Document Type : Research Article
Authors
Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, P.O. Box 35195-363, Semnan, Iran.
Abstract
Option pricing is a fundamental issue in financial markets, and barrier options are a popular type of options that can become valuable or worthless when the underlying asset price reaches a predetermined level. A double barrier option consist two barriers, one situated above and the other below the prevailing stock price. This particular option is categorized as path dependent because the return for the holder is influenced by the stock price’s breach of the two barriers. The double barrier option contract stipulates three specific payoffs, depending on whether the up-barrier or down-barrier is touched, or if there is no breach of either barrier during the entire duration of the option. In this paper, pricing of the double barrier options when the underlying asset price follows the exponential Ornstein-Uhlenbeck model is investigated, and also pricing formulas for different types of double barrier options (knock-in and knock-out) are derived by α-paths of uncertain differential equations in the uncertain environment.
Keywords
81(1973), 637–654.
[2] X. Chen, American option pricing formula for uncertain financial market, Int J Op Res,
8(2) (2011), 32–37.
[3] X. Chen and D. Ralescu, Liu process and uncertain calculus, J Uncertain Anal Appl, (2013),
1:3.
[4] X. Chen and B. Liu, Existence and uniqueness theorem for uncertain differential equations,
Fuzzy Optim Decis Mak, 9(1) (2010), 69–81.
[5] R. Cont and P. Tankov, Financial Modelling with Jump Processes (Chapman and Hall/CRC
Financial Mathematics Series), CRC Press: Boca Raton, FL, USA, 2004.
[6] P. Eloe, R.H. Liu and J.Y. Sun, Double barrier option under regime-switching exponential
mean-reverting process, Int J Comput Math, 86(6) (2009), 964–981.
[7] X. Ji and J. Zhou, Option pricing for an uncertain stock model with jumps, Soft Comput,
19(11) (2015), 3323–3329.
[8] L. Jia and W. Chen, Knock-in options of an uncertain stock model with floating interest
rate, Chaos, Solitons and Fractals 141,(2020), 110324.
[9] D. Jun and H. Ku, Analytic solution for American barrier options with two barriers, J
Math Anal Appl, 422(1) (2015), 408–423.
[10] D. Kahneman and A. Tversky, Prospect theory: an analysis of decision under risk, Econometrica, 47(2) (1979), 263–292.
[11] B. Liu, Uncertainty Theory, seconded, Springer-Verlag, Berlin, 2007.
[12] B. Liu, Fuzzy process, hybrid process and uncertain process, J Uncertain Syst, 2(1) (2008),
3–16.
[13] B. Liu, Some research problems in uncertainty theory, J Uncertain Syst, 3(1), (2009), 3–10.
[14] B. Liu, Uncertainty theory: a branch of mathematics for modeling human uncertainty,
Springer, Berlin, 2010.
[15] Y. Liu, An analytic method for solving uncertain differential equation, J Uncrtain Syst, 6(4)
(2012), 244–249.
[16] B. Liu, Toward uncertain finance theory, J Uncertain Anal Appl, 1, (2013), Article1.
[17] C.F. LO and C.H. Hui, Lie-algebraic approach for pricing moving barrier options with
time-dependent parameters, J Math Anal Appl, 323(2) (2006), 1455–1464.
[18] R.C. Merton, Theory of rational option pricing, Bell J Econ Manag Sci, 4, (1973), 141–183.
[19] K. Nouri, B. Abbasi, F. Omidi and L. Torkzadeh, Digital barrier options pricing: an improved Monte Carlo algorithm, J Math Sci, 10 (2016), 65–70.
[20] K. Nouri and B. Abbasi, Implementation of the modified Monte Carlo simulation for evaluate the barrier option prices, Journal of Taibah University for Science, 11 (2017), 233–240.
[21] J. Peng and K. Yao, A new option pricing model for stocks in uncertainty markets, Int J
Op Res, 8(2) (2011), 18–26.
[22] D.R. Rich, The mathematical foundations of barrier option-pricing theory, Adv Futur Opt
Res, 7 (1994), 267–312.
[23] G. Rong, L. Kaixiang, L. Zhiguo and L. Liying, American barrier option Pricing formulas
for currency model in uncertain environment, J Syst Sci Complex, 35 (2022), 283–312.
[24] X. Yang, Z. Zhang and X. Gao, Asian-barrier option pricing formulas of uncertain financial
market, Chaos, Solitons and Fractals, 123 (2019), 79–86.
[25] K. Yao, A type of nonlinear uncertain differential equations with analytic solution, J Uncertain Anal Appl, (2013), 1: 8.
[26] K. Yao and X. Chen, A numerical method for solving uncertain differential equations, J
Intell Fuzzy Syst, 25 (2013), 825–832.
[27] K. Yao, Uncertain contour process and its application in stock model with floating interest
rate, Fuzzy Optim Decis Mak, 14 (2015), 399–424.
[28] X. Yu, A stock model with jumps for uncertain markets, Int J Uncert Fuzz Knowl Syst,
20(3) (2012), 421–432.
)