Vulnerable European Call Option Pricing Based on Uncertain Fractional Differential Equation

doi:10.1007/s11424-023-1140-1

Journal of Systems Science and Complexity ›› 2023, Vol. 36 ›› Issue (1): 328-359.DOI: 10.1007/s11424-023-1140-1

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Vulnerable European Call Option Pricing Based on Uncertain Fractional Differential Equation

LEI Ziqi¹, ZHOU Qing¹, WU Weixing², WANG Zengwu³

1. School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China;
2. Research Center of Applied Finance and School of Finance and Banking, University of International Business and Economics, Beijing 100029, China;
3. Institute of Finance and Banking, Chinese Academy of Social Sciences, Beijing 100029, China

Received:2021-05-03 Revised:2021-12-24 Online:2023-01-25 Published:2023-02-09
Supported by:
The work of ZHOU Qing is supported by the National Natural Science Foundation of China under Grant Nos. 11871010 and 11971040, and by the Fundamental Research Funds for the Central Universities under Grant No. 2019XD-A11. The work of WU Weixing is supported by the National Natural Science Foundation of China under Grant No. 71073020.

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LEI Ziqi, ZHOU Qing, WU Weixing, WANG Zengwu. Vulnerable European Call Option Pricing Based on Uncertain Fractional Differential Equation[J]. Journal of Systems Science and Complexity, 2023, 36(1): 328-359.

This paper presents two new versions of uncertain market models for valuing vulnerable European call option. The dynamics of underlying asset, counterparty asset, and corporate liability are formulated on the basis of uncertain differential equations and uncertain fractional differential equations of Caputo type, respectively, and the solution to an uncertain fractional differential equation of Caputo type is presented by employing the Mittag-Leffler function and α-path. Then, the pricing formulas of vulnerable European call option based on the proposed models are investigated as well as some algorithms. Some numerical experiments are performed to verify the effectiveness of the results.

Key words: $\alpha$-path, uncertainty, uncertain fractional differential equation, vulnerable option pricing

[1] Merton R C, On the pricing of corporate debt:The risk structure of interest rates, Journal of Finance, 1974, 29(2):449-470.
[2] Johnson H and Stulz R, The pricing of options with default risk, Journal of Finance, 1987, 42(2):267-280.
[3] Hull J C and White A, The impact of default risk on the prices of options and other derivative securities, Journal of Banking and Finance, 1995, 19(2):299-322.
[4] Jarrow R A and Turnbull S, Pricing derivatives on financial securities subject to credit risk, Journal of Finance, 1995, 50(1):53-85.
[5] Don R, The valuation and behavior of Black-Scholes options subject to intertemporal default risk, Review of Derivatives Research, 1996, 1:25-29.
[6] Klein P, Pricing Black-Scholes options with correlated credit risk, Journal of Banking & Finance, 1996, 20(7):1211-1229.
[7] Manuel A, Credit Risk Valuation-Methods, Models and Applications, Springer, New York, 2001.
[8] Wang X C, Pricing vulnerable European options with stochastic correlation, Probability in the Engineering and Informational Sciences, 2018, 32(1):67-95.
[9] Wang G Y and Wang X C, Valuing vulnerable options with bond collateral, Applied Economics Letters, 2021, 28(2):115-118.
[10] Liu B, Uncertainty Theory, 2nd Ed, Springer-Verlag, Berlin, 2007.
[11] Liu B, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2008, 2(1):316.
[12] Liu B, Some research problems in uncertainty theory, Journal of Uncertain Systems, 2009, 3(1):310.
[13] Yao K and Chen X W, A numerical method for solving uncertain differential equations, Journal of Intelligent & Fuzzy Systems, 2013, 25(3):825-832.
[14] Black F and Scholes M, The pricing of options and corporate liabilities, Journal of Political Economy, 1975, 81(3):637-654.
[15] Liu B, Uncertain entailment and modus ponens in the framework of uncertain logic, Journal of Uncertain Systems, 2009, 3(4):243-251.
[16] Chen X, American option pricing formula for uncertain financial market, International Journal of Operations Research, 2011, 8(2):32-37.
[17] Peng J and Yao K, A new option pricing model for stocks in uncertainty markets, International Journal of Operations Research, 2010, 7(4):213-224.
[18] Yao K, Uncertain contour process and its application in stock model with floating interest rate, Fuzzy Optimization and Decision Making, 2015, 14(4):399-424.
[19] Sun Y and Su T, Mean-reverting stock model with floating interest rate in uncertain environment, Fuzzy Optimization and Decision Making, 2017, 16(2):235-255.
[20] Wang X, Pricing of European currency options with uncertain exchange rate and stochastic interest rates, Discrete Dynamics in Nature and Society, 2019, 2548592.
[21] Gao R, Li Z, and Lü R, American barrier option pricing formulas for stock model in uncertain environment, IEEE Access, 2019, 7:97846-97856.
[22] Zhang L D, Sun Y M, and Meng X B, European spread option pricing with the floating interest rate for uncertain financial market, Mathematical Problems in Engineering, 2020, 2015845.
[23] Zhu Y G, Uncertain fractional differential equations and an interest rate model, Mathematical Methods in the Applied Sciences, 2015, 38(15):3359-3368.
[24] Lu Z Q, Yan H Y, and Zhu Y G, European option pricing model based on uncertain frational differential equation, Fuzzy Optimization and Decision Making, 2019, 18(2):199-217.
[25] Jin T, Sun Y, and Zhu Y G, Extreme values for solution to uncertain fractional differential equation and application to American option pricing model, Physica A:Statistical Mechanics and Its Applications, 2019, 534:122357.
[26] Wang W W and Dan A R, Option pricing formulas based on uncertain fractional differential equation, Fuzzy Optimization and Decision Making, 2021, 20(4):471-495.
[27] Liu B, Uncertainty Theory, 4th Ed., Springer-Verlag, Berlin, 2015.
[28] Podlubny I, Fractional Differential Equation, Academic Press, San Diego, 1999.
[29] Kilbas A A, Srivastava H M, and Trujillo J J, Theory and Application of Fractional Differential Equation, Elsevier, Amsterdam, 2006.
[30] Tian L H, Wang G Y, Wang X C, et al., Pricing vulnerable options with correlated credit risk under jump-diffusion processes, The Journal of Futures Markets, 2014, 34(10):957-979.
[31] Wang G Y, Wang X C, and Zhou K, Pricing vulnerable options with stochastic volatility, Physica A:Statistical Mechanics and Its Applications, 2017, 485:91-103.
[32] Zhang Z and Liu W, Geometric average Asian optionpricing for uncertain financial market, Journal of Uncertain Systems, 2014, 8(4):317-320.
[33] Wang W W and Chen P, A mean-reverting currency model with floating interest rate in uncertain environment, Journal of Industrial and Management Optimization, 2019, 15(4):1921-1936.
[34] Chen X and Liu B, Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optimization and Decision Making, 2010, 9(1):69-81.
[35] Zhu Y G, Existence and uniqueness of the solution to uncertain fractional differential equation, Journal of Uncertainty Analysis and Applications, 2015, 3(5).

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