## 1. Introduction

Interest rate swap is one of the most popular interest derivatives. The interest rate swap began trading in 1981, and now owns a hundred billion dollar market. A swap is a contract in which two counterparties exchange cash flows at prearranged date, where the cash flows’ values are derived from some underlying assets, such as interest rate, equities, exchange rates, or commodities. When the underlying asset is interest, the swaps are called interest rate swaps.

In order to value the interest rate swaps, Murphy [

1] assumed that the floating interest rate follows a static model, and gave the classical valuation model of the interest rate swaps. Smith [

2] gave a static valuation model using overnight indexed swap (OIS) rates. Mitra [

3] extended the static model by assuming that the variations of the interest rate followed a stochastic differential equation. Yang [

4] considered the bilateral default risk and gave another valuation model. Li [

5] and Balsam [

6] focused on the corporate use of interest rate swaps. Since multi-factor interest models fit the term structure better than one-factor models, Fanelli [

7] and Ravi [

8] discussed the interest rate swaps under the assumption that the floating interest rate followed Heath-Jarrow-Morton and Cox-Ingersoll-Ross models, respectively.

The traditional models of valuing interest rate swaps are mainly based on probability theory. However, in a real financial market, the interest rate is affected by the timely policies and news (e.g., If the central bank suddenly announces that the country will practice tight monetary policy, then the interest rate will rise). These affections will lead the interest rate to deviate from the previous tendencies. At this time, we need a new interest rate model to describe the new pattern. However, there is little information about the new pattern, so in this situation, we can only employ experts to give the belief degrees of the interest rate.

In order to model belief degrees, uncertainty theory was founded by Liu [

9] in 2007, and refined by Liu [

10] in 2010. Uncertainty theory is a branch of mathematics based on normality, duality, subadditivity, and product axioms.To model the evolutionary uncertain phenomena, Liu [

11] proposed uncertain process in 2008. It can be seen as a sequence of uncertain variables indexed by time. What is more, Liu [

11] designed a special uncertain process called Liu process. Liu process is a Lipschitz continuous uncertain process with stationary and independent increments, and after that, he established uncertain calculus. Liu process and uncertain integral can be seen as uncertain counterparts of Brownian motion and stochastic integral. Liu [

11] initiated the study of uncertain differential equations. Afterwards, Chen and Liu [

12] proved the existence and uniqueness theorem for uncertain differential equations. Some stability theorems were proved by Yao et al. [

13]. Moreover, Yao and Chen [

14] found a relationship between an uncertain differential equation and a family of ordinary differential equations. This result is called Yao–Chen formula. Chen [

15] studied a special type of multi-dimensional uncertain differential equation called a nested uncertain differential equation. He extended the interest rate models to multi-factor term structure models.

As an application of uncertain differential equations, an uncertain stock model was proposed by Liu [

16], and various option pricing formulas were derived (e.g., European option [

16], American Option [

17], and Asian option [

18]). Chen and Gao [

19] also presented three types of interest rate models, which are the uncertain counterparts of Ho-Lee model, Vasicek model, and CIR Model. In this paper, we mainly discuss the valuation model of the interest rate swaps under the assumption that interest rate follows an uncertain differential equation. Firstly, we assume that the floating interest rate follows a mean-reversion one-dimensional uncertain differential equation, and give the explicit solution of the interest rate swap. Secondly, considering the hump effect of the term structure, we assume that the floating interest rate follows a nested-uncertain differential equation, and derive explicit solutions of interest rate swap. Since explicit solutions are difficult to calculate in many situations, a numerical method was also designed. Lastly, two examples are presented for illustrating purpose.

The organization of this paper is as follows. In section two, we gave the valuation model wherein the floating interest rate follows a mean-reversion uncertain differential equation. In section three, we gave the valuation model wherein the floating interest rate follows a nested uncertain differential equation. In section four, we designed a numerical method to calculate the value of the interest rate swap, and give two examples to show the applications.

## 2. The Valuation Model with Mean-Reversion Uncertain Differential Equation

An interest rate swap is a popular financial derivative instrument. It regulates a fixed interest rate r. It allows the two parties to exchange interest rate cash flows, based on a specified notional amount from a fixed rate r to a floating rate, or from one floating rate to another. So, the value of the swap for the fixed rate payer (denoted by ${V}_{fix}$) is the present value of the floating interest minus the present value of the fixed interest, and the value of the swap for the floating rate payer (denoted by ${V}_{float}$) is the present value of the fixed interest minus the present value of the floating interest. If these two legs are equal to zero, then the regulated fixed interest rate r is fair for the two parties.

