# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2018129

## A mean-reverting currency model with floating interest rates in uncertain environment

 School of Science, Nanjing University of Science and Technology, Nanjing 210094, China

* Corresponding author: Weiwei Wang

Received  March 2018 Revised  April 2018 Published  August 2018

Currency option is an important risk management tool in the foreign exchange market, which has attracted the attention of many researchers. Unlike the classical stochastic theory, we investigate the valuation of currency option under the assumption that the risk factors are described by uncertain processes. Considering the long-term fluctuations of the exchange rate and the changing of the interest rates from time to time, we propose a mean-reverting uncertain currency model with floating interest rates to simulate the foreign exchange market. Subsequently, European and American currency option pricing formulas for the new currency model are derived and some mathematical properties of the formulas are studied. Finally, some numerical algorithms are designed to calculate the prices of these options.

Citation: Weiwei Wang, Ping Chen. A mean-reverting currency model with floating interest rates in uncertain environment. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018129
##### References:
 [1] F. Black and M. Scholes, The pricing of option and corporate liabilities, J. Polit. Econ., 81 (1973), 637-654. doi: 10.1086/260062. [2] X. Chen and B. Liu, Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optim. Decis. Mak., 9 (2010), 69-81. doi: 10.1007/s10700-010-9073-2. [3] Y. Gao, Existence and uniqueness theorem on uncertain differential equations with local Lipschitz condition, J. Uncertain Syst., 6 (2012), 223-232. [4] R. Gao, Milne method for solving uncertain differential equations, Appl. Math. Comput., 274 (2016), 774-785. doi: 10.1016/j.amc.2015.11.043. [5] D. Kahneman and A. Tversky, Prospect theory: An analysis of decision making under risk, Econometrica, 47 (1979), 263-292. [6] B. Liu, Uncertainty Theory, 2nd edition, Springer-Verlag, Berlin, 2007. [7] B. Liu, Fuzzy process, hybrid process and uncertain process, J. Uncertain Syst., 2 (2008), 3-16. [8] B. Liu, Some research problems in uncertainty theory, J. Uncertain Syst., 3 (2009), 3-10. [9] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010. [10] B. Liu, Toward uncertain finance theory, J. Uncertain. Anal. Appl., 1 (2013), Article 1. doi: 10.1186/2195-5468-1-1. [11] Y. Liu, An analytic method for solving uncertain differential equations, J. Uncertain Syst., 6 (2012), 244-249. [12] Y. Liu, Semi-linear uncertain differential equation with its analytic solution, Inf: Int Interdiscip J., 16 (2013), 889-894. [13] H. Liu, H. Ke and W. Fei, Almost sure stability for uncertain differential equation, Fuzzy Optim. Decis. Mak., 13 (2014), 463-473. doi: 10.1007/s10700-014-9188-y. [14] Y. Liu, X. Chen and D. Ralescu, Uncertain currency model and currency option pricing, Int. J. Intell. Syst., 30 (2015), 40-51. doi: 10.1002/int.21680. [15] Y. Sheng and C. Wang, Stability in the p-th moment for uncertain differential equation, J. Intell. Fuzzy Syst., 26 (2014), 1263-1271. [16] Y. Shen and K. Yao, A mean-reverting currency model in an uncertain environment, Soft Comput., 20 (2016), 4131-4138. doi: 10.1007/s00500-015-1748-8. [17] Y. Sheng and J. Gao, Exponential stability of uncertain differential equation, Soft Comput., 20 (2016), 3673-3678. doi: 10.1007/s00500-015-1727-0. [18] Z. Wang, Analytic solution for a general type of uncertain differential equation, Inf: Int Interdiscip J., 16 (2013), 1003-1010. [19] X. Wang and Y. Ning, An uncertain currency model with floating interest rates, Soft Comput., 21 (2017), 6739-6754. doi: 10.1007/s00500-016-2224-9. [20] X. Yang and D. Ralescu, Adams method for solving uncertain differential equation, Appl. Math. Comput., 270 (2015), 993-1003. doi: 10.1016/j.amc.2015.08.109. [21] X. Yang and Y. Shen, Runge-Kutta method for solving uncertain differential equations, J. Uncertain. Anal. Appl., 3 (2015), Article 17. doi: 10.1186/s40467-015-0038-4. [22] K. Yao, A type of uncertain differential equations with analytic solution, J. Uncertain. Anal. Appl., 1 (2013), Article 8. doi: 10.1186/2195-5468-1-8. [23] K. Yao, Extreme values and integral of solution of uncertain differential equation, J. Uncertain. Anal. Appl., 1 (2013), Article 2. doi: 10.1186/2195-5468-1-2. [24] K. Yao, Uncertain contour process and its application in stock model with floating interest rate, Fuzzy Optim. Decis. Mak., 14 (2015), 399-424. doi: 10.1007/s10700-015-9211-y. [25] K. Yao and X. Chen, A numerical method for solving uncertain differential equations, J. Intell. Fuzzy Syst., 25 (2013), 825-832. [26] K. Yao, J. Gao and Y. Gao, Some stability theorems of uncertain differential equation, Fuzzy Optim. Decis. Mak., 12 (2013), 3-13. doi: 10.1007/s10700-012-9139-4. [27] K. Yao, H. Ke and Y. Sheng, Stability in mean for uncertain differential equation, Fuzzy Optim. Decis. Mak., 14 (2015), 365-379. doi: 10.1007/s10700-014-9204-2.

