An Interest Rate Model for Uncertain-Stochastic Financial Markets

Over the past decades, financial markets have increasingly exhibited features of both randomness and uncertainty, creating challenges for interest rate models that rely solely on stochastic or uncertain processes. These models often fail to adequately capture the dual nature of indeterminacy, limiting their relevance in volatile and unpredictable market conditions. This study aims to design and assess an interest rate model for uncertain-stochastic financial markets and to derive a framework for zero-coupon bond pricing under this setting. The methodology applies uncertain stochastic differential equations, which integrate elements of both probability theory and uncertainty theory, thereby accommodating aleatory and epistemic forms of indeterminacy. The proposed model extends the classical short-rate frameworks by introducing two sources of indeterminacy and provides theoretical derivations for bond pricing. Numerical illustrations are included to demonstrate the application of the model to zero-coupon bond valuation and to highlight differences from conventional approaches. The findings indicate that interest rates and zero-coupon bond prices in uncertain stochastic financial markets can be effectively modeled through uncertain random processes, leading to improved pricing accuracy and risk management in environments characterised by incomplete information and unpredictable shocks. The key conclusion is that incorporating uncertain stochastic differential equations into the interest rate and zero-coupon bonds’ prices modelling offers a more robust and flexible framework for uncertain stochastic markets. This study contributes to the growing body of uncertain stochastic finance by underscoring the need for hybrid models capable of guiding policymakers, investors and financial institutions in ensuring stability and resilience under future market uncertainties.

1. Chen Y., Li S., Kang R. Epistemic uncertainty quantification via uncertainty theory in the reliability evaluation of a system with failure Trigger effect. Reliability Engineering and System Safety. 2021;215:107896. URL: https://doi.org/10.1016/j.ress.2021.107896

2. Li X.-Y., Tao Z., Wu J.-P., Zhang W. Uncertainty theory based reliability modeling for fatigue. Engineering Failure Analysis. 2021;119:104931. URL: https://doi.org/10.1016/j.engfailanal.2020.104931

3. Kolmogoroff A. Grundbegriffe der wahrscheinlichkeitsrechnung. 1933. URL: https://doi.org/10.1007/978–3–642–49888–6

4. Liu B. Uncertainty Theory (2nd Ed.) Berlin: Springer-Verlag; 2007.

5. Liu B. Some research problems in uncertainty theory. Journal of Uncertain systems. 2009;3(1):3–10.

6. Matenda F. R., Chikodza E. A stock model with jumps for Itô-Liu financial markets. Soft Computing. 2019;23:4065–80. URL: https://doi.org/10.1007/s00500–018–3054–8

7. Li G., He G., Zheng M., Zheng A. Uncertain sensor-weapon-target allocation problem based on uncertainty theory. Symmetry. 2023;15(1):176. URL: https://doi.org/10.3390/sym15010176

8. Huang H., Ning Y. Risk-neutral pricing method of options based on uncertainty theory. Symmetry. 2021;13(12):2285. URL: https://doi.org/10.3390/sym13122285

9. Şanlıbaba İ. Uncredibility with fuzzy variables in uncertainty and its use in decision support in making sense of heart sounds. Applied Soft Computing. 2025;169:112641. URL: https://doi.org/10.1016/j.asoc.2024.112641

10. Yang T., Huang X. Two new mean-variance enhanced index tracking models based on uncertainty theory. The North American Journal of Economics and Finance. 2022;59:101622. URL: https://doi.org/10.1016/j.najef.2021.101622

11. Craigmile P., Herbei R., Liu G., Schneider G. Statistical inference for stochastic differential equations. Wiley Interdisciplinary Reviews: Computational Statistics. 2023;15(2): e1585. URL: https://doi.org/10.1002/wics.1585

12. Chirima J., Matenda F. R., Chikodza E., Sibanda M. Dynamic programming principle for optimal control of uncertain random differential equations and its application to optimal portfolio selection. Review of Business and Economics Studies. 2024;12(3):74–85. URL: https://doi.org/10.26794/2308–944X-2024–12–3–74–85

