Using the Shanghai Interbank Offered Rate data of overnight, 1 week, 2 week and 1 month, this paper provides a comparative analysis of some popular onefactor short rate models, including the Merton model, the geometric Brownian model, the Vasicek model, the CoxIngersollRoss model, and the meanreversion jumpdiffusion model. The parameter estimation and the model selection of these singlefactor short interest rate models are investigated. We document that the most successful model in capturing the Shanghai Interbank Offered Rate is the meanreversion jumpdiffusion model.
The shortterm interest rate is not only the fundamental importance to the management of interest rate risk, but also plays an important role in many areas of asset pricing studies. Thus it is important to understand and model the term structure of interest rates. In fact, there are many benefits from a better understanding of the shortterm structure of interest rates and the models associated with it. Historically, many popular models currently used by academic researchers and practitioners have been developed in a continuoustime setting, which provides a rich framework for specifying the dynamic behavior of the shortterm riskless rate. Among the existing models, onefactor models are a popular class of interest rate models which are used for these purposes, especially in the pricing of interest rate derivatives.
While the Black and Scholes [
Although the spot rate dynamics have been extensively examined in mature markets in the literature, there has been little study of interest rates in China and other emerging markets. As stated in Hong et al. [
Obviously, it is well known that Chinese interest rates are quite heavily managed by the authorities. For example, it is often found that Chinese interest rates are strongly subject to administrative control by the government, and their mechanism is quite different from that of other developed markets (see [
The structure of this paper is organized as follows. In Section
Understanding and modeling the dynamics of Chinese interest rates represents one of the most challenging topics of financial research. In the literature, onefactor models are a popular class of interest rate models which are used for pricing interest rate derivatives. In fact, a commonly studied problem in finance is the modeling of the dynamics of the shortterm riskless interest rate, often using the following general stochastic differential equation:
The stochastic differential equation given in (
Merton [
Inspired by the research of Black and Scholes [
Since the short rate
Despite the popularity of GBM, some scholars have examined the empirical evidence for meanreverting behavior in interest rates. Indeed, there are also compelling economic arguments in favor of mean reversion. When the rates are high, the economy tends to slow down and borrowers require less funds. Furthermore, the rates pull back to its equilibrium value and the rates decline. On the contrary when the rates are low, there tends to be high demand for funds on the part of the borrowers and rates tend to increase. This feature is particularly attractive without it, interest rates could drift permanently upward the way stock prices do, and this is simply not observed in practice. To capture this mean reversion of interest rates, Vasicek [
Let us mention that the Vasicek model was the first one to capture mean reversion, which is an essential characteristic of the interest rate. In fact, the Vasicek model is a so called “mean reverting process.” When
The major drawback of the Vasicek model is that the short rate
The CIR model is the square root process that is proposed by Cox and Ross [
The process followed by the short rate in the CIR model is also called a squareroot process. The good mean reversion property in the Vasicek model is preserved in the CIR model. The bad property of possible negativity in the Vasicek model is removed in the CIR model under some assumptions and hence ensuring that the origin is inaccessible to the process. On the other hand, the distribution of the short rate in the CIR model is neither normal nor lognormal but it possesses a noncentral Chisquared distribution.
Since models of interest rates mentioned above are pure diffusion processes, the proposed dynamics of the various short interest rates do not meet empirical evidence on the presence of discontinuities in the process of the interest rate. The pure diffusion processes may not be sufficient to capture all of the observed asymmetry in interest rate changes. Furthermore, various economic shocks, news announcements, and government interventions in bond markets have pronounced effects on the behavior of interest rates and tend to generate large jumps in interest rate data. Statistically many researchers have shown that diffusion models (even with stochastic volatility) cannot generate the excessive leptokurtosis exhibited by the changes of the spot rates and that jumpdiffusion models are a convenient way to generate excessive kurtosis or, more generally, heavy tails. For example, Das [
We can see that the second random term in (
We would like to mention that the models proposed in this paper are continuous stochastic processes. One can perform statistical inference of these models based on either one or many realizations of the process over a time period. However, in practice, it is virtually impossible to observe a process continuously over any given time period, for example, due to limitations on the precision of the measuring instruments or due to the unavailability of observations at every time point. One often encounters practical difficulties to obtain a complete continuous observation of the sample path, and only the discrete time observations are possible. In this section, we will discuss the problem of estimating unknown parameters in the models proposed in this paper based on sampled data. Moreover, we will also introduce the likelihood ratio test in the latter part of this section.
In applications usually the processes cannot be observed continuously. Only discrete time observations are available. Hence, in this section, from discrete observations, we describe the econometric approach used in estimating the parameters of these interest rate models proposed in this paper. In order to obtain the unknown parameters in these models mentioned in this paper, we should choose a suitable method to estimate the unknown parameters. In the literature, several heuristic methods are available for solving problems of this sort. The most popular approaches are either the maximum likelihood estimation or the least squares estimation. In this section, we will adopt the maximum likelihood approach. The reasons for choosing this method are threefold. First, this technique has been applied efficiently in a large set (see, e.g., [
