American Institute of Mathematical Sciences

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

Option pricing formulas for generalized fuzzy stock model

Cuilian You , and  Le Bo

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

* Corresponding author: Cuilian You

Received  June 2018 Revised  June 2018 Published  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.

Keywords: Liu process, fuzzy integral, fuzzy differential equation, option pricing.
Mathematics Subject Classification: Primary: 91B25, 91B26; Secondary: 91B24.
Citation: Cuilian You, Le Bo. Option pricing formulas for generalized fuzzy stock model. Journal of Industrial & 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.  Google Scholar

[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.   Google Scholar

[3]

W. Dai, Reflection principle of Liu process, 2007. Available from: http://orsc.edu.cn/process\/071110.pdf. Google Scholar

[4]

W. Dai, Lipschitz continuity of Liu process, 2008. Available from: http://orsc.edu.cn/process\/080831.pdf. Google Scholar

[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.  Google Scholar

[6]

J. Gao and X. Gao, A new stock model for credibilistic option pricing, Journal of Uncertain Systems, 2 (2008), 243-247.   Google Scholar

[7]

X. Gao and X. Chen, Option pricing formula for generalized stock models, 2008. Available from: http://orsc.edu.cn/process/080317.pdf. Google Scholar

[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.   Google Scholar

[9]

O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987), 301-317.  doi: 10.1016/0165-0114(87)90029-7.  Google Scholar

[10]

B. Liu, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2 (2008), 3-16.   Google Scholar

[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.   Google Scholar

[12]

R. Merton, Theory of rational option pricing, Bell Journal of Economics and Management Science, 4 (1973), 141-183.  doi: 10.2307/3003143.  Google Scholar

[13]

J. Peng, A general stock model for fuzzy markets, Journal of Uncertain Systems, 2 (2008), 248-254.   Google Scholar

[14]

Z. Qin and X. Li, Option pricing formula for fuzzy financial market, Journal of Uncertain System, 2 (2008), 17-21.   Google Scholar

[15]

Z. Qin and X. Li, Fuzzy calculus for finance, 2008. Available from: http://orsc.edu.cn/process\/fc.pdf. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[19]

C. You, W. Wang and H. Huo, Existence and unqiueness theorems for fuzzy differential equation, Journal of Uncertain Systems, 7 (2013), 303-315.   Google Scholar

[20]

L. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.  Google Scholar

show all references

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.  Google Scholar

[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.   Google Scholar

[3]

W. Dai, Reflection principle of Liu process, 2007. Available from: http://orsc.edu.cn/process\/071110.pdf. Google Scholar

[4]

W. Dai, Lipschitz continuity of Liu process, 2008. Available from: http://orsc.edu.cn/process\/080831.pdf. Google Scholar

[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.  Google Scholar

[6]

J. Gao and X. Gao, A new stock model for credibilistic option pricing, Journal of Uncertain Systems, 2 (2008), 243-247.   Google Scholar

[7]

X. Gao and X. Chen, Option pricing formula for generalized stock models, 2008. Available from: http://orsc.edu.cn/process/080317.pdf. Google Scholar

[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.   Google Scholar

[9]

O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987), 301-317.  doi: 10.1016/0165-0114(87)90029-7.  Google Scholar

[10]

B. Liu, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2 (2008), 3-16.   Google Scholar

[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.   Google Scholar

[12]

R. Merton, Theory of rational option pricing, Bell Journal of Economics and Management Science, 4 (1973), 141-183.  doi: 10.2307/3003143.  Google Scholar

[13]

J. Peng, A general stock model for fuzzy markets, Journal of Uncertain Systems, 2 (2008), 248-254.   Google Scholar

[14]

Z. Qin and X. Li, Option pricing formula for fuzzy financial market, Journal of Uncertain System, 2 (2008), 17-21.   Google Scholar

[15]

Z. Qin and X. Li, Fuzzy calculus for finance, 2008. Available from: http://orsc.edu.cn/process\/fc.pdf. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[19]

C. You, W. Wang and H. Huo, Existence and unqiueness theorems for fuzzy differential equation, Journal of Uncertain Systems, 7 (2013), 303-315.   Google Scholar

[20]

L. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.  Google Scholar

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