13. Gupta S., Jayannavar A. M. Stochastic resetting: A (very) brief review. Frontiers in Physics. 2022;10:789097. URL: https://doi.org/10.3389/fphy.2022.789097

14. Huang Y., Mabrouk Y., Gompper G., Sabass B. Sparse inference and active learning of stochastic differential equations from data. Scientific Reports. 2022;12(1):21691. URL: https://doi.org/10.1038/s41598–022–25638–9

15. Black F., Scholes M. The pricing of options and corporate liabilities. Journal of Political Economy. 1973;81(3):637–54. URL: https://doi.org/10.1086/260062

16. Merton R. C. Theory of rational option pricing. The Bell Journal of Economics and Management Science. 1973:141–83. URL: https://doi.org/10.2307/3003143

17. Vasicek O. An equilibrium characterization of the term structure. Journal of Financial Economics. 1977;5(2):177–88. URL: https://doi.org/10.1016/0304–405X(77)90016–2

18. Ho T. S., Lee S. B. Term structure movements and pricing interest rate contingent claims. The Journal of Finance. 1986;41(5):1011–29. URL: https://doi.org/10.1111/j.1540–6261.1986.tb02528.x

19. Hull J., White A. Pricing interest-rate-derivative securities. The Review of Financial Studies. 1990;3(4):573–92. URL: https://doi.org/10.1093/rfs/3.4.573

20. Cox J. C., Ingersoll Jr J. E., Ross S. A. An intertemporal general equilibrium model of asset prices. Econometrica: Journal of the Econometric Society. 1985:363–84. URL: https://doi.org/10.2307/1911241

21. Sun Y., Liu S. Interest‐Rate Products Pricing Problems with Uncertain Jump Processes. Discrete Dynamics in Nature and Society. 2021;2021(1):7398770. URL: https://doi.org/10.1155/2021/7398770

22. Yu S., Ning Y. An interest-rate model with jumps for uncertain financial markets. Physica A: Statistical Mechanics and its Applications. 2019;527:121424. URL: https://doi.org/10.1016/j.physa.2019.121424

23. Zhou J., Jiang Y., Pantelous A. A., Dai W. A systematic review of uncertainty theory with the use of scientometrical method. Fuzzy Optimization and Decision Making. 2023;22(3):463–518. URL: https://doi.org/10.1007/s10700–022–09400–4

24. Ye T., Liu B. Uncertain hypothesis test for uncertain differential equations. Fuzzy Optimization and Decision Making. 2023;22(2):195–211. URL: https://doi.org/10.1007/s10700–022–09389-w

25. Liu Z. Option Pricing Formulas in a New Uncertain Mean‐Reverting Stock Model with Floating Interest Rate. Discrete Dynamics in Nature and Society. 2020;2020(1):3764589. URL: https://doi.org/10.1155/2020/3764589

26. Hassanzadeh S., Mehrdoust F. Valuation of European option under uncertain volatility model. Soft Computing. 2018;22:4153–63. URL: https://doi.org/10.1007/s00500–017–2633–4

27. Sun Y., Su T. Mean-reverting stock model with floating interest rate in uncertain environment. Fuzzy Optimization and Decision Making. 2017;16:235–55. URL: https://doi.org/10.1007/s10700–016–9247–7

28. Li X., Xiao C., Chen X., Liu Y. Uncertain generalized mean reversion interest rate risk model with applications to financial instruments. Journal of Industrial and Management Optimization. 2025;21(4):2816–33. URL: https://doi.org/10.3934/jimo.2024195

29. Yang X., Ke H. Uncertain interest rate model for Shanghai interbank offered rate and pricing of American swaption. Fuzzy Optimization and Decision Making. 2023;22(3):447–62. URL: https://doi.org/10.1007/s10700–022–09399–8