To implement the maximum likelihood approach, first we need to define the likelihood function. Then we can obtain the parameter values that maximize the value of this likelihood function. Without loss of generality, we assume that the process of
Now, we focus on the maximum likelihood estimation for Merton model. Using ItôDoeblin formula and setting
Thus, the discretetime observation can be expressed in the form of vector as
Inspired by Hu et al. [
It is well known that the discretized version of (
It is easy to check that the MLEs of
Using ItôDoeblin formula, for any
Moreover, for
Now, the probability density function of the process (
Without loss of generality, the loglikelihood function of a set of observation can be derived from the conditional density function:
Then we can derive the MLEs for
In mathematical finance, the CIR model describes the evolution of interest rates. Actually, if
Thus we can easily derive the loglikelihood function:
Finally, we find MLEs of
To implement maximum likelihood estimation, an EulerMarayuma discretization is performed on (
Based on (
Then the MLEs are obtained by maximizing the loglikelihood function with respect to
Now, we have presented five alternative approaches to modeling SHIBOR using stochastic processes. One of the main challenges of the interest rate researches has been comparing different models and selecting the best one. This subsection is designed to answer the important question: What type of process performs best in capturing the behavior of SHIBOR? Actually, there are two classes of model selections: nested in a general model and nonnested models. Log likelihood test can be used to test a model whether it is nested in a more general model. Obviously, the models in previous sections are nonnested in terms of their functional forms. Consequently, here we employ the Vuong test [
The classical Vuong test proposed in [
Specifically, the Vuong test considers the average difference in the log likelihoods of two competing statistical models. The null hypothesis of the test is that this average difference is zero. Let
The null hypothesis simply states that the two models are equally close to the true specification. The expected value in the above hypothesis is unknown. Vuong demonstrates that under fairly general conditions
In fact, when the number of coefficients in two models is different, the Vuong test needs a correction for the degrees of freedom. The adjusted statistic is
Moreover, let us mention that the estimated variance is computed in the usual way (sum of the squares minus square of the sums)
Suitably normalized, the test statistic is normally distributed under the null hypothesis:
The Vuong test can be described in simple terms. If the null hypothesis is true, the average value of the loglikelihood ratio should be zero. If
In this section, for an illustration of the method derived in previous sections, we apply these approaches to the real data. We model the term structure of SHIBOR in China and propose estimation of the implied parameters using the approaches described above. Moreover, we will infer the best model among the proposed models based on the test of Vuong [
In January 2007, China established SHIBOR System with the aim of building a benchmark yield curve. The SHIBOR is the average interest rate at which term deposits are offered between prime banks in the Shanghai wholesale money market or interbank market. In fact, the SHIBOR rate is not determined in a funding market but is set in a similar way to LIBOR, with the rate calculated as an arithmetic average of Renminbi offered rates by participating banks (currently 16) and is a fixing at 11:30 a.m. on each business day. The SHIBOR is the average interest rate at which term deposits are offered between prime banks in the Shanghai wholesale money market or interbank market. In fact, three key shortterm interbank interest rates, namely, China Interbank Offered Rate, SHIBOR, and the repo rate, have followed each other very closely over the past few years, with volatility having increased substantially since late 2005. Given the greater liquidity in the repo market (with the turnover in the repo market far exceeding that in the uncollateralized China Interbank Offered Rate market) and the tight relationship between the SHIBOR and the China Interbank Offered Rate market rates with the repo, it seems little additional information is added by the SHIBOR rates for maturities and period when trading is active in either the interbank repo or loan market. The SHIBOR system does, however, provide a benchmark interest rate quote when interbank trading is limited.
It is well known that SHIBOR is calculated, announced, and named on the technological platform of the National Interbank Funding Center in Shanghai. It is a simple, noguarantee, wholesale interest rate calculated by arithmetically averaging all the interbank Renminbi lending rates offered by the price quotation group of banks with a high credit rating. Currently, the SHIBOR consists of eight maturities: overnight, 1 week, 2 weeks, 1 months, 3 months, 6 months, 9 months, and 1 year. The price quotation group of SHIBOR consists of 16 commercial banks. These quoting banks are primary dealers of open market operation or market makers in the FX market, with sound information disclosure and active Renminbi transactions in China’s money market. SHIBOR Working Group of PBC decides and adjusts the panel banks, supervises and administrates the SHIBOR operation, and regulates the behavior of the quoting banks and the specified publisher in accordance with the Implementation Rules of SHIBOR.
Actually, the data utilized in our empirical study are extracted from the web of
Let us now turn to the analysis of real financial data. As explained above, four indices are transformed into the logreturn format. Basic descriptive plots for SHIBOR are presented in the following figures. Figure
Some statistical figures of daily returns for the O/N SHIBOR rate from October 8th 2006 to September 10th 2012.
Daily closing prices
Daily log returns
QQ plot
Probability density of the return
Some statistical figures of daily returns for the 1week SHIBOR rate from October 8th 2006 to September 10th 2012.
Daily closing prices
Daily log returns
QQ plot
Probability density of the return
Some statistical figures of daily returns for the 2week SHIBOR rate from October 8th 2006 to September 10th 2012.
Daily closing prices
Daily log returns
QQ plot
Probability density of the return
Some statistical figures of daily returns for the 1month SHIBOR rate from October 8th 2006 to September 10th 2012.
Daily closing prices
Daily log returns
QQ plot
Probability density of the return
To give a brief insight into the properties of the selected data, Table
Summary statistics of daily returns for time series of the return of SHIBOR.
Index  Minimum  Maximum  Mean  Std. Dev.  Skewness  Kurtosis 

