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doi: 10.3934/jimo.2020090

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

 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  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.

Citation: Miao Tian, Xiangfeng Yang, Yi Zhang. Lookback option pricing problem of mean-reverting stock model in uncertain environment. Journal of Industrial & Management Optimization, 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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [4] Y. Gao, X. Yang and Z. Fu, Lookback option pricing problem of uncertain exponential Ornstein-Uhlenbeck model, Soft Computing, 22 (2018), 5647-5654.   Google Scholar [5] X. Ji and J. Zhou, Option pricing for an uncertain stock model with jumps, Soft Computing, 19 (2015), 3323-3329.   Google Scholar [6] A. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer-Verlag, Berlin-New York, 1973.  Google Scholar [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.  Google Scholar [8] B. Liu, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2 (2008), 3-16.   Google Scholar [9] B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems, 3 (2009), 3-10.   Google Scholar [10] B. Liu, Toward uncertain finance theory, J. Uncertain. Anal. Appl., 1 (2013). Google Scholar [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.   Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] X. Yang and J. Gao, Uncertain differential games with application to capitalism, J. Uncertain. Anal. Appl., 1 (2013), Art. 17. Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [20] K. Yao, No-arbitrage determinant theorems on mean-reverting stock model in uncertain market, Knowledge Based Systems, 35 (2012), 259-263.   Google Scholar [21] K. Yao, Extreme values and integral of solution of uncertain differential equation, J. Uncertain. Anal. Appl., 1 (2013), Art. 2. Google Scholar [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.  Google Scholar [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.  Google Scholar [24] K. Yao, Uncertain Differential Equations, Springer Uncertainty Research, Springer-Verlag, Berlin, 2016. doi: 10.1007/978-3-662-52729-0.  Google Scholar [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.  Google Scholar [26] Z. Zhang and W. Liu, Geometric average asian option pricing for uncertain financial market, Journal of Uncertain Systems, 8 (2014), 317-320.   Google Scholar [27] Y. Zhu, Uncertain optimal control with application to a portfolio selection model, Cybernetics and Systems, 41 (2010), 535-547.   Google Scholar

show all references

##### References:
 [1] X. Chen, Ametican option pricing formula for uncertain financial market, Int. J. Oper. Res. (Taichung), 8 (2011), 27-32.   Google Scholar [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.  Google Scholar [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.  Google Scholar [4] Y. Gao, X. Yang and Z. Fu, Lookback option pricing problem of uncertain exponential Ornstein-Uhlenbeck model, Soft Computing, 22 (2018), 5647-5654.   Google Scholar [5] X. Ji and J. Zhou, Option pricing for an uncertain stock model with jumps, Soft Computing, 19 (2015), 3323-3329.   Google Scholar [6] A. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer-Verlag, Berlin-New York, 1973.  Google Scholar [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.  Google Scholar [8] B. Liu, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2 (2008), 3-16.   Google Scholar [9] B. Liu, Some research problems in uncertainty theory, Journal of Uncertain Systems, 3 (2009), 3-10.   Google Scholar [10] B. Liu, Toward uncertain finance theory, J. Uncertain. Anal. Appl., 1 (2013). Google Scholar [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.   Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] X. Yang and J. Gao, Uncertain differential games with application to capitalism, J. Uncertain. Anal. Appl., 1 (2013), Art. 17. Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [20] K. Yao, No-arbitrage determinant theorems on mean-reverting stock model in uncertain market, Knowledge Based Systems, 35 (2012), 259-263.   Google Scholar [21] K. Yao, Extreme values and integral of solution of uncertain differential equation, J. Uncertain. Anal. Appl., 1 (2013), Art. 2. Google Scholar [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.  Google Scholar [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.  Google Scholar [24] K. Yao, Uncertain Differential Equations, Springer Uncertainty Research, Springer-Verlag, Berlin, 2016. doi: 10.1007/978-3-662-52729-0.  Google Scholar [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.  Google Scholar [26] Z. Zhang and W. Liu, Geometric average asian option pricing for uncertain financial market, Journal of Uncertain Systems, 8 (2014), 317-320.   Google Scholar [27] Y. Zhu, Uncertain optimal control with application to a portfolio selection model, Cybernetics and Systems, 41 (2010), 535-547.   Google Scholar
Lookback call option price $f_{call}$ with different parameters
Lookback put option price $f_{put}$ with different parameters
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