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Option pricing based on a type of fuzzy process

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Abstract

Liu process is a basic process in fuzzy environment. As an extension of Liu process, fractional Liu process has attracted the attention of many scholars. In this paper, a fuzzy stock model driven by fractional Liu process is established, and its European, American, Asian, power options pricing formulas are given. In order to better understand these formulas, we give a few numerical examples to illustrate the changes of European option price with different parameters when time is fixed. However, these examples are not based on real-life data since the lack of parameter estimation method for fuzzy differential equation driven by Liu process. Then the changes of American option price are given when time and parameters are both changed. At the same time, we study the parameter interval where the option price fluctuates greatly. Finally, the fuzzy stock model is extended to the generalized case, and the stock price is given.

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Acknowledgements

This work was supported by Natural Science Foundation of China Grant No.61773150, the Key Research Foundation of Education Bureau of Hebei province No. ZD2020172 and the Youth Research Foundation of Education Bureau of Hebei province No. QN2020124.

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Authors and Affiliations

  1. College of Mathematics and Information Science, Hebei University, Baoding, 071002, China

    Cuilian You & Le Bo

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  1. Cuilian You
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  2. Le Bo
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Appendix

Appendix

Table 5 shows the prices of European call option with different parameters.

Table 5 European call option price
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In order to study the value of the parameter \(\alpha \) when the option prices increase sharply, we obtain Table 6.

Table 6 European call option price when \(\alpha \in \) [0.64, 0.69]
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Table 6 shows that if \(\alpha \in \) (0.67, 0.68), the option prices increase sharply. That is to say, in reality, it may be better to choose \(\alpha \in \) [0.68, 1], since in which the option prices remain constant. To obtain more precise parameter interval, one can make analysis on \(\alpha \in \) (0.67, 0.68) in the same way above.

Table 7 shows the prices of European put option with different parameters.

Table 7 European put option price
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We do the same work to find the interval where the option prices increase sharply, see Table 8.

Table 8 European put option price when \(\alpha \in \) [0.61, 0.68]
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Table 8 shows that if \(\alpha \in \) (0.67, 0.68), the option prices increase significantly.

To obtain the exact interval where the change located, we used four tables to study the change situation of American call option prices (Tables 9 and 10), American put option prices (Tables 11 and 12).

Table 9 American call option price \(\alpha \in \) [0.71, 0.76]
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Table 10 American call option price \(\alpha \in \) [0.65, 0.7] and \(t=10\), \(t=11\)
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Table 11 American put option price \(\alpha \in \) [0.62, 0.68]
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Table 12 American put option price \(\alpha \in \) [0.70, 0.75] and \(t=9\), \(t=12\)
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You, C., Bo, L. Option pricing based on a type of fuzzy process. J Ambient Intell Human Comput 13, 3771–3785 (2022). https://doi.org/10.1007/s12652-021-03334-2

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