Because the floating interest rate changes with time, we assume that the floating interest rate follows an uncertain differential equation, as follows:

where

m,

a, and

σ are constants, and

σ > 0. This model can be seen as the counterpart of the Vasicek model. The process for

${r}_{t}$ is a mean-reversion process.

${C}_{t}$ is a Liu process. It is used for modeling unexpected market risk.

σ determines the volatility of the interest rate.

$m/a$ is the long-run equilibrium.

a measures the speed of reversion.

Let

${V}_{fix}$ and

${V}_{float}$ denote the fair value of the interest rate swap contract for the fixed interest rate payer and the floating interest rate payer, respectively, and

${S}_{0}$ is the nominal principle. At time

T, the fixed interest rate payer must pay the fixed rate interest and receive the floating rate interest. So, the payoff of the fixed interest rate payer at time

T is as follows:

Considering the time value of the money, the present value of the payoff is

So, the net return of the fixed interest rate payer is

On the other hand, the net return of the floating interest rate payer is

Therefore, the fair value of the interest rate swap for the fixed interest rate payer should make the fixed interest rate payer and the floating interest rate payer have an identical expected return

With the same analysis as the floating rate payer, the fair value of the interest rate swap for the floating interest rate payer should follow the equation below:

**Definition** **1.** The interest rate swap contract regulates that the notional principal amount is ${S}_{0}$, the fixed interest rate is r, and the floating interest rate is ${r}_{t}$, and ${r}_{t}$ is defined as Equation (1). The two counterparties need to exchange their cash flows at time $t=T$. The fair values for the fixed interest rate payer and the floating interest rate payer are as follows: **Theorem** **1.** Assume ${r}_{t}$ follows an uncertain differential Equation (1), and the interest rate swap contract is described in Definition 1. The fair value for the fixed interest rate payer and the floating interest rate payer are as follows:where **Proof** **of Theorem 1.** Solving the ordinary equation

where

$0<\alpha <1$ and

${\Phi}^{-1}\left(\alpha \right)$ is the inverse standard normal uncertainty distribution, we have

This means that the uncertain differential equation

has an

α-path

It follows Yao–Chen formula that

${r}_{t}$ has an inverse uncertainty distribution

Since

$\mathrm{exp}\left(-{\int}_{0}^{T}{r}_{t}\mathrm{d}t+rT\right)$ is a decreasing function for

${r}_{t}$, it has an inverse uncertainty distribution

## 3. The Valuation Model with Nested Uncertain Differential Equation

In the real capital market, the drift of interest rate may display a hump. However, the single-factor interest rate models are not rich enough to describe this phenomenon. Therefore, we added an uncertain reversion process into the drift term. Then, the previous uncertain differential equation evolved into a nested uncertain differential equation. Chen [

15] showed that this model guarantees a better fitting for the hump in the drift structure. So, in this part, we assume that the the interest rate follows a nested uncertain differential equation, as follows:

where

${u}_{t}$ follows an uncertain differential equation

The uncertain process ${u}_{t}$ reverses to 0 at rate b. ${C}_{1t}$ and ${C}_{2t}$ are two independent Liu processes.

**Theorem** **2.** Assume ${r}_{t}$ follows a nested uncertain differential Equation (19), and the interest rate swap contract is described in Definition 1. The fair value for the fixed interest rate payer and the floating interest rate payer are as follows:where ${\Psi}_{t}^{-1}\left(1-\alpha \right)={r}_{t}^{\alpha}$, and ${r}_{t}^{\alpha}$ is the solution for the following ordinary differential equation:${u}_{t}^{\alpha}$ is the solution for the following from the ordinary differential equation **Proof** **of Theorem 2.** According to Theorem A2, the nested uncertain differential equation

has an

α-path

${u}_{t}^{\alpha}$ is solving from

So,

${r}_{t}$ has an inverse uncertainty distribution

Since

$\mathrm{exp}\left(-{\int}_{0}^{T}{r}_{t}\mathrm{d}t+rT\right)$ is a decreasing function for

${r}_{t}$, it has an inverse uncertainty distribution

## 4. Numerical Examples

Based on Theorem A2, we find that there are explicit solutions for the fair values of the interest rate swaps. However, it is sometimes difficult to calculate. So, in this section, we first give a 99-method for solving the fair values of the interest rate swaps.

