Abstract
Uncertain process is an important tool to model dynamic uncertain systems. This paper proposes a special type of uncertain processes, named contour processes, whose sample paths can be classified by their inverse uncertainty distributions. It is shown that the set of contour processes is closed under the extreme value operator and the time integral operator as well as the monotone function. As an application, this paper considers an uncertain stock model with floating interest rate, in which both the interest rate and the stock price follow uncertain differential equations. By means of contour processes, some pricing formulas are derived for the European options, American options and Asian options of the stock model.
Similar content being viewed by others
References
Chen, X. (2011). American option pricing formula for uncertain financial market. International Journal of Operations Research, 8(2), 32–37.
Chen, X., & Gao, J. (2013). Uncertain term structure model of interest rate. Soft Computing, 17(4), 597–604.
Chen, X., & Liu, B. (2010). Existence and uniqueness theorem for uncertain differential equations. Fuzzy Optimization and Decision Making, 9(1), 69–81.
Chen, X., Liu, Y., & Ralescu, D. A. (2013). Uncertain stock model with periodic dividends. Fuzzy Optimization and Decision Making, 12(1), 111–123.
Jiao, D., & Yao, K. (2015). An interest rate model in uncertain environment. Soft Computing, 19(3), 775–780.
Liu, B. (2007). Uncertainty theory (2nd ed.). Berlin: Springer.
Liu, B. (2008). Fuzzy process, hybrid process and uncertain process. Journal of Uncertain Systems, 2(1), 3–16.
Liu, B. (2009). Some research problems in uncertainty theory. Journal of Uncertain Systems, 3(1), 3–10.
Liu, B. (2010). Uncertainty theory: A branch of mathematics for modeling human uncertainty. Berlin: Springer.
Liu, B. (2013a). Toward uncertain finance theory. Journal of Uncertainty Analysis and Applications, 1. doi:10.1186/2195-5468-1-1
Liu, B. (2013b). Extreme value theorems of uncertain process with application to insurance risk model. Soft Computing, 17(4), 549–556.
Liu, B. (2014). Uncertainty distribution and independence of uncertain processes. Fuzzy Optimization and Decision Making, 13(3), 259–271.
Liu, Y. (2012). An analytic method for solving uncertain differential equations. Journal of Uncertain Systems, 6(4), 244–249.
Liu, Y., & Ha, M. (2010). Expected value of function of uncertain variables. Journal of Uncertain Systems, 4(3), 181–186.
Liu, Y., Chen, X., & Ralescu, D. A. (2015). Uncertain currency model and currency option pricing. International Journal of Intelligent Systems, 30(1), 40–51.
Peng, J., & Yao, K. (2011). A new option pricing model for stocks in uncertainty markets. International Journal of Operations Research, 8(2), 18–26.
Sun, J. J., & Chen, X. (2013). Asian option pricing formula for uncertain financial market. http://orsc.edu.cn/online/130511
Yao, K. (2013a). Extreme values and integral of solution of uncertain differential equation. Journal of Uncertainty Analysis and Applications, 1. doi:10.1186/2195-5468-1-2
Yao, K. (2013b). A type of nonlinear uncertain differential equations with analytic solution. Journal of Uncertainty Analysis and Applications, 1. doi:10.1186/2195-5468-1-8.
Yao, K. (2015). A no-arbitrage theorem for uncertain stock model. Fuzzy Optimization and Decision Making. doi:10.1007/s10700-014-9198-9.
Yao, K., & Chen, X. (2013). A numerical method for solving uncertain differential equations. Journal of Intelligent and Fuzzy Systems, 25(3), 825–832.
Acknowledgments
This work was supported by National Natural Science Foundation of China (Grant No. 61403360).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yao, K. Uncertain contour process and its application in stock model with floating interest rate. Fuzzy Optim Decis Making 14, 399–424 (2015). https://doi.org/10.1007/s10700-015-9211-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10700-015-9211-y