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##### References:
 [1] F. Black and M. Scholes, The pricing of option and corporate liabilities, J. Polit. Econ., 81 (1973), 637-654. doi: 10.1086/260062. [2] X. Chen and B. Liu, Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optim. Decis. Mak., 9 (2010), 69-81. doi: 10.1007/s10700-010-9073-2. [3] Y. Gao, Existence and uniqueness theorem on uncertain differential equations with local Lipschitz condition, J. Uncertain Syst., 6 (2012), 223-232. [4] R. Gao, Milne method for solving uncertain differential equations, Appl. Math. Comput., 274 (2016), 774-785. doi: 10.1016/j.amc.2015.11.043. [5] D. Kahneman and A. Tversky, Prospect theory: An analysis of decision making under risk, Econometrica, 47 (1979), 263-292. [6] B. Liu, Uncertainty Theory, 2nd edition, Springer-Verlag, Berlin, 2007. [7] B. Liu, Fuzzy process, hybrid process and uncertain process, J. Uncertain Syst., 2 (2008), 3-16. [8] B. Liu, Some research problems in uncertainty theory, J. Uncertain Syst., 3 (2009), 3-10. [9] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010. [10] B. Liu, Toward uncertain finance theory, J. Uncertain. Anal. Appl., 1 (2013), Article 1. doi: 10.1186/2195-5468-1-1. [11] Y. Liu, An analytic method for solving uncertain differential equations, J. Uncertain Syst., 6 (2012), 244-249. [12] Y. Liu, Semi-linear uncertain differential equation with its analytic solution, Inf: Int Interdiscip J., 16 (2013), 889-894. [13] H. Liu, H. Ke and W. Fei, Almost sure stability for uncertain differential equation, Fuzzy Optim. Decis. Mak., 13 (2014), 463-473. doi: 10.1007/s10700-014-9188-y. [14] Y. Liu, X. Chen and D. Ralescu, Uncertain currency model and currency option pricing, Int. J. Intell. Syst., 30 (2015), 40-51. doi: 10.1002/int.21680. [15] Y. Sheng and C. Wang, Stability in the p-th moment for uncertain differential equation, J. Intell. Fuzzy Syst., 26 (2014), 1263-1271. [16] Y. Shen and K. Yao, A mean-reverting currency model in an uncertain environment, Soft Comput., 20 (2016), 4131-4138. doi: 10.1007/s00500-015-1748-8. [17] Y. Sheng and J. Gao, Exponential stability of uncertain differential equation, Soft Comput., 20 (2016), 3673-3678. doi: 10.1007/s00500-015-1727-0. [18] Z. Wang, Analytic solution for a general type of uncertain differential equation, Inf: Int Interdiscip J., 16 (2013), 1003-1010. [19] X. Wang and Y. Ning, An uncertain currency model with floating interest rates, Soft Comput., 21 (2017), 6739-6754. doi: 10.1007/s00500-016-2224-9. [20] X. Yang and D. Ralescu, Adams method for solving uncertain differential equation, Appl. Math. Comput., 270 (2015), 993-1003. doi: 10.1016/j.amc.2015.08.109. [21] X. Yang and Y. Shen, Runge-Kutta method for solving uncertain differential equations, J. Uncertain. Anal. Appl., 3 (2015), Article 17. doi: 10.1186/s40467-015-0038-4. [22] K. Yao, A type of uncertain differential equations with analytic solution, J. Uncertain. Anal. Appl., 1 (2013), Article 8. doi: 10.1186/2195-5468-1-8. [23] K. Yao, Extreme values and integral of solution of uncertain differential equation, J. Uncertain. Anal. Appl., 1 (2013), Article 2. doi: 10.1186/2195-5468-1-2. [24] K. Yao, Uncertain contour process and its application in stock model with floating interest rate, Fuzzy Optim. Decis. Mak., 14 (2015), 399-424. doi: 10.1007/s10700-015-9211-y. [25] K. Yao and X. Chen, A numerical method for solving uncertain differential equations, J. Intell. Fuzzy Syst., 25 (2013), 825-832. [26] K. Yao, J. Gao and Y. Gao, Some stability theorems of uncertain differential equation, Fuzzy Optim. Decis. Mak., 12 (2013), 3-13. doi: 10.1007/s10700-012-9139-4. [27] K. Yao, H. Ke and Y. Sheng, Stability in mean for uncertain differential equation, Fuzzy Optim. Decis. Mak., 14 (2015), 365-379. doi: 10.1007/s10700-014-9204-2.
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