Paper:

# Lookback Option Pricing Problems of Uncertain Mean-Reverting Stock Model

## Zhaopeng Liu^{†}

School of Mathematics and Statistics, Suzhou University

East Campus of Suzhou University, Education Park, Suzhou, Anhui 234000, China

^{†}Corresponding author

A lookback option is a path-dependent option, offering a payoff that depends on the maximum or minimum value of the underlying asset price over the life of the option. This paper presents a new mean-reverting uncertain stock model with a floating interest rate to study the lookback option price, in which the processing of the interest rate is assumed to be the uncertain counterpart of the Cox–Ingersoll–Ross (CIR) model. The CIR model can reflect the fluctuations in the interest rate and ensure that such rate is positive. Subsequently, lookback option pricing formulas are derived through the α-path method and some mathematical properties of the uncertain option pricing formulas are discussed. In addition, several numerical examples are given to illustrate the effectiveness of the proposed model.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.25, No.5, pp. 539-545, 2021.

- [1] M. B. Goldman, H. B. Sosin, and M. A. Gatto, “Path dependent options: Buy at the low, sell at the high,” The J. of Finance, Vol.34, No.5, pp. 1111-1127, 1979.
- [2] R. C. Heynen and H. M. Kat, “Lookback options with discrete and partial monitoring of the underlying price,” Applied Mathematical Finance, Vol.2, No.4, pp. 273-284, 1995.
- [3] H. Y. Wong and Y. K. Kwok, “Sub-replication and replenishing premium: Efficient pricing of multi-state lookbacks,” Review of Derivatives Research, Vol.6, pp. 83-106, 2003.
- [4] M. Dai, H. Y. Wong, and Y. K. Kwok, “Quanto lookback options,” Mathematical Finance, Vol.14, No.3, pp. 445-467, 2004.
- [5] B. Liu, “Uncertainty Theory,” Springer, 2007.
- [6] B. Liu, “Some research problems in uncertainty theory,” J. of Uncertain Systems, Vol.3, No.1, pp. 3-10, 2009.
- [7] X. Chen, “American option pricing formula for uncertain financial market,” Int. J. of Operation Research, Vol.8, No.2, pp. 32-37, 2011.
- [8] J. Peng and K. Yao, “A new option pricing model for stocks in uncertainty markets,” Int. J. of Operation Research, Vol.8, No.2, pp. 18-26, 2011.
- [9] K. Yao, “No-arbitrage determinant theorems on mean reverting stock model in uncertain market,” Knowledge-Based Systems, Vol.35, pp. 259-263, 2012.
- [10] Z. Zhang and W. Liu, “Geometric average Asian option pricing for uncertain financial market,” J. of Uncertain Systems, Vol.8, No.4, pp. 317-320, 2014.
- [11] Y. Gao, X. Yang, and Z. Fu, “Lookback option pricing problem of uncertain exponential Ornstein–Uhlenbeck model,” Soft Computing, Vol.22, No.17, pp. 5647-5654, 2018.
- [12] W. Wang and P. Chen, “Pricing Asian Options in an Uncertain Stock Model with floating interest rate,” Int. J. for Uncertainty Quantification, Vol.8, No.6, pp. 543-557, 2018.
- [13] Z. Zhang, H. Ke, and W. Liu, “Lookback options pricing for uncertain financial market,” Soft Computing, Vol.23, pp. 5537-55463, 2019.
- [14] Z. Liu, “Option pricing formulas in a new uncertain mean- reverting stock model with floating interest rate,” Discrete Dynamics in Nature and Society, Vol.2020, Article ID: 3764589, 2020.
- [15] Y. Sun, T. Su, “Mean-reverting stock model with floating interest rate in uncertain environment,” Fuzzy Optimization and Decision Making, Vol.16, pp. 235-255, 2017.
- [16] B. Liu, “Uncertainty Theory: A branch of mathematics for modeling human uncertainty,” Springer, 2010.
- [17] K. Yao and X. Chen, “A numerical method for solving uncertain differential equations,” J. of Intelligent and Fuzzy Systems,Vol.25, No.3, pp. 825-832, 2013.
- [18] B. Liu, “Uncertainty distribution and independence of uncertain processes,” Fuzzy Optimization and Decision Making, Vol.13, No.3, pp. 259-271, 2014.
- [19] K. Yao, “Uncertain contour process and its application in stock model with floating interest rate,” Fuzzy Optimization and Decision Making, Vol.14, No.4, pp. 399-424, 2015.
- [20] B. Liu, “Fuzzy process, hybrid process and uncertain process,” J. of Uncertain System, Vol.2, No.1, pp. 3-16, 2008.

This article is published under a Creative Commons Attribution-NoDerivatives 4.0 Internationa License.