# American Institute of Mathematical Sciences

January  2020, 16(1): 387-396. doi: 10.3934/jimo.2018158

## Option pricing formulas for generalized fuzzy stock model

 College of Mathematics and Information Science, Hebei University, Baoding 071002, China

* Corresponding author: Cuilian You

Received  June 2018 Revised  June 2018 Published  January 2020 Early access  September 2018

Fund Project: The first author is supported by NSFC grant (No.61773150) and Key Lab. of Machine Learning and Computational Intelligence, College of Mathematics and Information Science, Hebei University, Baoding, 071002, China.

Fuzzy stock model has been studied by many scholars in recent years, in which option pricing problem is the most important part. In this paper, we studied option pricing for a new generalized fuzzy stock model. Based on credibility theory, pricing formulas of European option and American option were obtained.

Citation: Cuilian You, Le Bo. Option pricing formulas for generalized fuzzy stock model. Journal of Industrial and Management Optimization, 2020, 16 (1) : 387-396. doi: 10.3934/jimo.2018158
##### References:
 [1] F. Black and M. Scholes, The pricing of option and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062. [2] X. Chen and Z. Qin, A new existence and uniqueness theorem for fuzzy differential equation, International Journal of Fuzzy Systems, 13 (2011), 148-151. [3] W. Dai, Reflection principle of Liu process, 2007. Available from: http://orsc.edu.cn/process\/071110.pdf. [4] W. Dai, Lipschitz continuity of Liu process, 2008. Available from: http://orsc.edu.cn/process\/080831.pdf. [5] Z. Ding, M. Ma and A. Kandel, Exsitence of the solutions of fuzzy differential equations with parameters, Information Sciences, 99 (1999), 205-217.  doi: 10.1016/S0020-0255(96)00279-4. [6] J. Gao and X. Gao, A new stock model for credibilistic option pricing, Journal of Uncertain Systems, 2 (2008), 243-247. [7] X. Gao and X. Chen, Option pricing formula for generalized stock models, 2008. Available from: http://orsc.edu.cn/process/080317.pdf. [8] H. Hu, Power option pricing model for stock price follow geometric fractional Liu process, Journal of Henan Normal University (Natural Science Edition), 41 (2013), 1-5. [9] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987), 301-317.  doi: 10.1016/0165-0114(87)90029-7. [10] B. Liu, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2 (2008), 3-16. [11] B. Liu and Y. K. Liu, Expected value of fuzzy variable and fuzzy expected value models, IEEE Transactions on Fuzzy Systems, 10 (2002), 445-450. [12] R. Merton, Theory of rational option pricing, Bell Journal of Economics and Management Science, 4 (1973), 141-183.  doi: 10.2307/3003143. [13] J. Peng, A general stock model for fuzzy markets, Journal of Uncertain Systems, 2 (2008), 248-254. [14] Z. Qin and X. Li, Option pricing formula for fuzzy financial market, Journal of Uncertain System, 2 (2008), 17-21. [15] Z. Qin and X. Li, Fuzzy calculus for finance, 2008. Available from: http://orsc.edu.cn/process\/fc.pdf. [16] C. You, H. Huo and W. Wang, Multi-dimensional Liu process, differential and integral, East Asian Mathematical Journal, 29 (2013), 13-22.  doi: 10.7858/eamj.2013.002. [17] C. You, H. Ma and H. Huo, A new kind of generalized fuzzy integrals, Journal of Nonlinear Science and Applications, 9 (2016), 1396-1401.  doi: 10.22436/jnsa.009.03.63. [18] C. You and G. Wang, Properties of a new kind of fuzzy integral, Journal of Hebei University (Natural Science Edition), 31 (2011), 337-340. [19] C. You, W. Wang and H. Huo, Existence and unqiueness theorems for fuzzy differential equation, Journal of Uncertain Systems, 7 (2013), 303-315. [20] L. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.

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##### References:
 [1] F. Black and M. Scholes, The pricing of option and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062. [2] X. Chen and Z. Qin, A new existence and uniqueness theorem for fuzzy differential equation, International Journal of Fuzzy Systems, 13 (2011), 148-151. [3] W. Dai, Reflection principle of Liu process, 2007. Available from: http://orsc.edu.cn/process\/071110.pdf. [4] W. Dai, Lipschitz continuity of Liu process, 2008. Available from: http://orsc.edu.cn/process\/080831.pdf. [5] Z. Ding, M. Ma and A. Kandel, Exsitence of the solutions of fuzzy differential equations with parameters, Information Sciences, 99 (1999), 205-217.  doi: 10.1016/S0020-0255(96)00279-4. [6] J. Gao and X. Gao, A new stock model for credibilistic option pricing, Journal of Uncertain Systems, 2 (2008), 243-247. [7] X. Gao and X. Chen, Option pricing formula for generalized stock models, 2008. Available from: http://orsc.edu.cn/process/080317.pdf. [8] H. Hu, Power option pricing model for stock price follow geometric fractional Liu process, Journal of Henan Normal University (Natural Science Edition), 41 (2013), 1-5. [9] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987), 301-317.  doi: 10.1016/0165-0114(87)90029-7. [10] B. Liu, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2 (2008), 3-16. [11] B. Liu and Y. K. Liu, Expected value of fuzzy variable and fuzzy expected value models, IEEE Transactions on Fuzzy Systems, 10 (2002), 445-450. [12] R. Merton, Theory of rational option pricing, Bell Journal of Economics and Management Science, 4 (1973), 141-183.  doi: 10.2307/3003143. [13] J. Peng, A general stock model for fuzzy markets, Journal of Uncertain Systems, 2 (2008), 248-254. [14] Z. Qin and X. Li, Option pricing formula for fuzzy financial market, Journal of Uncertain System, 2 (2008), 17-21. [15] Z. Qin and X. Li, Fuzzy calculus for finance, 2008. Available from: http://orsc.edu.cn/process\/fc.pdf. [16] C. You, H. Huo and W. Wang, Multi-dimensional Liu process, differential and integral, East Asian Mathematical Journal, 29 (2013), 13-22.  doi: 10.7858/eamj.2013.002. [17] C. You, H. Ma and H. Huo, A new kind of generalized fuzzy integrals, Journal of Nonlinear Science and Applications, 9 (2016), 1396-1401.  doi: 10.22436/jnsa.009.03.63. [18] C. You and G. Wang, Properties of a new kind of fuzzy integral, Journal of Hebei University (Natural Science Edition), 31 (2011), 337-340. [19] C. You, W. Wang and H. Huo, Existence and unqiueness theorems for fuzzy differential equation, Journal of Uncertain Systems, 7 (2013), 303-315. [20] L. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.
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