# An uncertain currency model with floating interest rates

- 290 Downloads
- 3 Citations

## Abstract

Considering the uncertain fluctuations in the financial market from time to time, we propose a currency model with floating interest rates within the framework of uncertainty theory. Different from the classical stochastic currency models, this paper is assumed that the domestic interest rate, the foreign interest rate and the exchange rate follow uncertain differential equations. After that, the pricing formulas of European and American currency options are derived. The simulation experiments presented in this paper illustrate the performance of the proposed model, and the relationship between the option pricing formulas and all relevant parameters is analyzed.

### Keywords

Currency model Option pricing formula Floating interest rate Uncertain differential equation## Notes

### Acknowledgments

This work is supported by Natural Science Foundation of Shandong Province (ZR2014GL002).

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest to this work.

### References

- Amin K, Jarrow R (1991) Pricing foreign currency options under stochastic interest rates. J Int Money Finance 10(3):310–329CrossRefGoogle Scholar
- Black F, Scholes M (1973) The pricing of options and corporate liabilities. J Polit Econ 81(1):637–654MathSciNetCrossRefMATHGoogle Scholar
- Bollen N, Rasiel E (2003) The performance of alternative valuation models in the OTC currency options market. J Int Money Finance 22(1):33–64CrossRefGoogle Scholar
- Carr P, Wu L (2007) Stochastic skew in currency options. J Financ Econ 86(1):213–247CrossRefGoogle Scholar
- Chen X, Gao J (2013) Uncertain term structure model of interest rate. Soft Comput 17(4):597–604CrossRefMATHGoogle Scholar
- Cox J, Ingersoll J, Ross S (1985) An intertemporal general equilibrium model of asset prices. Econometrica 53:363–382MathSciNetCrossRefMATHGoogle Scholar
- Garman M, Kohlhagen S (1983) Foreign currency option values. J Int Money Finance 2(3):231–237CrossRefGoogle Scholar
- Grabbe J (1983) The pricing of call and put options on foreign exchange. J Int Money Finance 2(3):239–253CrossRefGoogle Scholar
- Heston S (1993) A closed-form solution for options with stochastic volatility with applications to bonds and currency options. Rev Financ Stud 6(2):327–343CrossRefGoogle Scholar
- Hilliard J, Madura J, Tucker A (1991) Currency option pricing with stochastic domestic and foreign interest rates. J Financ Quant Anal 26(2):139–151CrossRefGoogle Scholar
- Ho T, Lee S (1986) Term structure movements and pricing interest rate contingent claims. J Finance 41(5):1011–1029CrossRefGoogle 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, BerlinMATHGoogle Scholar
- Liu B (2008) Fuzzy process, hybrid process and uncertain process. J Uncertain Syst 2(1):3–16MathSciNetGoogle Scholar
- Liu B (2009) Some research problems in uncertainty theory. J Uncertain Syst 3(1):3–10MathSciNetGoogle Scholar
- Liu B (2010) Uncertainty theory: a branch of mathematics for modeling human uncertainty. Springer, BerlinCrossRefGoogle Scholar
- Liu B (2014) Uncertainty distribution and independence of uncertain processes. Fuzzy Optim Decis Mak 13(3):259–271MathSciNetCrossRefGoogle Scholar
- Liu Y, Ha M (2010) Expected value of function of uncertain variables. J Uncertain Syst 4(3):181–186Google Scholar
- Liu Y, Chen X, Ralescu DA (2015) Uncertain currency model and currency option pricing. Int J Intell Syst 30(1):40–51CrossRefGoogle Scholar
- Melino A, Turnbull S (1990) Pricing foreign currency options with stochastic volatility. J Econom 45(1):239–265CrossRefMATHGoogle Scholar
- Sarwar G, Krehbiel T (2000) Empirical performance of alternative pricing models of currency options. J Future Mark 20(2):265–291CrossRefGoogle Scholar
- Sun L (2013) Pricing currency options in the mixed fractional Brownian motion. Phys A 392(iss.16):3441–3458Google Scholar
- Swishchuk A, Tertychnyi M, Elliott R (2014) Pricing currency derivatives with Markov-modulated Lévy dynamics. Insur Math Econ 57:67–76CrossRefMATHGoogle Scholar
- van Haastrecht A, Pelsser A (2011) Generic pricing of FX, inflation and stock options under stochastic interest rates and stochastic volatility. Quant Finance 11(5):665–691MathSciNetCrossRefMATHGoogle Scholar
- Vasicek O (1977) An equilibrium characterization of the term structure. J Financ Econ 5(2):177–188CrossRefMATHGoogle Scholar
- Wang X, Ning Y, Moughal T, Chen X (2015) Adams-Simpson method for solving uncertain differential equation. Appl Math Comput 271:209–219MathSciNetGoogle Scholar
- Xiao W, Zhang W, Zhang X, Wang Y (2010) Pricing currency options in a fractional Brownian motion with jumps. Econ Model 27(iss. 5):935–942CrossRefGoogle Scholar
- Xu G (2006) Analysis of pricing European call foreign currency option under the Vasicek interest rate model. J Tongji Univ (Nat Sci) 34(4):552–556MathSciNetGoogle Scholar
- Yao K, Chen X (2013) A numerical method for solving uncertain differential equations. J Intell Fuzzy Syst 25(3):825–832MathSciNetMATHGoogle 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
- Zhu Y (2015) Uncertain fractional differential equations and an interest rate model. Math Methods Appl Sci 38(15):3359–3368MathSciNetCrossRefMATHGoogle Scholar