# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2022033
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## Pricing of European call option under fuzzy interest rate

 1 College of Mathematics and Information Science, Hebei University, Baoding 071002, China 2 Hebei Key Lab of Machine Learning and Computation Intelligence Hebei University, Baoding 071002, China

*Corresponding author: Cuilian You

Received  April 2021 Revised  January 2022 Early access March 2022

Fund Project: The first author is supported by Science and Technology Project of Hebei Education Department No. ZD2020172 and No. QN2020124

Option pricing under fuzzy environment is a hot research topic nowadays. Traditionally, option pricing were made in the case of fixed interest rate, while the fluctuate of interest rate may result in profit loss or bring unexpected risk. Thus, based on credibility theory, a new option pricing model under fuzzy interest rate are constructed in this paper. In fact, almost all fuzzy option pricing uses expected value method. In this paper, a new pricing method, fuzzy term structure and fuzzy affine term structure method, is adopted, and two European call option pricing formulas are obtained, one is that the fuzzy interest rate coefficients are constants, the other is that the fuzzy interest rate drift coefficient is a fuzzy process.

Citation: Cuilian You, Le Bo. Pricing of European call option under fuzzy interest rate. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022033
##### References:
 [1] F. Black and M. Scholes, The pricing of option and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654. doi: 10.1086/260062. [2] L. Bo and C. You, Fuzzy interest rate term structure equation, International Journal of Fuzzy Systems, 22 (2020), 999-1006.  doi: 10.1007/s40815-020-00810-3. [3] J. Cox, J. Ingersoll and S. Ross, A theory of the term structure of interest rates, Econometrica, 53(1985) 385–408. doi: 10.2307/1911242. [4] X. Chen and Z. Qin, A new existence and uniqueness theorem for fuzzy differential equation, International Journal of Fuzzy Systems, 13 (2011), 148-151. [5] X. Gao and X. Chen, Option pricing formula for generalized stock models, 2008., Available from: http://orsc.edu.cn/process/080317.pdf. [6] J. Gao and X. Gao, A new stock model for credibilistic option pricing, Journal of Uncertain Systems, 2 (2008), 243-247. [7] Z. Guo, option pricing under the Merton model of the short rate in subdiffusive Brownian motion regime, Journal of Statistical Computation and Simulation, 87 (2017), 519-529. doi: 10.1080/00949655.2016.1218880. [8] H. Hu, Power option pricing model for stock price follow geometric fractional Liu process, Journal of Henan Normal University (Natural Science Edition), 41 (2013), 1-5. [9] J. Hull and A. White, Pricing interest-rate-derivative securities, The Review of Financial Studies, 3 (1990), 573-592.  doi: 10.1093/rfs/3.4.573. [10] B. Liu, Uncertainty theory, 2$^{nd}$ edition, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-39987-2. [11] B. Liu and Y. K. Liu, Expected value of fuzzy variable and fuzzy expected value models, IEEE Transactions on Fuzzy Systems, 10 (2002), 445-450. doi: 10.1016/S0020-0255(03)00079-3. [12] B. Liu, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2 (2008), 3-16. [13] J. Kung and L. Lee, Option pricing under the Merton model of the short rate, Mathematics and Computers Simulatiom, 80 (2009), 378-386. doi: 10.1016/j.matcom.2009.07.006. [14] M. Magdziarz, Black-Scholes formula in subdiffusive regime, Journal of Statistical Physics, 136 (2009), 553-564. doi: 10.1007/s10955-009-9791-4. [15] R. Merton, A dynamic general equilibrium model of the assset market and its application to the pricing of the captial structure of the firm, Available from: Working paper 497, Sloan School of management, MIT, Cambridge, 1970. [16] R. Merton, Theory of rational option pricing, Bell Journal of Economics and Management Science, 4 (1973), 141-183. doi: 10.2307/3003143. [17] J. Peng, A general stock model for fuzzy markets, Journal of Uncertain Systems, 2 (2008), 248-254. [18] Z. Qin and X. Li, Option pricing formula for fuzzy financial market, Journal of Uncertain System, 2 (2008), 17-21. [19] Z. Qin and X. Gao, Fractional Liu process with application to finance, Mathematical and Computer Modelling, 50 (2009), 1538-1543. doi: 10.1016/j.mcm.2009.08.031. [20] C. You and L. Bo, Option pricing formula for generalized fuzzy stock model, Journal of Industrial and Management Optimization, 16 (2020), 387-396. doi: 10.3934/jimo.2018158. [21] O. Vasicek, An equilibriun characterization of the term structure, Journal of Financial Economics, 5 (1977), 177-188. [22] C. You, W. Wang and H. Huo, Existence and unqiueness theorems for fuzzy differential equation, Journal of Uncertain Systems, 7 (2013), 303-315. [23] C. You and Y. Hao, Fuzzy Euler approximation and its local convergence, Journal of Computational and Applied Mathematics, 343 (2018), 55-61. doi: 10.1016/j.cam.2018.04.031. [24] C. You and Y. Hao, Stability in mean for fuzzy differential equation, Journal of Industrial and Management Optimization, 15 (2019), 1375-1385. doi: 10.1103/physrevd.99.032002. [25] C. You and Y. Hao, Numerical solution of fuzzy differential equation based on Taylor expansion, Journal of Hebei University (Nature Science), 38 (2018), 113-118. [26] L. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353. doi: 10.1016/S0019-9958(65)90241-X. [27] L. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, 1 (1978), 3-28. doi: 10.1016/0165-0114(78)90029-5.

