDE approach for the simultaneous pricing and delta-gamma hedging of large portfolios consisting of high-dimensional multi-asset Bermudan options, arXiv preprint, 2025. http://dx.doi.org/10.48550/arXiv.2502.11706 |
| [31] | N. K. Pande, P. Pasricha, A. Kumar, A. K. Gupta, European option pricing in regime switching framework via physics-informed residual learning, Expert Syst. Appl., 2025. http://dx.doi.org/10.1016/j.eswa.2025.128226 |
| [32] |
E. Pardoux, S. Peng, Adapted solution of a backward stochastic differential equation, Syst. Control Lett., 14 (1990), 55–61. http://dx.doi.org/10.1016/0167-6911(90)90082-6 doi: 10.1016/0167-6911(90)90082-6
|
| [33] |
Y. Qiu, N. Bridges, P. Chen, Derivative-enhanced deep operator network, Adv. Neur. Inform. Process. Syst., 37 (2024), 20945–20981. https://doi.org/10.52202/079017-0660 doi: 10.52202/079017-0660
|
| [34] |
M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., 378 (2019), 686–707. http://dx.doi.org/10.1016/j.jcp.2018.10.045 doi: 10.1016/j.jcp.2018.10.045
|
| [35] |
I. Rani, C. K. Verma, G-pinns: A Bayesian-optimized gru-enhanced physics-informed neural network for advancing short-rate model predictions, Eng. Anal. Bound. Elem., 179 (2025), 106396. http://dx.doi.org/10.1016/j.enganabound.2025.106396 doi: 10.1016/j.enganabound.2025.106396
|
| [36] | S. Shalev-Shwartz, S. Ben-David, Understanding machine learning: From theory to algorithms, Cambridge University Press, 2014. http://dx.doi.org/10.1017/CBO9781107298019 |
| [37] | S. E. Shreve, Stochastic calculus for finance II: Continuous-time models, Springer, 11 (2004). http://dx.doi.org/10.1007/978-1-4757-4296-1 |
| [38] | H. Wang, H. Chen, A. Sudjianto, R. Liu, Q. Shen, Deep learning-based BSDE solver for LIBOR market model with application to Bermudan swaption pricing and hedging, arXiv preprint, 2018. http://dx.doi.org/10.48550/arXiv.1807.06622 |
| [39] |
X. Wang, J. Li, J. Li, A deep learning based numerical PDE method for option pricing, Comput. Econ., 62 (2023), 149–164. http://dx.doi.org/10.1007/s10614-022-10279-x doi: 10.1007/s10614-022-10279-x
|
| [40] | B. Yu, X. Xing, A. Sudjianto, Deep-learning based numerical BSDE method for barrier options, arXiv preprint, 2019. http://dx.doi.org/10.48550/arXiv.1904.05921 |
Figures(8) / Tables(5)
Sanghyun Lee, Jeonggyu Huh, Seungwon Jeong. DeepONet-based surrogate modeling for bond option pricing[J]. AIMS Mathematics, 2026, 11(3): 5853-5896. doi: 10.3934/math.2026242
DownLoad: