Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2021 / Article

Research Article | Open Access

Volume 2021 |Article ID 7398770 | https://doi.org/10.1155/2021/7398770

Yiyao Sun, Shiqin Liu, "Interest-Rate Products Pricing Problems with Uncertain Jump Processes", Discrete Dynamics in Nature and Society, vol. 2021, Article ID 7398770, 8 pages, 2021. https://doi.org/10.1155/2021/7398770

Interest-Rate Products Pricing Problems with Uncertain Jump Processes

Academic Editor: Filippo Cacace
Received13 Apr 2021
Revised26 May 2021
Accepted10 Jun 2021
Published19 Jun 2021

Abstract

Uncertain differential equations (UDEs) with jumps are an essential tool to model the dynamic uncertain systems with dramatic changes. The interest rates, impacted heavily by human uncertainty, are assumed to follow UDEs with jumps in ideal markets. Based on this assumption, two derivatives, namely, interest-rate caps (IRCs) and interest-rate floors (IRFs), are investigated. Some formulas are presented to calculate their prices, which are of too complex forms for calculation in practice. For this reason, numerical algorithms are designed by using the formulas in order to compute the prices of these structured products. Numerical experiments are performed to illustrate the effectiveness and efficiency, which also show the prices of IRCs are strictly increasing with respect to the diffusion parameter while the prices of IRFs are strictly decreasing with respect to the diffusion parameter.

1. Introduction

The stochastic differential equations have been widely applied to mathematical finance since Black and Scholes [1] derived the option pricing formulas under the assumption that the price of a stock is an exponential Brownian motion. Hull and White [2] discussed the stochastic volatilities of the option derivatives and modeled the volatility with a geometric stochastic differential equation. Amin and Jarrow [3] built a framework to price contingent claims on foreign currencies and obtained closed form solutions for currency futures. Ninomiya and Victoir [4] investigated the Asian option pricing problems of a stochastic volatility model. Recently, Wang et al. [5] developed a novel hybrid model that integrates fractal interpolation to forecast the time series of stock price indexes and proposed a method to calculate the vertical scaling factors of the fractal interpolation iterated function system. Aiming at researching the dynamic behavior of the fractional Black–Scholes model, He and Zhang [6] developed a full-discrete numerical scheme. Through investing in the measurement of interest-rate risk, Tang and Du [7] considered the existence of embedded options under the assumption that the fluctuation of interest rate follows fuzzy stochastic process.

The abovementioned financial models are all based on the assumption that the prices of the assets are only subject to random fluctuations. However, the real financial markets often involve human uncertain factors in addition to stochastic factors. For modelling the human subjective uncertainty in the view of mathematics, Liu [8, 9] founded and refined the uncertainty theory on the basis of normality axiom, duality axiom, subadditivity axiom, and product axioms. Uncertain variables are mainly a tool of modelling quantities whose possible values are estimated according to human belief degree, and uncertain processes, which are a sequence of uncertain variables that evolve with the time, describe ever-changing uncertain systems. Frequently used uncertain processes include the Liu processes [9] and the renewal processes [10].

Driven by the uncertain processes, the UDEs are widely applied in finance. For example, the UDE was firstly introduced to the financial models by Liu [9], in which the price of a stock is described by an exponential Liu process, and the European options of the stock are priced. Then, Sun and Su [11] modeled the price of a stock and the interest rate in long run by means of mean-reverting processes. In order to ensure the interest rate can always remain positive, Liu [12] made some improvements on the basis of Sun and Su’s [11] model. Sheng and Shi [13] and Gao et al. [14] calculated the prices of Asian options and lookback options, respectively. These works were generalized by Hassanzadeh and Mehrdoust [15] and Yang et al. [16]. The UDEs were also used to model the interest rate in the uncertain environment by Chen and Gao [17], in which the zero-coupon bond is priced in an analytic form. Following that, Zhang et al. [18] investigated the interest-rate ceilings and floors based on these interest-rate models. For the purpose of deriving the European currency options, Wang [19] supposed a currency model in which interest rate and exchange rate follow stochastic process and uncertain process, respectively. By assuming that exchange rate, volatility, domestic interest rate, and foreign interest rate all follow uncertain processes, Li et al. [20] researched the pricing problem of foreign currency options and designed Runge–Kutta-99 hybrid method for solving nested uncertain differential equations. Option pricing problems based on uncertain fractional-order differential equation with Caputo type were investigated by Jin et al. [21].

Driven by both Liu processes and renewal processes, the UDEs with jumps that were proposed by Yao [22] describe dynamic systems with uncertain sharp drifts. The stability properties of the equations were investigated by Yao [23], Gao [24], Ji and Ke [25], and Ma et al. [26]. As applications of UDEs with jumps, Ji and Wu [27] described the exchange rates between different currencies which are greatly impacted by the uncertain factors. Recently, Yu and Ning [28] presented an interest-rate model with jumps to describe the severe perturbation of interest rate and calculated the prices of zero-coupon bonds. Based on this model, we compute the price of another two widely used derivatives in this paper, namely, the IRCs and the IRFs. The remainder of this paper is structured as follows. Section 2 introduces the uncertain interest-rate model with jumps. Then, the IRCs and the IRFs are priced in analytic but extremely complex forms in Sections 3 and 4, respectively. After that, some algorithms are designed to compute the prices numerically, and experiments are performed to illustrate the effectiveness of the algorithms in Section 5. Finally, some remarks are made in Section 6.

2. Preliminaries

The uncertainty theory is a branch of axiomatic mathematics to deal with the information involving human uncertainty. The uncertain variables are used to model the quantities whose possible values are assigned based on human experience, and the UDEs are widely applied to describe dynamic systems with human uncertainty. For more about uncertain variables and UDEs, please refer to Liu [29] and Yao [30].

In order to model the interest rate in the ideal market which may change dramatically, Yu and Ning [28] suggestedwhere denotes the interest rate, and are constant parameters, is a canonical Liu process, and is an uncertain renewal process. Let denote the standard normal uncertainty distribution (UD) and denote the UD of interarrival times. Then, the solution,of equation (1) possesses an UD

Yu and Ning [28] further showed that possesses the inverse uncertainty distribution (IUD)where denotes the maximal integer less than or equal to . In fact, they gave a more powerful result as follows.

Theorem 1. (see [28]). Let denote the standard normal UD and denote the UD of interarrival times. Then, the solution of equation (1) satisfies

3. Interest-Rate Caps

The IRC is a contract between a lending institute and a borrower that prevents the institute charging more than a certain level of interest from the borrower. It is a benefit for the borrowers in variable interest-rate products.

Let an uncertain process denote the interest rate. Then, the price of IRC, as suggested in Zhang et al. [18], iswhere is the maximum interest rate preset in the contract and is the expiration date of the contract. Now, based on the assumption of the interest rate in Yu and Ning [28], that is, it follows the UDE with jumps (1), we derive pricing formula of the IRCs.

Theorem 2. Assume the interest rate follows an UDE with jumpsLet denote the standard normal UD and denote the UD of interarrival times. Then, the IRC, which is of an expiration date and a maximum interest rate , is priced by

Proof. Denote the IUD of with , i.e.,Then, according to Theorem 1, we haveSincewe haveSimilarly, we haveFurthermore, sincewe getthat is,has an IUDHence,has an IUDas the function is decreasing with . Then,The theorem is proved.

4. Interest-Rate Floors

The IRF is a contract between a lending institute and a borrower that prevents the borrower repaying less than a certain level of interest to the lending institute. It is a benefit for the lending institute in variable interest-rate products.

Let an uncertain process denote the interest rate. Then, Zhang et al. [18] suggested computing the price of the IRF viawhere is the minimum interest rate preset in the contract and is the expiration date of the contract. Now, based on the assumption of the interest rate in Yu and Ning [28], that is, it follows the UDE with jumps (1), we derive the pricing formula of the IRFs.

Theorem 3. Assume the interest rate follows an UDE with jumpsLet denote the standard normal UD and denote the UD of interarrival times. Then, the IRF, which is of an expiration date and a minimum interest rate , is priced by

Proof. Denote the IUD of with , i.e.,Then, according to Theorem 1, we haveSincewe haveSimilarly, we haveFurthermore, sincewe getthat is,has an IUDHence,has an IUDas the function is increasing with . Then,The theorem is proved.

5. Numerical Algorithms

The numerical algorithms to compute the prices of the IRCs and IRFs are designed based on formulas (8) and (23) and illustrated via some experiments in the subsections.

5.1. Algorithm for Interest-Rate Caps
Step 1: according to the precision degree, preset two large numbers and . Set and , , .Step 2: set .Step 3: set .Step 4: set .Step 5: set .Step 6: computeIf , return to Step 5.Step 7: computeIf , return to Step 3.Step 8: compute

Example 1. Consider the uncertain interest-rate model with an initial interest rate . Assume the parameters are , , and ; the standard normal UD isand the UD of interarrival times isThen, the IRC with an expiration date and a maximum value has a price .

