Binary option pricing formulas for fuzzy financial market based on the exponential Ornstein-Uhlenbeck model

Document Type : Research Paper


1 Hebei University

2 College of Mathematics and Information Science, Hebei University, Baoding , China


Binary option is an exotic option which is popular in Over the Counter market for hedging and speculation. According to their different payoff, there are two types of binary options, that is, cash-or-nothing and asset-or-nothing option. This paper investigates the fuzzy financial market based on the exponential Ornstein-Uhlenbeck model and derives binary option pricing formulas. In order to better understand the mathematical properties of these formulas, we give a few numerical examples and some figures to illustrate the changes of binary option price with different parameters when others are fixed.


Main Subjects

[1] F. Black, M. Scholes, The pricing of option and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654.
[2] A. Carlos, V. Mejia, Calibration of the expomemtial ornstein-uhlenbeck process when spot prices are visible through the maximum log-likelihood method. Example with gold prices, Advances in Di erence Equations, 2018 (2018), 269.
[3] X. Chen, Z. Qin, A new existence and unqueness theorem for fuzzy di erential equation, International Journal of Fuzzy Systems, 13(2) (2011), 148-151.
[4] Y. Cheng, C. You, Convergence of numerical methods for fuzzy di erential equations, Journal of Intelligent and Fuzzy Systems, 38(4) (2020), 5257-5266.
[5] L. Dai, Z. Fu, Z. Huang, Option pricing formulas for uncertain  nancial market based on the exponential Ornstein-Uhlenbeck model, Journal of Intelligent Manufacturing, 28(3) (2017), 597-604.
[6] J. Gao, Credibilistic option pricing: A new model, Journal of Uncertain Systems, 2(4) (2008), 243-247.
[7] L. Ji, C. You, Milstein method for solving fuzzy di erential equation, Iranian Journal of Fuzzy Systems, 18(3) (2021), 129-141.
[8] L. Kang, S. Chen, Binary option pricing with jump based on mixed fractional Brownian motion, Advances in Applied Mathematics, 8(5) (2019), 883-891.
[9] W. Li, J. Li, K. Qiao, The martingale pricing for convertible bond under stochastic interest and exponential Ornstein-Uhlenbeck process model, Journal of Quantitative Economics, 28(4) (2011), 71-74.
[10] B. Liu, Uncertainty theory: An introduction to its axiomatic foundations, Berlin, Springer-Verlag, 2004.
[11] B. Liu, Uncertainty theory, 2nd ed., Berlin, Springer-Verlag, 2007.
[12] B. Liu, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2(1) (2008), 3-16.
[13] B. Liu, Y. K. Liu, Expected value of fuzzy variable and fuzzy expected value models, IEEE Transactions on Fuzzy Systems, 10(4) (2002), 445-450.
[14] R. Merton, The theory of rational option pricing, Bell Journal of Economics and Management Science, 4(1) (1973), 141-183.
[15] J. Peng, A general stock model for fuzzy markets, Journal of Uncertain Systems, 2(4) (2008), 248-254.
[16] Z. Qin, X. Gao, Fracrional Liu process with application to  nance, Mathematical and Computer Modelling, 50(9-10) (2009), 1538-1543.
[17] X. Qin, X. Lin, Q. Shang, Fuzzy pricing of binary option based on the long memory property of  nancial markets,  Journal of Intelligent and Fuzzy Systems, 38 (2020), 4889-4900.
[18] E. Schwartz, The stochastic behavior of commodity prices: Implications for valuation and hedging, Journal of Finance, 52(3) (1997), 923-973.
[19] A. Thavaneswaran, S. Appadoo, J. Frank, Binary option pricing using fuzzy numbers, Applied Mathematics Letters, 26 (2013), 65-72.
[20] H. Wang, Promotion of the pricing formula for binary option, Journal of Sanming University, 33(4) (2016), 1-5.
[21] Y. Wu, J. He, The pricing model of binary option and its solution, Journal of Industrial Engineering, 16(4) (2002), 108-110.
[22] C. You, L. Bo, Option pricing formulas for generalized fuzzy stock model, Journal of Industrial and Management optimization, 16(1) (2020), 387-396.
[23] C. You, L. Bo, Option pricing based on a type of fuzzy process, Journal of Ambient Intelligence and Humanized Computing, 13(8) (2022), 3771-3785.
[24] C. You, Y. Hao, Fuzzy Euler approximation and its local convergence, Journal of Computational and Applied Mathematics, 343 (2018), 55-61.
[25] C. You, Y. Hao, Numerical solution of fuzzy di erential equation based on Taylor expansion, Journal of Hebei University (Nature Science), 38(2) (2018), 113-118.
[26] C. You, W. Wang, H. Huo, Existence and uniqueness theorems for fuzzy di erential equations, Journal of Uncertain Systems, 7(4) (2013), 303-315.
[27] L. Zadeh, Fuzzy sets, Information and Control, 8(3) (1965), 338-353.
[28] L. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, 1 (1978), 3-28.
[29] Y. Zhang, R.Wang, B. Song, A study on pricing of Kanzhangbao income documeny product based on binary options, Advances in Applied Mathematics, 7(7) (2018), 934-946.
[30] Y. Zhang, C. You, Option pricing formula for a new stock model, Journal of Applied Mathematics Progress, 7(10) (2018), 1225-1232.
[31] P. Zhao, Pricing of power European options based on the exponential Ornstein-Uhlenback process, Journal of Guizhou Normal University (Natural Sciences), 32(1) (2014), 44-47.