Analysis and Numerical Approximation of a Rainfall-Derivative Pricing Equation

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-24-2026-299
Published: Sep 21, 2026
Cite as:
Tareq Saeed, Javed Hussain, Ashraf M. Zenkour, Abdullah O. Alshammari, Analysis and Numerical Approximation of a Rainfall-Derivative Pricing Equation, Int. J. Anal. Appl., 24 (2026), 299.

Main Article Content

Tareq Saeed, Javed Hussain, Ashraf M. Zenkour, Abdullah O. Alshammari

Abstract

Rainfall derivatives are useful when precipitation risk affects agricultural production, water management, or other climate-sensitive activities, but their pricing differs from the pricing of traded financial assets. We study a rainfall-index contract whose state variables are the current rainfall intensity and an accumulated deficit index. The rainfall intensity follows a seasonally forced Ornstein–Uhlenbeck model, and the index records the time spent below a reference rainfall level. After choosing an equivalent pricing measure through a constant market price of rainfall risk, we obtain a backward valuation equation with diffusion in the rainfall variable and one-sided transport in the index variable. The paper analyses this parabolic–transport problem and then uses the analysis to guide computation. We prove an energy estimate, continuous dependence on the data, well-posedness, comparison, positivity, and monotonicity in the cumulative index. We also construct an explicit finite-difference benchmark with an upwind treatment of the transported variable and compare it with a residual-based parametric approximation. The error estimate relates the approximation error to the PDE residual and the boundary and initial mismatches. The calibrated examples use seven Saudi rainfall regimes. Across these cases the residual-based approximation remains close to the finite-difference benchmark, with relative \(L^2\)-errors below \(9\times10^{-2}\). The sensitivity results are not uniform across regions. Arid interior cities show weak volatility exposure, Makkah and Medina are more responsive to mean reversion and volatility, and Abha has negative sensitivities under the put payoff convention. These differences support regime-specific calibration rather than a single national rainfall parameter set.