30. Liu Y., Jing H., Ye T. A new uncertain interest rate model with application to Hibor. Symmetry. 2022;14(7):1344. URL: https://doi.org/10.3390/sym14071344

31. Chen X., Gao J. Uncertain term structure model of interest rate. Soft Computing. 2013;17:597–604. URL: https://doi.org/10.1007/s00500–012–0927–0

32. Zhang Z., Ralescu D. A., Liu W. Valuation of interest rate ceiling and floor in uncertain financial market. Fuzzy Optimization and Decision Making. 2016;15:139–54. URL: https://doi.org/10.1007/s10700–015–9223–7

33. Zhu Y. Uncertain fractional differential equations and an interest rate model. Mathematical Methods in the Applied Sciences. 2015;38(15):3359–68. URL: https://doi.org/10.1002/mma.3335

34. Sun Y., Yao K., Fu Z. Interest rate model in uncertain environment based on exponential Ornstein–Uhlenbeck equation. Soft Computing. 2018;22:465–75. URL: https://doi.org/10.1007/s00500–016–2337–1

35. Li Z., Liu Y.-J., Zhang W.-G. Quasi-closed-form solution and numerical method for currency option with uncertain volatility model. Soft Computing. 2020;24(19):15041–57. URL: https://doi.org/10.1007/s00500–020–04854–3

36. Wang X. Pricing of European currency options with uncertain exchange rate and stochastic interest rates. Discrete Dynamics in Nature and Society. 2019;2019(1):2548592. URL: https://doi.org/10.1155/2019/2548592

37. Liu Y. Uncertain random variables: A mixture of uncertainty and randomness. Soft computing. 2013;17:625–34. URL: https://doi.org/10.1007/s00500–012–0935–0

38. Wang X., Shi G., Sheng Y. Delayed renewal process with uncertain random inter-arrival times. Symmetry. 2021;13(10):1943. URL: https://doi.org/10.3390/sym13101943

39. Shi G., Sheng Y., Ahmadzade H., Tahmasebi S. Extropy: Dual of entropy for uncertain random variables and iIts applications. Journal of Industrial and Management Optimization. 2025;21(5):4025–40. URL: https://doi.org/10.3934/jimo.2025041

40. Hu F., Fu X., Qu Z. Uncertain random variables and laws of large numbers under U-C chance space. Fuzzy Sets and Systems. 2024;493:109086. URL: https://doi.org/10.1016/j.fss.2024.109086

41. Gao J., Yao K. Some concepts and theorems of uncertain random process. International Journal of Intelligent Systems. 2015;30(1):52–65. URL: https://doi.org/10.1002/int.21681

42. Fei W. On existence and uniqueness of solutions to uncertain backward stochastic differential equations. Applied Mathematics-A Journal of Chinese Universities. 2014;29(1):53–66. URL: https://doi.org/10.1007/s11766–014–3048-y

43. Liu B. Uncertainty Theory (5th Ed.). China: Uncertainty Theory Laboratory; 2024.

44. Chirima J., Chikodza E., Hove-Musekwa S. D. Uncertain stochastic option pricing in the presence of uncertain jumps. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems. 2019;27(04):613–35. URL: https://doi.org/10.1142/S0218488519500272

45. Wu C., Yang L., Zhang C. Uncertain stochastic optimal control with jump and its application in a portfolio game. Symmetry. 2022;14(9):1885. URL: https://doi.org/10.3390/sym14091885

46. Zhou J., Yang F., Wang K. Multi-objective optimization in uncertain random environments. Fuzzy Optimization and Decision Making. 2014;13:397–413. URL: https://doi.org/10.1007/s10700–014–9183–3

47. Fei W. Optimal control of uncertain stochastic systems with Markovian switching and its applications to portfolio decisions. Cybernetics and Systems. 2014;45(1):69–88. URL: https://doi.org/10.1080/01969722.2014.862445

48. Chen X. Theory of uncertain finance. Beijing, China: Uncertainty Theory Laboratory; 2016.