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September  2021, 17(5): 2703-2714. doi: 10.3934/jimo.2020090

Lookback option pricing problem of mean-reverting stock model in uncertain environment

Miao Tian 1, , Xiangfeng Yang 2,, and  Yi Zhang 3,

1. 

School of Mathematics, Renmin University of China, Beijing 100872, China
2. 

School of Information Technology & Management, University of International, Business & Economics, Beijing 100029, China
3. 

School of Economics & Management, Beijing University of Chemical, Technology, Beijing 100029, China

* Corresponding author: Xiangfeng Yang

Received  February 2019 Revised  January 2020 Published  September 2021 Early access  May 2020

Fund Project: The second author is supported by the Program for Young Excellent Talents in UIBE (No.18YQ06).

A lookback option is an exotic option that allows investors to look back at the underlying prices occurring over the life of the option, and to exercise the right at assets optimal point. This paper proposes a mean-reverting stock model to investigate the lookback option in an uncertain environment. The lookback call and put options pricing formulas of the stock model are derived, and the corresponding numerical algorithms are designed to compute the prices of these two options.

Keywords: Uncertainty theory, uncertain differential equation, mean-reverting stock model, lookback option.
Mathematics Subject Classification: Primary: 91G99; Secondary: 91G80.
Citation: Miao Tian, Xiangfeng Yang, Yi Zhang. Lookback option pricing problem of mean-reverting stock model in uncertain environment. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2703-2714. doi: 10.3934/jimo.2020090
References:
[1]

X. Chen, Ametican option pricing formula for uncertain financial market, Int. J. Oper. Res. (Taichung), 8 (2011), 27-32. 

[2]

X. Chen and B. Liu, Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optim. Decis. Mak., 9 (2010), 69-81.  doi: 10.1007/s10700-010-9073-2.

[3]

X. Chen, Y. Liu and D. A. Ralesu, Uncertain stock model with periodic dividends, Fuzzy Optim. Decis. Mak., 12 (2013), 111-123.  doi: 10.1007/s10700-012-9141-x.

[4]

Y. Gao, X. Yang and Z. Fu, Lookback option pricing problem of uncertain exponential Ornstein-Uhlenbeck model, Soft Computing, 22 (2018), 5647-5654. 

[5]

X. Ji and J. Zhou, Option pricing for an uncertain stock model with jumps, Soft Computing, 19 (2015), 3323-3329. 

[6]

A. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer-Verlag, Berlin-New York, 1973.

[7]

B. Liu, Uncertainty theory. An introduction to its axiomatic foundations, in Studies in Fuzziness and Soft Computing, 154, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-540-39987-2.

[8]

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

[9]

B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems, 3 (2009), 3-10. 

[10]

B. Liu, Toward uncertain finance theory, J. Uncertain. Anal. Appl., 1 (2013).

[11]

Y. Liu, X. Chen and D. A. Ralescu, Uncertain currency model and currency option pricing, International Journal of Intelligent Systems, 30 (2015), 40-51. 

[12]

J. Peng and K. Yao, A new option pricing model for stocks in uncertainty markets, Int. J. Oper. Res. (Taichung), 8 (2011), 18-26. 

[13]

Y. Sun and T. Su, Mean-reverting stock model with floating interest rate in uncertain environment, Fuzzy Optim. Decis. Mak., 16 (2017), 235-255.  doi: 10.1007/s10700-016-9247-7.

[14]

M. Tian, X. Yang and Y. Zhang, Barrier option pricing problem of mean-reverting stock model in uncertain environment, Math. Comput. Simulation, 166 (2019), 126-143.  doi: 10.1016/j.matcom.2019.04.009.

[15]

M. Tian, X. Yang and S. Kar, Equity warrants pricing problem of mean-reverting model in uncertain environment, Phys. A, 531 (2019), 9 pp. doi: 10.1016/j.physa.2019.121593.

[16]

X. Yang and J. Gao, Uncertain differential games with application to capitalism, J. Uncertain. Anal. Appl., 1 (2013), Art. 17.

[17]

X. Yang and J. Gao, Linear-quadratic uncertain differential games with application to resource extraction problem, IEEE Transactions on Fuzzy Systems, 24 (2016), 819-826. 

[18]

X. Yang and K. Yao, Uncertain partial differential equation with application to heat conduction, Fuzzy Optim. Decis. Mak., 16 (2017), 379-403.  doi: 10.1007/s10700-016-9253-9.

[19]

X. Yang, Z. Zhang and X. Gao, Asian-barrier option pricing formulas of uncertain financial market, Chaos Solitons Fractals, 123 (2019), 79-86.  doi: 10.1016/j.chaos.2019.03.037.

[20]

K. Yao, No-arbitrage determinant theorems on mean-reverting stock model in uncertain market, Knowledge Based Systems, 35 (2012), 259-263. 

[21]

K. Yao, Extreme values and integral of solution of uncertain differential equation, J. Uncertain. Anal. Appl., 1 (2013), Art. 2.

[22]

K. Yao and X. Chen, A numerical method for solving uncertain differential equations, J. Intell. Fuzzy Systems, 25 (2013), 825-832.  doi: 10.3233/IFS-120688.

[23]

K. Yao, Uncertain contour process and its application in stock model with floating interest rate, Fuzzy Optim. Decis. Mak., 14 (2015), 399-424.  doi: 10.1007/s10700-015-9211-y.