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##### References:
 [1] F. Black and M. Scholes, The pricing of option and corporate liabilities, Journal of Political Economy, 81 (1973), 637-654. doi: 10.1086/260062. [2] L. Bo and C. You, Fuzzy interest rate term structure equation, International Journal of Fuzzy Systems, 22 (2020), 999-1006.  doi: 10.1007/s40815-020-00810-3. [3] J. Cox, J. Ingersoll and S. Ross, A theory of the term structure of interest rates, Econometrica, 53(1985) 385–408. doi: 10.2307/1911242. [4] X. Chen and Z. Qin, A new existence and uniqueness theorem for fuzzy differential equation, International Journal of Fuzzy Systems, 13 (2011), 148-151. [5] X. Gao and X. Chen, Option pricing formula for generalized stock models, 2008., Available from: http://orsc.edu.cn/process/080317.pdf. [6] J. Gao and X. Gao, A new stock model for credibilistic option pricing, Journal of Uncertain Systems, 2 (2008), 243-247. [7] Z. Guo, option pricing under the Merton model of the short rate in subdiffusive Brownian motion regime, Journal of Statistical Computation and Simulation, 87 (2017), 519-529. doi: 10.1080/00949655.2016.1218880. [8] H. Hu, Power option pricing model for stock price follow geometric fractional Liu process, Journal of Henan Normal University (Natural Science Edition), 41 (2013), 1-5. [9] J. Hull and A. White, Pricing interest-rate-derivative securities, The Review of Financial Studies, 3 (1990), 573-592.  doi: 10.1093/rfs/3.4.573. [10] B. Liu, Uncertainty theory, 2$^{nd}$ edition, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-39987-2. [11] B. Liu and Y. K. Liu, Expected value of fuzzy variable and fuzzy expected value models, IEEE Transactions on Fuzzy Systems, 10 (2002), 445-450. doi: 10.1016/S0020-0255(03)00079-3. [12] B. Liu, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2 (2008), 3-16. [13] J. Kung and L. Lee, Option pricing under the Merton model of the short rate, Mathematics and Computers Simulatiom, 80 (2009), 378-386. doi: 10.1016/j.matcom.2009.07.006. [14] M. Magdziarz, Black-Scholes formula in subdiffusive regime, Journal of Statistical Physics, 136 (2009), 553-564. doi: 10.1007/s10955-009-9791-4. [15] R. Merton, A dynamic general equilibrium model of the assset market and its application to the pricing of the captial structure of the firm, Available from: Working paper 497, Sloan School of management, MIT, Cambridge, 1970. [16] R. Merton, Theory of rational option pricing, Bell Journal of Economics and Management Science, 4 (1973), 141-183. doi: 10.2307/3003143. [17] J. Peng, A general stock model for fuzzy markets, Journal of Uncertain Systems, 2 (2008), 248-254. [18] Z. Qin and X. Li, Option pricing formula for fuzzy financial market, Journal of Uncertain System, 2 (2008), 17-21. [19] Z. Qin and X. Gao, Fractional Liu process with application to finance, Mathematical and Computer Modelling, 50 (2009), 1538-1543. doi: 10.1016/j.mcm.2009.08.031. [20] C. You and L. Bo, Option pricing formula for generalized fuzzy stock model, Journal of Industrial and Management Optimization, 16 (2020), 387-396. doi: 10.3934/jimo.2018158. [21] O. Vasicek, An equilibriun characterization of the term structure, Journal of Financial Economics, 5 (1977), 177-188. [22] C. You, W. Wang and H. Huo, Existence and unqiueness theorems for fuzzy differential equation, Journal of Uncertain Systems, 7 (2013), 303-315. [23] C. You and Y. Hao, Fuzzy Euler approximation and its local convergence, Journal of Computational and Applied Mathematics, 343 (2018), 55-61. doi: 10.1016/j.cam.2018.04.031. [24] C. You and Y. Hao, Stability in mean for fuzzy differential equation, Journal of Industrial and Management Optimization, 15 (2019), 1375-1385. doi: 10.1103/physrevd.99.032002. [25] C. You and Y. Hao, Numerical solution of fuzzy differential equation based on Taylor expansion, Journal of Hebei University (Nature Science), 38 (2018), 113-118. [26] L. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353. doi: 10.1016/S0019-9958(65)90241-X. [27] L. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, 1 (1978), 3-28. doi: 10.1016/0165-0114(78)90029-5.
European call option expected price under fuzzy short rate with certain coefficients
 $\alpha$ 0.5 1 1.5 2 2.5 $f$ 58.4275 69.3088 88.428 123.534 178.465
 $\alpha$ 0.5 1 1.5 2 2.5 $f$ 58.4275 69.3088 88.428 123.534 178.465
European call option expected price under fuzzy short rate with certain coefficients
 $\alpha$ 0.5 1 1.5 2 2.5 $f$ 63.3709 75.8918 92.82 116.099 148.833
 $\alpha$ 0.5 1 1.5 2 2.5 $f$ 63.3709 75.8918 92.82 116.099 148.833
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