**Step** **0:**Set $i=1$, $j=1$, ${s}_{i}=0$, ${\alpha}_{j}$ = 0, n = 100, step length = $T/n$.

**Step** **1:**Set ${\alpha}_{j}={\alpha}_{j}+0.01$, ${s}_{i}={s}_{i}+T/n$.

**Step** **2:**Solving the corresponding ordinary differential equations

and

respectively. Then we obtain

${r}_{{s}_{i}}^{1-{\alpha}_{j}}$ and

${u}_{{s}_{i}}^{{\alpha}_{j}}$. It is suggested to employ a numerical method to solve the equation when an analytic solution is unavailable.

**Step** **3:**Repeat Step 1 and Step 2 99 times.

**Step** **4:**The solution

${r}_{{s}_{i}}$ has a 99-table,

${\alpha}_{j}$ | 0.01 | 0.02 | ⋯ | 0.99 |

${r}_{{s}_{i}}^{1-{\alpha}_{j}}$ | ${r}_{{s}_{i}}^{0.99}$ | ${r}_{{s}_{i}}^{0.98}$ | ⋯ | ${r}_{{s}_{i}}^{0.01}$ |

This table gives an approximate uncertainty distribution of

${r}_{{s}_{i}}$; i.e, for any

α =

$i/100$,

$i=1,2,\cdots ,99$.

**Step** **5:**Set $i=i+1$, $j=j+1$, ${\alpha}_{j}=0$, and repeat Step 1 to Step 4 n times.

**Step** **6:**

**Example** **1.** Consider that there are two firms, owning the same floating interest rate bond. ${S}_{0}$ denotes the face value of the bond, and ${S}_{0}$ is 1 million. The bond will expire the next year. Both of the firms think that they face the interest rate risk. In order to hedge this kind of risk, they decide to sign an interest rate swap contract. The one year floating-for-fixed interest rate swap contract regulated that the fixed interest rate is 0.07. These two companies want to know whether the regulated fixed rate is reasonable. Firm 1 thinks that the drift of the interest rate will not display a hump. So, it models the floating interest rate by the model proposed in Section 2. That is,where a = 0.4, $m/a$ = 0.07, and σ = 0.0025. After calculation, the fair value for the fixed interest rate payer is ${V}_{fix}$ = 0.1366, and the fair value for the floating interest rate payer is ${V}_{float}$ = −0.1366.

Firm 2, on the contrary, thinks that the drift of the interest rate will display a hump. So, it models the floating interest rate by the model proposed in Section 3. That is,and ${u}_{t}$ followswhere a = 0.07, k = 0.4, b = 0.01, ${\sigma}_{1}$ = ${\sigma}_{2}$ = 0.0025. After calculation, the fair value for the fixed interest rate payer is ${V}_{fix}$ = 0.01, and the fair value for the floating interest rate payer is ${V}_{float}$ = −0.01.

Through the comparison, we can find that the hump effect of the interest rate drift will definitely influence the interest rate. Whether the floating interest rate consists dump effect is judged by the experiences of the decision-maker. No matter what his judgement it is, however, this paper provides him with powerful tools.

**Example** **2.** On 2 November 1953, The Procter & Gamble company (P&G) signed an interest swap contract with Bankers Trust New York Corporation (BT). The contract regulated that the nominal principle is 2 hundred millions. This contract would last for 5 years. The two companies would exchange the interest semiannually. The BT is the fixed interest rate payer, who promised to pay at 5.3%, while P&G is the floating interest rate payer, who promised to pay at 1 month average interest rate of commercial bills minus 75 bp. However, on 17 May 1954, the Federal Reserve System decided to raise its interest rate; this would directly raise the interest rate of the commercial bills. For this new interest rate pattern, we had not enough samples, so we can only use belief degrees to model it. If we thought that there is no hump effect of the interest rate drift, we could assume that the interest rate followed the mean-reversion uncertain differential equation.where a = 0.4, $m/a$ = 5.55%, and σ = 0.0025. Then, we can estimate that This meant that P&G might face a loss in the future. However, the management of P&G did not pay enough attention to the potential loss, and this led the company to lose about 157 million in April 1994.