[24]

K. Yao, Uncertain Differential Equations, Springer Uncertainty Research, Springer-Verlag, Berlin, 2016. doi: 10.1007/978-3-662-52729-0.

[25]

X. Yu, A stock model with jumps for uncertain markets, Internat. J. Uncertain. Fuzziness Knowledge-Based Systems, 20 (2012), 421-432.  doi: 10.1142/S0218488512500213.

[26]

Z. Zhang and W. Liu, Geometric average asian option pricing for uncertain financial market, Journal of Uncertain Systems, 8 (2014), 317-320. 

[27]

Y. Zhu, Uncertain optimal control with application to a portfolio selection model, Cybernetics and Systems, 41 (2010), 535-547. 

show all references

References:
[1]

X. Chen, Ametican option pricing formula for uncertain financial market, Int. J. Oper. Res. (Taichung), 8 (2011), 27-32. 

[2]

X. Chen and B. Liu, Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optim. Decis. Mak., 9 (2010), 69-81.  doi: 10.1007/s10700-010-9073-2.

[3]

X. Chen, Y. Liu and D. A. Ralesu, Uncertain stock model with periodic dividends, Fuzzy Optim. Decis. Mak., 12 (2013), 111-123.  doi: 10.1007/s10700-012-9141-x.

[4]

Y. Gao, X. Yang and Z. Fu, Lookback option pricing problem of uncertain exponential Ornstein-Uhlenbeck model, Soft Computing, 22 (2018), 5647-5654. 

[5]

X. Ji and J. Zhou, Option pricing for an uncertain stock model with jumps, Soft Computing, 19 (2015), 3323-3329. 

[6]

A. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer-Verlag, Berlin-New York, 1973.

[7]

B. Liu, Uncertainty theory. An introduction to its axiomatic foundations, in Studies in Fuzziness and Soft Computing, 154, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-540-39987-2.

[8]

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

[9]

B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems, 3 (2009), 3-10. 

[10]

B. Liu, Toward uncertain finance theory, J. Uncertain. Anal. Appl., 1 (2013).

[11]

Y. Liu, X. Chen and D. A. Ralescu, Uncertain currency model and currency option pricing, International Journal of Intelligent Systems, 30 (2015), 40-51. 

[12]

J. Peng and K. Yao, A new option pricing model for stocks in uncertainty markets, Int. J. Oper. Res. (Taichung), 8 (2011), 18-26. 

[13]

Y. Sun and T. Su, Mean-reverting stock model with floating interest rate in uncertain environment, Fuzzy Optim. Decis. Mak., 16 (2017), 235-255.  doi: 10.1007/s10700-016-9247-7.

[14]

M. Tian, X. Yang and Y. Zhang, Barrier option pricing problem of mean-reverting stock model in uncertain environment, Math. Comput. Simulation, 166 (2019), 126-143.  doi: 10.1016/j.matcom.2019.04.009.

[15]

M. Tian, X. Yang and S. Kar, Equity warrants pricing problem of mean-reverting model in uncertain environment, Phys. A, 531 (2019), 9 pp. doi: 10.1016/j.physa.2019.121593.

[16]

X. Yang and J. Gao, Uncertain differential games with application to capitalism, J. Uncertain. Anal. Appl., 1 (2013), Art. 17.

[17]

X. Yang and J. Gao, Linear-quadratic uncertain differential games with application to resource extraction problem, IEEE Transactions on Fuzzy Systems, 24 (2016), 819-826. 

[18]

X. Yang and K. Yao, Uncertain partial differential equation with application to heat conduction, Fuzzy Optim. Decis. Mak., 16 (2017), 379-403.  doi: 10.1007/s10700-016-9253-9.

[19]

X. Yang, Z. Zhang and X. Gao, Asian-barrier option pricing formulas of uncertain financial market, Chaos Solitons Fractals, 123 (2019), 79-86.  doi: 10.1016/j.chaos.2019.03.037.

[20]

K. Yao, No-arbitrage determinant theorems on mean-reverting stock model in uncertain market, Knowledge Based Systems, 35 (2012), 259-263. 

[21]

K. Yao, Extreme values and integral of solution of uncertain differential equation, J. Uncertain. Anal. Appl., 1 (2013), Art. 2.

[22]

K. Yao and X. Chen, A numerical method for solving uncertain differential equations, J. Intell. Fuzzy Systems, 25 (2013), 825-832.  doi: 10.3233/IFS-120688.

[23]

K. Yao, Uncertain contour process and its application in stock model with floating interest rate, Fuzzy Optim. Decis. Mak., 14 (2015), 399-424.  doi: 10.1007/s10700-015-9211-y.

[24]

K. Yao, Uncertain Differential Equations, Springer Uncertainty Research, Springer-Verlag, Berlin, 2016. doi: 10.1007/978-3-662-52729-0.

[25]

X. Yu, A stock model with jumps for uncertain markets, Internat. J. Uncertain. Fuzziness Knowledge-Based Systems, 20 (2012), 421-432.  doi: 10.1142/S0218488512500213.

[26]

Z. Zhang and W. Liu, Geometric average asian option pricing for uncertain financial market, Journal of Uncertain Systems, 8 (2014), 317-320. 

[27]

Y. Zhu, Uncertain optimal control with application to a portfolio selection model, Cybernetics and Systems, 41 (2010), 535-547. 

Figure 1.  Lookback call option price $ f_{call} $ with different parameters
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Figure 2.  Lookback put option price $ f_{put} $ with different parameters
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