By using the algorithm mentioned above, we can numerically investigate the relationships of the IRC price with respect to the parameters in the pricing formula (8) such as , , , , and . The next example plots the function of with respect to when the other parameters are given.

Example 2. Consider the uncertain interest-rate model with an initial interest rate . Assume the parameters are and ; the standard normal UD isand the UD of interarrival times isThe IRC has an expiration date and a maximum value . Then, the function of the IRC price with respect to the parameter is shown in Figure 1. Apparently, the price is a strictly increasing function with respect to .

5.2. Algorithm for Interest-Rate Floors
Step 1: according to the precision degree, preset two large numbers and . Set and , , .Step 2: set .Step 3: set .Step 4: set .Step 5: set .Step 6: computeIf , return to Step 5.Step 7: computeIf , return to Step 3.Step 8: compute

Example 3. Consider the uncertain interest-rate model with an initial interest rate . Assume the parameters are , , and ; the standard normal UD isand the UD of interarrival times isThen, the IRF with an expiration date and a minimum value has a price .

By using the algorithm mentioned above, we can numerically investigate the relationships of the IRF price with respect to the parameters in the pricing formula (23) such as , , , , and . The next example plots the function of with respect to when the other parameters are given.

Example 4. Consider the uncertain interest-rate model with an initial interest rate . Assume the parameters are and ; the standard normal UD isand the UD of interarrival times isThe IRF has an expiration date and a minimum value . Then, the function of the IRF price with respect to the parameter is shown in Figure 2. Apparently, the price is a strictly decreasing function with respect to .

6. Conclusions

Based on an interest-rate model described by an UDE with jumps, pricing problems of IRCs and IRFs were considered. Some pricing formulas were obtained in analytic but extremely complex forms. Numerical algorithms were designed based on the pricing formulas, and numerical experiments were performed to test the effectiveness of the presented algorithms. Further research may consider the structured product pricing problems based on interest-rate models with both positive and negative jumps.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (Grant No. 61702165), the Hebei Provincial Natural Science Foundation, China (Grant No. F2020111001), and the Institute of Applied Mathematics, Hengshui University (2018YJ04).

