Abstract
Lookback options are among the most popular path-dependent options in financial market. In this paper, the option pricing problem of lookback options is investigated under the assumption that the underlying stock price follows an uncertain differential equation driven by Liu process instead of stochastic differential equation, and the lookback options pricing formulae are derived under this assumption. Several numerical examples are also discussed to illustrate the pricing formula.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Black F, Scholes M (1973) The pricing of option and corporate liabilities. J Polit Econ 81:637–654
Chen XW (2011) American option pricing formula for uncertain financial market. Int J Oper Res 8(2):32–37
Chen XW, Gao J (2013) Uncertain term structure model of interest rate. Soft Comput 17(4):597–604
Chen XW, Liu YH, Ralescu DA (2013) Uncertain stock model with periodic dividends. Fuzzy Optim Decis Mak 12(1):111–123
Conze A, Viswanathan R (1991) Path dependent options: the case of lookback options. J Finance 46:1893–1907
Dai M, Wong HY, Kwok YK (2004) Quanto lookback options. Math Finance 14:445–467
Goldman MB, Sosin HB, Gatto MA (1979) Path dependent options: buy at the low, sell at the high. J Finance 34:1111–1127
Heynen RC, Kat HM (1995) Lookback options with discrete and partial monitoring of the underlying price. Appl Math Finance 2:273–284
Liu B (2007) Uncertainty theory, 2nd edn. Springer, Berlin
Liu B (2009) Some research problems in uncertainty theory. J Uncertain Syst 3(1):3–10
Liu B (2010) Uncertainty theory: a branch of mathematics for modeling human uncertainty. Springer, Berlin
Liu YH, Ha MH (2010) Expected value of function of uncertain variables. J Uncertain Syst 4(3):181–186
Liu YH, Chen XW, Ralescu DA (2015) Uncertain currency model and currency option pricing. Int J Intell Syst 30:40–51
Longstaff FA (1995) How much can marketability affect security values? J Finance 50:1767–1774
Peng J, Yao K (2011) A new option pricing model for stocks in uncertainty markets. Int J Oper Res 8(2):18–26
Wong HY, Kwok YK (2003) Sub-replication and replenishing premium: efficient pricing of multi-state lookbacks. Rev Deriv Res 6:83–106
Yao K (2012) No-arbitrage determinant theorems on mean-reverting stock model in uncertain market. Knowl Based Syst 35:259–263
Yao K (2013) Extreme values and integral of solution of uncertain differential equation. J Uncertain Anal Appl 1:2
Yao K, Chen XW (2013) A numerical method for solving uncertain differential equations. J Intell Fuzzy Syst 25(3):825–832
Zhang ZQ, Liu WQ (2014) Geometric average Asian option pricing for uncertain financial market. J Uncertain Syst 8(4):317–320
Zhang ZQ, Ralescu DA, Liu WQ (2016) Valuation of interest rate ceiling and floor in uncertain financial market. Fuzzy Optim Decis 15(2):139–154
Zhang ZQ, Liu WQ, Ding JH (2017a) Valuation of stock loan under uncertain environment. Soft Comput. https://doi.org/10.1007/s00500-017-2591-x
Zhang ZQ, Liu WQ, Zhang XD (2017b) Valuation of convertible bond under uncertain mean-reverting stock model. J Amb Intel Hum Comput 8(5):641–650
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 71371113, 71371141, 71001080) and Doctoral Fund of Shanxi Datong University (No. 2016-B-03).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that there is no conflict of interest.
Additional information
Communicated by V. Loia.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zhang, Z., Ke, H. & Liu, W. Lookback options pricing for uncertain financial market. Soft Comput 23, 5537–5546 (2019). https://doi.org/10.1007/s00500-018-3211-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-018-3211-0