October  2019, 15(4): 1921-1936. 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  October 2019 Early access  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, 2019, 15 (4) : 1921-1936. doi: 10.3934/jimo.2018129
References:
[1]

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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.  Google Scholar

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Y. Gao, Existence and uniqueness theorem on uncertain differential equations with local Lipschitz condition, J. Uncertain Syst., 6 (2012), 223-232.   Google Scholar

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R. Gao, Milne method for solving uncertain differential equations, Appl. Math. Comput., 274 (2016), 774-785.  doi: 10.1016/j.amc.2015.11.043.  Google Scholar

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B. Liu, Uncertainty Theory, 2nd edition, Springer-Verlag, Berlin, 2007. Google Scholar

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B. Liu, Fuzzy process, hybrid process and uncertain process, J. Uncertain Syst., 2 (2008), 3-16.   Google Scholar

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B. Liu, Some research problems in uncertainty theory, J. Uncertain Syst., 3 (2009), 3-10.   Google Scholar

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B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010. Google Scholar

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B. Liu, Toward uncertain finance theory, J. Uncertain. Anal. Appl., 1 (2013), Article 1. doi: 10.1186/2195-5468-1-1.  Google Scholar

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Y. Liu, An analytic method for solving uncertain differential equations, J. Uncertain Syst., 6 (2012), 244-249.   Google Scholar

[12]

Y. Liu, Semi-linear uncertain differential equation with its analytic solution, Inf: Int Interdiscip J., 16 (2013), 889-894.   Google Scholar

[13]

H. LiuH. 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.  Google Scholar

[14]

Y. LiuX. Chen and D. Ralescu, Uncertain currency model and currency option pricing, Int. J. Intell. Syst., 30 (2015), 40-51.  doi: 10.1002/int.21680.  Google Scholar

[15]

Y. Sheng and C. Wang, Stability in the p-th moment for uncertain differential equation, J. Intell. Fuzzy Syst., 26 (2014), 1263-1271.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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Z. Wang, Analytic solution for a general type of uncertain differential equation, Inf: Int Interdiscip J., 16 (2013), 1003-1010.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

K. Yao and X. Chen, A numerical method for solving uncertain differential equations, J. Intell. Fuzzy Syst., 25 (2013), 825-832.   Google Scholar

[26]

K. YaoJ. 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.  Google Scholar

[27]

K. YaoH. 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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

Y. Gao, Existence and uniqueness theorem on uncertain differential equations with local Lipschitz condition, J. Uncertain Syst., 6 (2012), 223-232.   Google Scholar

[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.  Google Scholar

[5]

D. Kahneman and A. Tversky, Prospect theory: An analysis of decision making under risk, Econometrica, 47 (1979), 263-292.   Google Scholar

[6]

B. Liu, Uncertainty Theory, 2nd edition, Springer-Verlag, Berlin, 2007. Google Scholar

[7]

B. Liu, Fuzzy process, hybrid process and uncertain process, J. Uncertain Syst., 2 (2008), 3-16.   Google Scholar

[8]

B. Liu, Some research problems in uncertainty theory, J. Uncertain Syst., 3 (2009), 3-10.   Google Scholar

[9]

B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010. Google Scholar

[10]

B. Liu, Toward uncertain finance theory, J. Uncertain. Anal. Appl., 1 (2013), Article 1. doi: 10.1186/2195-5468-1-1.  Google Scholar

[11]

Y. Liu, An analytic method for solving uncertain differential equations, J. Uncertain Syst., 6 (2012), 244-249.   Google Scholar

[12]

Y. Liu, Semi-linear uncertain differential equation with its analytic solution, Inf: Int Interdiscip J., 16 (2013), 889-894.   Google Scholar

[13]

H. LiuH. 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.  Google Scholar

[14]

Y. LiuX. Chen and D. Ralescu, Uncertain currency model and currency option pricing, Int. J. Intell. Syst., 30 (2015), 40-51.  doi: 10.1002/int.21680.  Google Scholar

[15]

Y. Sheng and C. Wang, Stability in the p-th moment for uncertain differential equation, J. Intell. Fuzzy Syst., 26 (2014), 1263-1271.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

Z. Wang, Analytic solution for a general type of uncertain differential equation, Inf: Int Interdiscip J., 16 (2013), 1003-1010.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

K. Yao and X. Chen, A numerical method for solving uncertain differential equations, J. Intell. Fuzzy Syst., 25 (2013), 825-832.   Google Scholar

[26]

K. YaoJ. 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.  Google Scholar

[27]

K. YaoH. 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.  Google Scholar

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