Article Details

References

  1. 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. https://doi.org/10.1016/j.jcp.2018.10.045.
  2. L. Yang, X. Meng, G.E. Karniadakis, B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data, J. Comput. Phys. 425 (2021), 109913. https://doi.org/10.1016/j.jcp.2020.109913.
  3. A. Dhiman, Y. Hu, Physics Informed Neural Network for Option Pricing, arXiv:2312.06711, 2023. https://doi.org/10.48550/arXiv.2312.06711.
  4. D. Hainaut, A. Casas, Option Pricing in the Heston Model with Physics Inspired Neural Networks, Ann. Financ. 20 (2024), 353–376. https://doi.org/10.1007/s10436-024-00452-7.
  5. D. Hainaut, Valuation of Guaranteed Minimum Accumulation Benefits (GMABs) with Physics-Inspired Neural Networks, Ann. Actuar. Sci. 18 (2024), 442–473. https://doi.org/10.1017/s1748499524000095.
  6. K.M. Graczyk, K. Witkowski, Bayesian Reasoning for Physics Informed Neural Networks, arXiv:2308.13222, 2023. https://doi.org/10.48550/arXiv.2308.13222.
  7. Y. Ding, S. Chen, H. Miyake, X. Li, Physics-Informed Neural Networks with Fourier Features for Seismic Wavefield Simulation in Time-Domain Nonsmooth Complex Media, arXiv:2409.03536, 2024. https://doi.org/10.48550/arXiv.2409.03536.
  8. Y. Hou, X. Li, J. Wu, Y.-G. Wang, Enhanced BPINN Training Convergence in Solving General and Multi-scale Elliptic PDEs with Noise, arXiv:2408.09340, 2024. https://doi.org/10.48550/arXiv.2408.09340.
  9. D.d.S. Santos, T.A.E. Ferreira, Neural Network Learning of Black-Scholes Equation for Option Pricing, arXiv:2405.05780, 2024. https://doi.org/10.48550/arXiv.2405.05780.
  10. X. Wang, J. Li, J. Li, A Deep Learning Based Numerical PDE Method for Option Pricing, Comput. Econ. 62 (2023), 149–164. https://doi.org/10.1007/s10614-022-10279-x.
  11. F. Gatta, V.S. Di Cola, F. Giampaolo, F. Piccialli, S. Cuomo, Meshless Methods for American Option Pricing Through Physics-Informed Neural Networks, Eng. Anal. Bound. Elem. 151 (2023), 68–82. https://doi.org/10.1016/j.enganabound.2023.02.040.
  12. A.Q. Ibrahim, S. Götschel, D. Ruprecht, Parareal with a Physics-Informed Neural Network as Coarse Propagator, in: Euro-Par 2023: Parallel Processing, Lecture Notes in Computer Science, Springer Nature Switzerland, Cham, 2023, pp. 649–663. https://doi.org/10.1007/978-3-031-39698-4_44.
  13. Y. Bai, T. Chaolu, S. Bilige, The Application of Improved Physics-Informed Neural Network (IPINN) Method in Finance, Nonlinear Dyn. 107 (2022), 3655–3667. https://doi.org/10.1007/s11071-021-07146-z.
  14. P. Li, Pricing Weather Derivatives with Partial Differential Equations of the Ornstein–Uhlenbeck Process, Comput. Math. Appl. 75 (2018), 1044–1059. https://doi.org/10.1016/j.camwa.2017.10.030.
  15. W. Tang, S. Chang, A Semi-Lagrangian Method for the Weather Options of Mean-Reverting Brownian Motion with Jump–Diffusion, Comput. Math. Appl. 71 (2016), 1045–1058. https://doi.org/10.1016/j.camwa.2015.12.040.
  16. C. Nhangumbe, E. Sousa, Numerical Solutions of an Option Pricing Rainfall Weather Derivatives Model, Comput. Math. Appl. 153 (2024), 43–55. https://doi.org/10.1016/j.camwa.2023.11.011.
  17. C. Harris, The Valuation of Weather Derivatives Using Partial Differential Equations, MSc Dissertation, University of Reading, 2003.
  18. H. Hamisultane, Which Method for Pricing Weather Derivatives?, (2008). https://shs.hal.science/halshs-00355856v1.
  19. E. Broni-Mensah, Numerical Solutions of Weather Derivatives and Other Incomplete Market Problems, PhD Thesis, The University of Manchester, 2012.
  20. P. Li, The Valuation of Weather Derivatives Using One-Sided Crank–Nicolson Schemes, Comput. Econ. 58 (2021), 825–847. https://doi.org/10.1007/s10614-020-10052-y.
  21. S. Bansal, P. Boro, S. Natesan, Physics-Informed Neural Network for Option Pricing Weather Derivatives Model, Comput. Math. Appl. 200 (2025), 1–21. https://doi.org/10.1016/j.camwa.2025.09.001.
  22. F. Black, M. Scholes, The Pricing of Options and Corporate Liabilities, J. Polit. Econ. 81 (1973), 637–654. https://doi.org/10.1086/260062.
  23. J.S. Butler, B. Schachter, Unbiased Estimation of the Black/Scholes Formula, J. Financ. Econ. 15 (1986), 341–357. https://doi.org/10.1016/0304-405X(86)90025-5.
  24. J.D. Macbeth, L.J. Merville, An Empirical Examination of the Black-Scholes Call Option Pricing Model, J. Finance 34 (1979), 1173–1186. https://doi.org/10.2307/2327242.
  25. J.P. Villarino, A. Leitao, J.A. García Rodríguez, Boundary-Safe PINNs Extension: Application to Non-Linear Parabolic PDEs in Counterparty Credit Risk, J. Comput. Appl. Math. 425 (2023), 115041. https://doi.org/10.1016/j.cam.2022.115041.
  26. L. Zeng, Weather Derivatives and Weather Insurance: Concept, Application, and Analysis, Bull. Am. Meteorol. Soc. 81 (2000), 2075–2082. https://doi.org/10.1175/1520-0477(2000)081<2075:WDAWIC>2.3.CO;2.
  27. F.E. Benth, J. Šaltytė-Benth, Modeling and Pricing in Financial Markets for Weather Derivatives, World Scientific, 2012. https://doi.org/10.1142/8457.
  28. R. Carmona, P. Diko, Pricing Precipitation Based Derivatives, Int. J. Theor. Appl. Finance 8 (2005), 959–988. https://doi.org/10.1142/S0219024905003311.
  29. M. Hess, On the Pricing and Hedging of Precipitation Derivatives, Probab. Uncertain. Quant. Risk 9 (2024), 499–528. https://doi.org/10.3934/puqr.2024021.
  30. M. Baljon, S.K. Sharma, Rainfall Prediction Rate in Saudi Arabia Using Improved Machine Learning Techniques, Water 15 (2023), 826. https://doi.org/10.3390/w15040826.
  31. M. Almazroui, M.N. Islam, P.D. Jones, H. Athar, M.A. Rahman, Recent Climate Change in the Arabian Peninsula: Seasonal Rainfall and Temperature Climatology of Saudi Arabia for 1979–2009, Atmos. Res. 111 (2012), 29–45. https://doi.org/10.1016/j.atmosres.2012.02.013.