# Two-factor term structure model with uncertain volatility risk

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## Abstract

This paper aims to study two-factor uncertain term structure model where the volatility of the uncertain interest rate is driven by another uncertain differential equation. In order to solve this model, the nested uncertain differential equation method is employed. This paper is also devoted to the study of the numerical solutions for the proposed nested uncertain differential equation using the \(\alpha \)-path methods. We also use the built two-factor term structure model to value the bond price with the help of proposed numerical method. Finally, we give a numerical example where the price of a zero-coupon bond is calculated based on the \(\alpha \)-path methods.

## Keywords

Two-factor term structure Uncertain short interest rate Volatility risk Bond pricing Uncertain differential equation## Notes

### Acknowledgements

The author gratefully acknowledges the financial support provided by National Natural Science Foundation of China (Grant Nos. 61673225, 61304182 and 61374082), Distinguished Young Scholar Project of Renmin University of China and China Scholarship Council under Grant 201606365008.

### Compliance with ethical standards

### Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

### Ethical approval

This article does not contain any studies with human participants performed by any of the authors.

## References

- Chen X, Liu B (2010) Existence and uniqueness theorem for uncertain differential equations. Fuzzy Optim Decis Mak 9(1):69–81MathSciNetCrossRefzbMATHGoogle Scholar
- Chen X, Gao J (2013) Uncertain term structure model of interest rate. Soft Comput 17(4):597–604CrossRefzbMATHGoogle Scholar
- Chen X (2015) Uncertain calculus with finite variation processes. Soft Comput 19(10):2905–2912CrossRefzbMATHGoogle Scholar
- Chen X (2016) Nested uncertain differential equations and its application to multi-factor term structure model. http://orsc.edu.cn/online/160104.pdf
- Dothan L (1978) On the term structure of interest rates. J Financ Econ 6:59–69CrossRefGoogle Scholar
- James J, Webber N (2000) Interest rate modelling. Wiley-Blackwell Publishing, OxfordzbMATHGoogle Scholar
- Jiao D, Yao K (2015) An interest rate model in uncertain environment. Soft Comput 19(3):775–780CrossRefGoogle Scholar
- Liu B (2007) Uncertainty theory, 2nd edn. Springer, BerlinzbMATHGoogle Scholar
- Liu B (2008) Fuzzy process, hybrid process and uncertain process. J Uncertain Syst 2(1):3–16Google Scholar
- Liu B (2009a) Some research problems in uncertainty theory. J Uncertain Syst 3(1):3–10Google Scholar
- Liu B (2009b) Theory and practice of uncertain programming, 2nd edn. Springer, BerlinGoogle Scholar
- Liu B (2010) Uncertainty theory: a branch of mathematics for modeling human uncertainty. Springer, BerlinCrossRefGoogle Scholar
- Liu Y, Chen X, Ralescu DA (2015) Uncertain currency model and currency option pricing. Int J Intell Syst 30(1):40–51CrossRefGoogle Scholar
- Marsh T, Rosenfeld E (1983) Stochastic processes for interest rates and equilibrium bond prices. J Financ 38(2):635–646CrossRefGoogle Scholar
- Maruyama G (1955) Continuous Markov processes and stochastic equations. Rend Circolo Mat Palermo 4(1):48–90MathSciNetCrossRefzbMATHGoogle Scholar
- Merton R (1973) Theory of rational option pricing. Bell J Econ Manag Sci 4:141–183MathSciNetCrossRefzbMATHGoogle Scholar
- Milstein GN (1974) Approximate integration of stochastic differential equations. Theory Probab Appl 19(3):557–562MathSciNetGoogle Scholar
- Peng J, Yao K (2011) A new option pricing model for stocks in uncertainty markets. Int J Oper Res 8(2):18–26MathSciNetGoogle Scholar
- Rumelin W (1982) Numerical treatment of stochastic differential equations. SIAM J Numer Anal 19(3):604–613MathSciNetCrossRefGoogle Scholar
- Vasicek O (1977) An equilibrium characterization of the term structure. J Financ Econ 5:177–188CrossRefzbMATHGoogle Scholar
- Yao K (2012) Uncertain calculus with renewal process. Fuzzy Optim Decis Mak 11(3):285–297MathSciNetCrossRefzbMATHGoogle Scholar
- Yao K, Chen X (2013) A Numerical method for solving uncertain differential equations. J Intell Fuzzy Syst 25(3):825–832MathSciNetzbMATHGoogle Scholar
- Yao K, Gao J, Gao Y (2013) Some stability theorems of uncertain differential equation. Fuzzy Optim Decis Mak 12(1):3–13MathSciNetCrossRefGoogle Scholar
- Yao K (2014) Multi-dimensional uncertain calculus with Liu process. J Uncertain Syst 8(4):244–254Google Scholar
- Yao K (2015) Uncertain contour process and its application in stock model with floating interest rate. Fuzzy Optim Decis Mak 14(4):399–424MathSciNetCrossRefGoogle Scholar
- Zhang Z, Ralescu D, Liu W (2016) Valuation of interest rate ceiling and floor in uncertain financial market. Fuzzy Optim Decis Mak 15(2):139–154MathSciNetCrossRefGoogle Scholar
- Zhu Y (2010) Uncertain optimal control with application to a portfolio selection model. Cybern Syst 41(7):535–547CrossRefzbMATHGoogle Scholar
- Zhu Y (2015) Uncertain fractional differential equations and an interest rate model. Math Methods Appl Sci 38(15):3359–3368MathSciNetCrossRefzbMATHGoogle Scholar