Uncertain interest rate model for Shanghai interbank offered rate and pricing of American swaption

Abstract

In the framework of uncertainty theory, this paper investigates the pricing problem of American swaption. By assuming that the floating interest rate obeys an uncertain differential equation, the pricing formula of American swaption is derived. Furthermore, parameter estimation of the uncertain interest rate model is given, and the uncertain hypothesis test shows that the uncertain interest rate model fits the Shanghai interbank offered rate well. Finally, as a byproduct, this paper also indicates that stochastic differential equations cannot model real-world interest rates.

Appendix A Stochastic interest rate model

Let us reconsider overnight SHIBOR from April 1, 2021 to March 15, 2022 (see Table 1). Assume the interest rate obeys a stochastic differential equation

$$\begin{aligned} \mathrm{d}r_t=(m-ar_t)\mathrm{d}t+\sigma \mathrm{d}W_t \end{aligned}$$

where m, a and $\sigma$ are unknown parameters, and $W_t$ is a Wiener process. For any fixed parameters $m,\ a,\ \sigma$ and $i\ (2\le i\le 238)$, we solve the updated stochastic differential equation with an initial value $r_{i-1}$

$$\begin{aligned} \mathrm{d}r_t=(m-ar_t)\mathrm{d}t+\sigma \mathrm{d}W_t, \end{aligned}$$

and get the probability distribution of normal random variable $r_{i}$

$$\begin{aligned} F_i(x)=\frac{1}{\nu \sqrt{2\pi }}\int _{-\infty }^x\exp \left( -\frac{(y-\mu _i)^2}{2\nu ^2}\right) \mathrm{d}y \end{aligned}$$

where $\mu _i$ is the expected value, i.e.,

$$\begin{aligned} \mu _i=\frac{m}{a}+\exp (-a)\left( x_{i-1}-\frac{m}{a}\right) , \end{aligned}$$

and $\nu ^2$ is the variance, i.e.,

$$\begin{aligned} \nu ^2=\frac{\sigma ^2}{2a}+(1-\exp (-2a)). \end{aligned}$$

Define the i-th residual

$$\begin{aligned} \varepsilon _i(m,a,\sigma ):=F_i(r_i). \end{aligned}$$

Then, $\varepsilon _i(m,a,\sigma )\in (0,1)$ can be regarded as a sample of uniform probability distribution ${{\mathcal {U}}}(0,1)$.

Since the number of unknown parameters is three and the first three moments of the uniform probability distribution ${{\mathcal {U}}}(0,1)$ are 1/2, 1/3, and 1/4, we have the following equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \frac{1}{237}\sum _{i=2}^{238}\varepsilon _i(m,a,\sigma )=\frac{1}{2}\\ \displaystyle \frac{1}{237}\sum _{i=2}^{238}\varepsilon _i^2(m,a,\sigma )=\frac{1}{3}\\ \displaystyle \frac{1}{237}\sum _{i=2}^{238}\varepsilon _i^3(m,a,\sigma )=\frac{1}{4}, \end{array}\right. } \end{aligned}$$

(13)

whose root is

$$\begin{aligned} m=0.0303,\quad a=1.5261,\quad \sigma =0.0031. \end{aligned}$$

Thus we obtain a stochastic interest rate model,

$$\begin{aligned} \mathrm{d}r_t=(0.0303-1.5261r_t)\mathrm{d}t+0.0031\mathrm{d}W_t \end{aligned}$$

(14)

Let us test whether the stochastic interest rate model (14) fits SHIBOR interest rates. That is, we should test whether the uniform probability distribution ${{\mathcal {U}}}(0,1)$ fits the 237 residuals

$$\begin{aligned} \varepsilon _i(0.0229,1.1591,0.0032),\ i=2,3,\cdots ,238. \end{aligned}$$

See Fig. 5. To check if the residuals are from the same population ${{\mathcal {U}}}(0,1)$, we apply the “Chi-square goodness-of-fit test” (Snedecor & Cochran, 1989) with a significance level 0.05. Then using the function ‘chi2gof’ in Matlab, we obtain the p-value as 0.0011, indicating that the residuals do not come from the same population ${{\mathcal {U}}}(0,1)$. Therefore, the stochastic interest rate model (14) does not fit overnight SHIBOR.

Fig. 5

Residual plot of stochastic interest rate model (14)