References

  1. F. Black and M. Scholes, “The pricing of options and corporate liabilities,” Journal of Political Economy, vol. 81, no. 3, pp. 637–654, 1973. View at: Publisher Site | Google Scholar
  2. J. Hull and A. White, “The pricing of options on assets with stochastic volatilities,” The Journal of Finance, vol. 42, no. 2, pp. 281–300, 1987. View at: Publisher Site | Google Scholar
  3. K. I. Amin and R. A. Jarrow, “Pricing foreign currency options under stochastic interest rates,” Journal of International Money and Finance, vol. 10, no. 3, pp. 310–329, 1991. View at: Publisher Site | Google Scholar
  4. S. Ninomiya and N. Victoir, “Weak approximation of stochastic differential equations and application to derivative pricing,” Applied Mathematical Finance, vol. 15, no. 2, pp. 107–121, 2008. View at: Publisher Site | Google Scholar
  5. H. Wang, H. Li, and J. Y. Shen, “A novel hybrid fractal interpolation-SVM model for forecasting stock price indexes,” Fractals, vol. 27, no. 4, Article ID 1950055, 2019. View at: Publisher Site | Google Scholar
  6. J. He and A. Zhang, “Finite difference/fourier spectral for a time fractional BlackCScholes model with option pricing,” Mathematical Problems in Engineering, vol. 2020, Article ID 1393456, 2020. View at: Publisher Site | Google Scholar
  7. E. Tang and W. Du, “Measuring interest rate risk with embedded option using HPL-MC method in fuzzy and stochastic environment,” Journal of Mathematics, vol. 2020, Article ID 7410909, 13 pages, 2020. View at: Publisher Site | Google Scholar
  8. B. Liu, Uncertainty Theory, Springer, Berlin, Heidelberg, Germany, 2nd edition, 2007.
  9. B. Liu, “Some research problems in uncertainty theory,” Journal of Uncertain Systems, vol. 3, no. 1, pp. 3–10, 2009. View at: Google Scholar
  10. K. Yao, Uncertain Renewal Processes, Springer, Singapore, 2019.
  11. Y. Sun and T. Su, “Mean-reverting stock model with floating interest rate in uncertain environment,” Fuzzy Optimization and Decision Making, vol. 16, no. 2, pp. 235–255, 2017. View at: Publisher Site | Google Scholar
  12. Z. Liu, “Option pricing formulas in a new uncertain mean-reverting stock model with floating interest rate,” Discrete Dynamics in Nature and Society, vol. 2020, Article ID 3764589, 2020. View at: Publisher Site | Google Scholar
  13. Y. Sheng and G. Shi, “Mean-reverting stock model and pricing rules for Asian currency option,” Journal of Intelligent and Fuzzy Systems, vol. 34, no. 6, pp. 4261–4268, 2018. View at: Publisher Site | Google Scholar
  14. Y. Gao, X. Yang, and Z. Fu, “Lookback option pricing problem of uncertain exponential Ornstein-Uhlenbeck model,” Soft Computing, vol. 22, no. 17, pp. 5647–5654, 2018. View at: Publisher Site | Google Scholar
  15. S. Hassanzadeh and F. Mehrdoust, “Valuation of European option under uncertain volatility model,” Soft Computing, vol. 22, no. 12, pp. 4153–4163, 2018. View at: Publisher Site | Google Scholar
  16. X. Yang, Z. Zhang, and X. Gao, “Asian-barrier option pricing formulas of uncertain financial market,” Chaos, Solitons & Fractals, vol. 123, pp. 79–86, 2019. View at: Publisher Site | Google Scholar
  17. X. Chen and J. Gao, “Uncertain term structure model of interest rate,” Soft Computing, vol. 17, no. 4, pp. 597–604, 2013. View at: Publisher Site | Google Scholar
  18. Z. Zhang, D. A. Ralescu, and W. Liu, “Valuation of interest rate ceiling and floor in uncertain financial market,” Fuzzy Optimization and Decision Making, vol. 15, no. 2, pp. 139–154, 2016. View at: Publisher Site | Google Scholar
  19. X. Wang, “Pricing of European currency options with uncertain exchange rate and stochastic interest rates,” Discrete Dynamics in Nature and Society, vol. 2019, Article ID 2548592, 2019. View at: Publisher Site | Google Scholar
  20. Z. Li, Y.-J. Liu, and W.-G. Zhang, “Quasi-closed-form solution and numerical method for currency option with uncertain volatility model,” Soft Computing, vol. 24, no. 19, pp. 15041–15057, 2020. View at: Publisher Site | Google Scholar
  21. T. Jin, X. Yang, H. Xia, and H. Ding, “Reliability index and option pricing formulas of the first hitting time model based on the uncertain fractional-order differential equation with Caputo type,” Fractals, vol. 29, no. 1, Article ID 2150012, 2020. View at: Publisher Site | Google Scholar
  22. K. Yao, “Uncertain calculus with renewal process,” Fuzzy Optimization and Decision Making, vol. 11, no. 3, pp. 285–297, 2012. View at: Publisher Site | Google Scholar
  23. K. Yao, “Uncertain differential equation with jumps,” Soft Computing, vol. 19, no. 7, pp. 2063–2069, 2015. View at: Publisher Site | Google Scholar
  24. R. Gao, “Stability in mean for uncertain differential equation with jumps,” Applied Mathematics and Computation, vol. 346, pp. 15–22, 2019. View at: Publisher Site | Google Scholar
  25. X. Ji and H. Ke, “Almost sure stability for uncertain differential equation with jumps,” Soft Computing, vol. 20, no. 2, pp. 547–553, 2016. View at: Publisher Site | Google Scholar
  26. W. Ma, L. Liu, and X. Zhang, “Stability in p-th moment for uncertain differential equation with jumps,” Journal of Intelligent and Fuzzy Systems, vol. 33, no. 3, pp. 1375–1384, 2017. View at: Publisher Site | Google Scholar
  27. X. Ji and H. Wu, “A currency exchange rate model with jumps in uncertain environment,” Soft Computing, vol. 21, no. 18, pp. 5507–5514, 2017. View at: Publisher Site | Google Scholar
  28. S. Yu and Y. Ning, “An interest-rate model with jumps for uncertain financial markets,” Physica A: Statistical Mechanics and its Applications, vol. 527, Article ID 121424, 2019. View at: Publisher Site | Google Scholar
  29. B. Liu, Uncertainty Theory, Springer, Berlin, Heidelberg, Germany, 4th edition, 2015.
  30. K. Yao, Uncertain Differential Equations, Springer, Berlin, Heidelberg, Germany, 2016.

Copyright © 2021 Yiyao Sun and Shiqin Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Related articles

No related content is available yet for this article.
 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views95
Downloads203
Citations

Related articles

No related content is available yet for this article.

Article of the Year Award: Outstanding research contributions of 2021, as selected by our Chief Editors. Read the winning articles.