American barrier option pricing with floating interest rate based on uncertain fractional differential equations

Abstract

Pricing barrier options pose a significant challenge in financial derivative valuation because they are activated only when the underlying asset reaches a predetermined barrier. The first-hitting time model was employed to characterize the activation process. In addition, the pricing of American barrier options with a floating interest rate is dynamically represented by an uncertain fractional differential equation. The study derives price formulas for various American barrier options, including up-and-in call, down-and-in put, up-and-output, and down-and-out call options. The proposed model enhances the accuracy of capturing the long-tail distribution and tail risk of financial markets, thereby addressing their complexity and nonlinearity. Furthermore, the predictor-corrector method is utilized to compute the numerical prices for the barrier options with floating interest rates, supplemented by illustrative numerical examples.

Keywords

Uncertainty theory / Uncertain fractional differential equation / First-hitting time model / Floating interest rate / American barrier option

References

[1] Black, F. and Scholes, M., The pricing of option and corporate liabilities, Journal of Political Economy, 1973, 81(3): 637-654.

[2] Chen, X., American option pricing formula for uncertain financial market, International Journal of Operational Research, 2011, 8(2): 32-37.

[3] Diethelm, K., Ford, N. and Freed, A., A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynamics, 2002, 29(1-4): 3-22.

[4] Gao, R., Liu, K., Li, Z. and Lv, R., American barrier option pricing formulas for stock model in uncertain environment, IEEE Access, 2019, 7: 97846-97856.

[5] Geman, H. and Yor, M., Pricing and hedging double barrier options: A probabilistic approach, Mathematical Finance, 1996, 6(4): 365-378.

[6] Howison, S. and Steinberg, M., A matched asymptotic expansions approach to continuity corrections for discretely sampled options, Applied Mathematical Finance, 2007, 14(1): 63-89.

[7] Jin, T., Sun, Y. and Zhu, Y., 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.

[8] Jin, T., Sun, Y. and Zhu, Y., Time integral about solution of an uncertain fractional order differential equation and application to zero-coupon bond model, Applied Mathematics and Computation, 2020, 372: 124991.

[9] Jin, T. and Zhu, Y., First hitting time about solution for an uncertain fractional differential equation and application to an uncertain risk index model, Chaos, Solitons & Fractals, 2020, 137: 109836.

[10] Jin, T., Ding, H., Xia, H. and Bao, J., Reliability index and Asian barrier option pricing formulas of the uncertain fractional first-hitting time model with Caputo type, Chaos, Solitons & Fractals, 2021, 142: 110409.

[11] Jin, T. and Yang, X., Monotonicity theorem for the uncertain fractional differential equation and application to uncertain financial market, Mathematics and Computers in Simulation, 2021, 190: 203-221.

[12] Kilbas, A., Srivastava, H. and Trujillo, J., Theory and Application of Fractional Differential Equation, Elsevier Science B. V., Amsterdam, 2006.

[13] Liu, B.,Uncertainty Theory, 2nd ed., Springer, Berlin, 2007.

[14] Lu, Z. and Zhu, Y., Numerical approach for solution to an uncertain fractional differential equation, Applied Mathematics and Computation, 2019, 343: 137-148.

[15] Peng, J. and Yao, K., A new option pricing model for stocks in uncertainty markets, International Journal of Operational Research, 2011, 8(2): 18-26.

[16] Podlubny, I., Fractional Differential Equation, Academic Press, San Diego, 1999.

[17] Rich, D., The mathematical foundations of barrier option pricing theory, Advances in Futures and Options Research, 1994, 7(1): 267-311.

[18] 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.

[19] Tian, M., Yang, X. and Zhang, Y., Barrier option pricing of mean-reverting stock model in uncertain environment, Mathematics and Computers in Simulation, 2019, 166: 126-143.

[20] Tian, M., Yang, X. and Zhang, Y., Lookback option pricing problem of mean-reverting stock model in uncertain environment, Journal of Industrial and Management Optimization, 2021, 17(5): 2703-2714.