American Institute of Mathematical Sciences

Advanced
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  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.

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

[1]

Cuilian You, Yangyang Hao. Stability in mean for fuzzy differential equation. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1375-1385. doi: 10.3934/jimo.2018099
[2]

Natalia Skripnik. Averaging of fuzzy integral equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1999-2010. doi: 10.3934/dcdsb.2017118
[3]

Seiyed Hadi Abtahi, Hamidreza Rahimi, Maryam Mosleh. Solving fuzzy volterra-fredholm integral equation by fuzzy artificial neural network. Mathematical Foundations of Computing, 2021, 4 (3) : 209-219. doi: 10.3934/mfc.2021013
[4]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3529-3539. doi: 10.3934/dcdss.2020432
[5]

Andrej V. Plotnikov, Tatyana A. Komleva, Liliya I. Plotnikova. The averaging of fuzzy hyperbolic differential inclusions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1987-1998. doi: 10.3934/dcdsb.2017117
[6]

Lisha Wang, Huaming Song, Ding Zhang, Hui Yang. Pricing decisions for complementary products in a fuzzy dual-channel supply chain. Journal of Industrial & Management Optimization, 2019, 15 (1) : 343-364. doi: 10.3934/jimo.2018046
[7]

Muslim Malik, Anjali Rose, Anil Kumar. Controllability of Sobolev type fuzzy differential equation with non-instantaneous impulsive condition. Discrete & Continuous Dynamical Systems - S, 2022, 15 (2) : 387-407. doi: 10.3934/dcdss.2021068
[8]

Editorial Office. Retraction: Xiao-Qian Jiang and Lun-Chuan Zhang, A pricing option approach based on backward stochastic differential equation theory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 969-969. doi: 10.3934/dcdss.2019065
[9]

Dariush Mohamadi Zanjirani, Majid Esmaelian. An integrated approach based on Fuzzy Inference System for scheduling and process planning through multiple objectives. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1235-1259. doi: 10.3934/jimo.2018202
[10]

Junjie Peng, Ning Chen, Jiayang Dai, Weihua Gui. A goethite process modeling method by Asynchronous Fuzzy Cognitive Network based on an improved constrained chicken swarm optimization algorithm. Journal of Industrial & Management Optimization, 2021, 17 (3) : 1269-1287. doi: 10.3934/jimo.2020021
[11]

Feyza Gürbüz, Panos M. Pardalos. A decision making process application for the slurry production in ceramics via fuzzy cluster and data mining. Journal of Industrial & Management Optimization, 2012, 8 (2) : 285-297. doi: 10.3934/jimo.2012.8.285
[12]

Huiqin Zhang, JinChun Wang, Meng Wang, Xudong Chen. Integration of cuckoo search and fuzzy support vector machine for intelligent diagnosis of production process quality. Journal of Industrial & Management Optimization, 2022, 18 (1) : 195-217. doi: 10.3934/jimo.2020150
[13]

Jiaquan Zhan, Fanyong Meng. Cores and optimal fuzzy communication structures of fuzzy games. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1187-1198. doi: 10.3934/dcdss.2019082
[14]

Xiaodong Liu, Wanquan Liu. The framework of axiomatics fuzzy sets based fuzzy classifiers. Journal of Industrial & Management Optimization, 2008, 4 (3) : 581-609. doi: 10.3934/jimo.2008.4.581
[15]

Juan J. Nieto, M. Victoria Otero-Espinar, Rosana Rodríguez-López. Dynamics of the fuzzy logistic family. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 699-717. doi: 10.3934/dcdsb.2010.14.699
[16]

Purnima Pandit. Fuzzy system of linear equations. Conference Publications, 2013, 2013 (special) : 619-627. doi: 10.3934/proc.2013.2013.619
[17]

İsmail Özcan, Sirma Zeynep Alparslan Gök. On cooperative fuzzy bubbly games. Journal of Dynamics & Games, 2021, 8 (3) : 267-275. doi: 10.3934/jdg.2021010
[18]

Harish Garg. Some robust improved geometric aggregation operators under interval-valued intuitionistic fuzzy environment for multi-criteria decision-making process. Journal of Industrial & Management Optimization, 2018, 14 (1) : 283-308. doi: 10.3934/jimo.2017047
[19]

Erik Kropat, Gerhard Wilhelm Weber. Fuzzy target-environment networks and fuzzy-regression approaches. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 135-155. doi: 10.3934/naco.2018008
[20]

Wei Wang, Xiao-Long Xin. On fuzzy filters of Heyting-algebras. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1611-1619. doi: 10.3934/dcdss.2011.4.1611

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (248)
  • HTML views (991)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy