^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

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.

The stochastic differential equations have been widely applied to mathematical finance since Black and Scholes [

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 [

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 [

Driven by both Liu processes and renewal processes, the UDEs with jumps that were proposed by Yao [

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 [

In order to model the interest rate in the ideal market which may change dramatically, Yu and Ning [

Yu and Ning [

(see [

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

Assume the interest rate

Let

Denote the IUD of

Then, according to Theorem

Since

Similarly, we have

Furthermore, since

Hence,

The theorem is proved.

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

Assume the interest rate

Let

Denote the IUD of

Then, according to Theorem

Since

Similarly, we have

Furthermore, since

Hence,

The theorem is proved.

The numerical algorithms to compute the prices of the IRCs and IRFs are designed based on formulas (

Step 1: according to the precision degree, preset two large numbers

Step 2: set

Step 3: set

Step 4: set

Step 5: set

Step 6: compute

If

Step 7: compute

If

Step 8: compute

Consider the uncertain interest-rate model

Then, the IRC with an expiration date

By using the algorithm mentioned above, we can numerically investigate the relationships of the IRC price

Consider the uncertain interest-rate model

The IRC has an expiration date

Relationship between

Step 1: according to the precision degree, preset two large numbers

Step 2: set

Step 3: set

Step 4: set

Step 5: set

Step 6: compute

If

Step 7: compute

If

Step 8: compute

Consider the uncertain interest-rate model

Then, the IRF with an expiration date

By using the algorithm mentioned above, we can numerically investigate the relationships of the IRF price

Consider the uncertain interest-rate model

The IRF has an expiration date

Relationship between

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.

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

The authors declare that they have no conflicts of interest.

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).