An Uncertain Renewal Stock Model for Barrier Options Pricing with Floating Interest Rate

Subject Areas : Financial Mathematics

Behzad Abbasi ¹ , Kazem Nouri ²

1 - Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, P.O. 35195-363, Semnan, Iran.(PHD Student)
2 - Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, P.O. Box 35195-363, Semnan, Iran.(Professor)

Received: 2024-01-15 Accepted : 2024-04-23 Published : 2024-05-26

Keywords: Uncertain Process, Renewal Process, Barrier Options Pricing , Floating Interest Rate , Uncertain Differential Equation(UDE),

Abstract :

Option pricing is a main topic in contemporary financial theories, captivating the attention of numerous financial analysts and economists. Barrier option, classified as an exotic option, derives its value from the behavior of an underlying asset. The outcome of this option is based on whether or not the price of the underlying asset has reached a predetermined barrier level. Over the years, the stock price has been represented through continuous stochastic processes, with the prominent one being the Brownian motion process. Correspondingly, the widely used Black-Scholes model has been employed. Nevertheless, it has become evident that utilizing stochastic differential equations to characterize the stock price process is unsuitable and leads to a perplexing paradox. As a result, many researchers have turned to incorporating fuzzy or uncertain environments in such situations. This study presents a methodology for pricing barrier options on stocks in an uncertain environment, in which the interarrival times are uncertain variables. The approach employs the Liu process and renewal uncertain process, considering the interest rate as dynamic and floating. The pricing formulas for knock-in barrier options are derived using α-paths of uncertain differential equations with jumps.

References:

[1] Black, F., Scholes, M., The pricing of option and corporate liabilities, J. Polit. Econ, 1973; 81, 637-654.
Doi: jstor.org/stable/1831029.
[2] Merton, R.C., Theory of rational option pricing, Bell J. Econ. Manag. Sci, 4, 1973; 141-183. Doi: 10.2307/3003143
[3] Nouri, K., Abbasi, B., Omidi, F., Torkzadeh, L., Digital barrier options pricing: an improved Monte Carlo algorithm, J. Math Sci, 2016; 10: 65-70. Doi: 10.1007/s40096-016-0179-8
[4] Nouri, K., Abbasi, B., Implementation of the modified Monte Carlo simulation for evaluate the barrier option prices, Journal of Taibah University for Science, 2017; 11: 233-240. Doi:10.1016/j.jtusci.2015.02.010
[5] LO, C.F., Hui, C.H., Lie-algebraic approach for pricing moving barrier options with time-dependent parameters, J. Math. Anal. Appl, 2006; 323(2): 1455-1464. Doi: 10.1016/j.jmaa.2005.11.068
[6] Jun, D., Ku, H., Analytic solution for American barrier options with two barriers, J. Math. Anal. Appl, 2015; 422(1): 408-423. Doi: 10.1016/j.jmaa.2014.08.047
[7] Liu, B., Toward uncertain finance theory. J. Uncertain. Anal. Appl, 1, 2013; 1. Doi: 10.1186/2195-5468-1-1
[8] Ji, X., Zhou, J., Option pricing for an uncertain stock model with jumps, Soft Comput, 2015; 19(11): 3323-3329. Doi: 10.1007/s00500-015-1635-3
[9] Liu, Z., Option Pricing Formulas in a New Uncertain Mean-Reverting Stock Model with Floating Inter-est Rate, Discrete Dynamics in Nature and Society, 2020; 3764589. Doi: 10.1155/2020/3764589
[10] Mashhadizadeh, M., Dastgir, M., Salahshour, S., Economic Appraisal of Investment Projects in Solar Energy under Uncertainty via Fuzzy Real Option Approach, Advances in Mathematical Finance & Appli-cations, 2018; 3 (4): 29-51. Doi: 10.22034/AMFA.2019.574157.1116
[11] Kahneman, D., Tversky, A., Prospect theory: an analysis of decision under risk. Econometrica, 1979; 47(2): 263-292. Doi: 10.2307/1914185
[12] Cont, R., Tankov, P., Financial Modelling with Jump Processes; Chapman and Hall/CRC Financial Mathematics Series, CRC Press: Boca Raton, FL, USA, 2004.
[13] Liu, B., Uncertainty Theory, seconded, Springer-Verlag, Berlin, 2007.
[14] Liu, B., Fuzzy process, hybrid process and uncertain process, J Uncertain Syst, 2008; 2(1): 3-16. Doi: http://orsc.edu.cn/liu
[15] Chen, X., Liu, B., Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optim Decis mak, 2010; 9(1): 69-81. Doi: 10.1007/s10700-010-9073-2
[16] Yao, K., A type of nonlinear uncertain differential equations with analytic solution, J Uncertain Anal Appl, 2013; 1: 8. Doi: 10.1186/2195-5468-1-8
[17] Yao, K., Chen, X., A numerical method for solving uncertain differential equations, J Intell Fuzzy Syst, 2013; 25: 825-832. Doi: 10.3233/IFS-120688
[18] Liu, B., Some research problems in uncertainty theory. J Uncertain Syst, 2009; 3(1): 3-10. Doi: www.jus.org.uk
[19] Chen, X., Ralescu, D., Liu process and uncertain calculus. J Uncertain Anal Appl, 2013; 1:3.
Doi: 10.1186/2195-5468-1-3
[20] Jafari, H., Farahani, H., Paripour, M., An anticipating Class of Fuzzy Stochastic Differential Equations. Advances in Mathematical Finance & Applications, 2023;8(2), P. 449-462.
Doi: 10.22034/AMFA.2022.1873842.1554
[21] Peng, J., Yao, K., A new option pricing model for stocks in uncertainty markets, Int J Op Res, 2011; 8(2): 18-26.
[22] Yao, K., Uncertain contour process and its application in stock model with floating interest rate, Fuzzy Optim Decis Mak, 2015; 14: 399-424. Doi: 10.1007/s10700-015-9211-y
[23] Yu, X., A stock model with jumps for uncertain markets, Int J Uncert Fuzz Knowl Syst, 2012; 20(3): 421-432. Doi: 10.1142/S0218488512500213
[24] Chen, X., American option pricing formula for uncertain financial market, Int J Op Res, 2011; 8(2): 32-37.
[25] Liu, Y.H., Uncertain random variables: a mixture of uncertainty and randomness ,Soft Comput, 2013; 17(4): 625-634. Doi: 10.1007/s00500-012-0935-0
[26] Yao, K., Uncertain calculus with renewal process, Fuzzy Opt Decis Mak, 2012; 11(3): 285-297. Doi: 10.1007/s10700-012-9132-y
[27] Jia, L., Chen, W., Knock-in options of an uncertain stock model with floating interest rate. Chaos, Solitons and Fractals 141, 2020; 110324. Doi:10.1016/j.chaos.2020.110324.
[28] Gao, R., Liu, K., Li, Z., Liying, L., American Barrier Option Pricing Formulas for Currency Model in Uncertain Environment. J Syst Sci Complex, 2020; 35: 283-312. Doi: 10.1007/s11424-021-0039-y
[29] Liu, B., Uncertainty theory: a branch of mathematics for modeling human uncertainty, Springer, Berlin 2010.