DDNSDiscrete Dynamics in Nature and Society1607-887X1026-0226Hindawi10.1155/2021/73987707398770Research ArticleInterest-Rate Products Pricing Problems with Uncertain Jump Processeshttps://orcid.org/0000-0002-9453-3646SunYiyao12https://orcid.org/0000-0002-7193-595XLiuShiqin3CacaceFilippo1Postdoctoral Scientific Research WorkstationPostal Savings Bank of ChinaBeijing 100808Chinapsbc.com2Postdoctoral Research StationSchool of FinanceRenmin University of ChinaBeijing 100872Chinaruc.edu.cn3College of Mathematics and Computer ScienceHengshui UniversityHebei 053000China20211962021202113420212652021106202119620212021Copyright © 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.

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.

National Natural Science Foundation of China61702165Natural Science Foundation of Hebei ProvinceF2020111001Institute of Applied Mathematics, Hengshui University2018YJ04
1. Introduction

The stochastic differential equations have been widely applied to mathematical finance since Black and Scholes  derived the option pricing formulas under the assumption that the price of a stock is an exponential Brownian motion. Hull and White  discussed the stochastic volatilities of the option derivatives and modeled the volatility with a geometric stochastic differential equation. Amin and Jarrow  built a framework to price contingent claims on foreign currencies and obtained closed form solutions for currency futures. Ninomiya and Victoir  investigated the Asian option pricing problems of a stochastic volatility model. Recently, Wang et al.  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  developed a full-discrete numerical scheme. Through investing in the measurement of interest-rate risk, Tang and Du  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  and the renewal processes .

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 , 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  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  made some improvements on the basis of Sun and Su’s  model. Sheng and Shi  and Gao et al.  calculated the prices of Asian options and lookback options, respectively. These works were generalized by Hassanzadeh and Mehrdoust  and Yang et al. . The UDEs were also used to model the interest rate in the uncertain environment by Chen and Gao , in which the zero-coupon bond is priced in an analytic form. Following that, Zhang et al.  investigated the interest-rate ceilings and floors based on these interest-rate models. For the purpose of deriving the European currency options, Wang  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.  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. .

Driven by both Liu processes and renewal processes, the UDEs with jumps that were proposed by Yao  describe dynamic systems with uncertain sharp drifts. The stability properties of the equations were investigated by Yao , Gao , Ji and Ke , and Ma et al. . As applications of UDEs with jumps, Ji and Wu  described the exchange rates between different currencies which are greatly impacted by the uncertain factors. Recently, Yu and Ning  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  and Yao .

In order to model the interest rate in the ideal market which may change dramatically, Yu and Ning  suggested(1)dXt=μXtdt+σXtdCt+δXtdNt,where Xt denotes the interest rate, μ,σ>0 and δ>0 are constant parameters, Ct is a canonical Liu process, and Nt is an uncertain renewal process. Let Ψ denote the standard normal uncertainty distribution (UD) and Φ denote the UD of interarrival times. Then, the solution,(2)Xt=X0expμt+σCt1+δNt,of equation (1) possesses an UD(3)Θty=maxk0Ψlnyμtkln1+δlnX0σt1Φtk+1.

Yu and Ning  further showed that Xt possesses the inverse uncertainty distribution (IUD)(4)Θt1β=X0expμt+σΨ1βt+ln1+δtΦ11β1,where z denotes the maximal integer less than or equal to z. In fact, they gave a more powerful result as follows.

Theorem 1.

(see ). Let Ψ denote the standard normal UD and Φ denote the UD of interarrival times. Then, the solution Xt of equation (1) satisfies(5)MXtX0expμt+σΨ1βt+ln1+δtΦ11β1,t0=β,MXt>X0expμt+σΨ1βt+ln1+δtΦ11β1,t0=1β.

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 Xt denote the interest rate. Then, the price of IRC, as suggested in Zhang et al. , is(6)fc=1Eexp0TXtC+dt,where C is the maximum interest rate preset in the contract and T is the expiration date of the contract. Now, based on the assumption of the interest rate in Yu and Ning , that is, it follows the UDE with jumps (1), we derive pricing formula of the IRCs.

Theorem 2.

Assume the interest rate Xt follows an UDE with jumps(7)dXt=μXtdt+σXtdCt+δXtdNt.

Let Ψ denote the standard normal UD and Φ denote the UD of interarrival times. Then, the IRC, which is of an expiration date T and a maximum interest rate C, is priced by(8)fc=101exp0TX0expμt+σΨ1βt+ln1+δtΦ11β1C+dtdβ.

Proof.

Denote the IUD of Xt with Θt1β, i.e.,(9)Θt1β=X0expμt+σΨ1βt+ln1+δtΦ11β1.

Then, according to Theorem 1, we have(10)MXtΘt1β,t0=β,MXt>Θt1β,t0=1β.

Since(11)0TXtC+dt0TΘt1βC+dtXtΘt1β,t0,T,we have(12)M0TXtC+dt0TΘt1βC+dtMXtΘt1β,t0,Tβ.

Similarly, we have(13)M0TXtC+dt>0TΘt1βC+dtMXt>Θt1β,t0,T1β.

Furthermore, since(14)M0TXtC+dt0TΘt1βC+dt+M0TXtC+dt>0TΘt1βC+dt=1,we get(15)M0TXtC+dt0TΘt1βC+dt=β,that is,(16)0TXtC+dt,has an IUD(17)0TΘt1βC+dt.

Hence,(18)exp0TXtC+dt,has an IUD(19)exp0TΘt11βC+dt,as the function expz is decreasing with z. Then,(20)fc=1Eexp0TXtC+dt=101exp0TΘt11βC+dtdβ=101exp0TΘt1βC+dtdβ.

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 Xt denote the interest rate. Then, Zhang et al.  suggested computing the price of the IRF via(21)fl=Eexp0TLXt+dt1,where L is the minimum interest rate preset in the contract and T is the expiration date of the contract. Now, based on the assumption of the interest rate in Yu and Ning , that is, it follows the UDE with jumps (1), we derive the pricing formula of the IRFs.

Theorem 3.

Assume the interest rate Xt follows an UDE with jumps(22)dXt=μXtdt+σXtdCt+δXtdNt.

Let Ψ denote the standard normal UD and Φ denote the UD of interarrival times. Then, the IRF, which is of an expiration date T and a minimum interest rate L, is priced by(23)fl=01exp0TLX0expμt+σΨ1βt+ln1+δtΦ11β1+dtdβ1.

Proof.

Denote the IUD of Xt with Θt1β, i.e.,(24)Θt1β=X0expμt+σΨ1βt+ln1+δtΦ11β1.

Then, according to Theorem 1, we have(25)MXt<Θt11β,t0=1β,MXtΘt11β,t0=β.

Since(26)0TLXt+dt0TLΘt11β+dtXtΘt11β,t0,T,we have(27)M0TLXt+dt0TLΘt11β+dtMXtΘt11β,t0,Tβ.

Similarly, we have(28)M0TLXt+dt>0TLΘt11β+dtMXt<Θt11β,t0,T1β.

Furthermore, since(29)M0TLXt+dt0TLΘt11β+dt+M0TLXt+dt>0TLΘt11β+dt=1,we get(30)M0TLXt+dt0TLΘt11β+dt=β,that is,(31)0TLXt+dt,has an IUD(32)0TLΘt11β+dt.

Hence,(33)exp0TLXt+dt,has an IUD(34)exp0TLΘt11β+dt,as the function expz is increasing with z. Then,(35)fl=Eexp0TLXt+dt1=01exp0TLΘt11β+dtdβ1=01exp0TLΘt1β+dtdβ1.

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 K and J. Set βk=k/K and tj=jT/J, k=1,2,,K1, j=1,2,,J.

Step 2: set k=0.

Step 3: set kk+1.

Step 4: set j=0.

Step 5: set jj+1.

Step 6: compute(36)Hk,j=max0,X0expμtj+σΨ1βktj+ln1+δtjΦ11βk1C.

Step 7: compute(37)expTJj=1JHk,j.

Step 8: compute

(38)fc=11K1k=1K1expTJj=1JHk,j.Example 1.

Consider the uncertain interest-rate model dXt=μXtdt+σXtdCt+δXtdNt with an initial interest rate X0=0.04. Assume the parameters are μ=0.05, σ=0.03, and δ=0.01; the standard normal UD is(39)Ψz=1+expπz31,and the UD of interarrival times is(40)Φz=1+expπ2lnz31.

Then, the IRC with an expiration date T=4 and a maximum value C=0.05 has a price fc=0.0017.

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

Example 2.

Consider the uncertain interest-rate model dXt=μXtdt+σXtdCt+δXtdNt with an initial interest rate X0=0.04. Assume the parameters are σ=0.03 and δ=0.01; the standard normal UD is(41)Ψz=1+expπz31,and the UD of interarrival times is(42)Φz=1+expπ2lnz31.

The IRC has an expiration date T=4 and a maximum value C=0.05. Then, the function of the IRC price fc with respect to the parameter μ is shown in Figure 1. Apparently, the price fc is a strictly increasing function with respect to μ.

Relationship between fc and μ in Example 2.

5.2. Algorithm for Interest-Rate Floors

Step 1: according to the precision degree, preset two large numbers K and J. Set βk=k/K and tj=jT/J, k=1,2,,K1, j=1,2,,J.

Step 2: set k=0.

Step 3: set kk+1.

Step 4: set j=0.

Step 5: set jj+1.

Step 6: compute(43)Hk,j=max0,LX0expμtj+σΨ1βktj+ln1+δtjΦ11βk1.

Step 7: compute(44)expTJj=1JHk,j.

Step 8: compute(45)fl=1K1k=1K1expTJj=1JHk,j1.

Example 3.

Consider the uncertain interest-rate model dXt=μXtdt+σXtdCt+δXtdNt with an initial interest rate X0=0.04. Assume the parameters are μ=0.02, σ=0.03, and δ=0.01; the standard normal UD is(46)Ψz=1+expπz31,and the UD of interarrival times is(47)Φz=1+expπ2lnz31.

Then, the IRF with an expiration date T=4 and a minimum value L=0.04 has a price fl=0.0013.

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

Example 4.

Consider the uncertain interest-rate model dXt=μXtdt+σXtdCt+δXtdNt with an initial interest rate X0=0.04. Assume the parameters are σ=0.03 and δ=0.01; the standard normal UD is(48)Ψz=1+expπz31,and the UD of interarrival times is(49)Φz=1+expπ2lnz31.

The IRF has an expiration date T=4 and a minimum value L=0.05. Then, the function of the IRF price fl with respect to the parameter μ is shown in Figure 2. Apparently, the price fl is a strictly decreasing function with respect to μ.

Relationship between fl and μ in Example 4.

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).

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