O/N 






1 week 






2 weeks 






1 month 






The basic stylized facts are as follows: near nonstationary behavior (slow mean reversion) can be observed; large changes and small changes are clustered together; the volatility increases with the level (level effect); obviously larger skewness and great kurtosis indicate high peaks and fat tails (leptokurtic). Hence, a pronounced feature of these data is that the SHIBOR is more volatile.
Based on the situation of discrete observations mentioned in the previous sections, we now proceed to estimate all the unknown parameters of stochastic models mentioned above based on the combination of the maximum likelihood approach and the selected data. Table
Parameter estimates for the singlefactor diffusion models.
Series  Model name  Parameters  Log likelihood  






 
O/N  Merton  0.386810  0.366409  —  —  —  —  −116.5897 
GBM  0.004838  0.102661  —  —  —  —  −79.4846  
Vasicek  0.059131  0.357140  2.182412  —  —  —  −67.1018  
CIR  0.185532  0.248036  5.304625  —  —  —  −38.2561  
MRJD  0.002212  0.045752  2.001096  0.054475  −0.047486  0.745071  −16.3986  


1Week  Merton  0.505131  0.452737  —  —  —  —  −108.7844 
GBM  0.006433  0.113952  —  —  —  —  −93.3939  
Vasicek  0.058609  0.442345  2.732211  —  —  —  −86.9711  
CIR  0.043466  0.219753  2.735201  —  —  —  −58.2525  
MRJD  0.003102  0.147420  2.408793  0.035441  −0.147026  1.165968  −13.5398  


2Week  Merton  0.529391  0.491417  —  —  —  —  −186.2086 
GBM  0.046521  0.093887  —  —  —  —  −106.2416  
Vasicek  0.058597  0.451624  2.731857  —  —  —  −105.7956  
CIR  0.027683  0.211341  2.803126  —  —  —  −68.6953  
MRJD  0.005202  0.210520  2.112525  0.042165  −0.22365  1.236651  −13.8344  


1Month  Merton  0.619768  0.252719  —  —  —  —  −164.5811 
GBM  0.013203  0.044481  —  —  —  —  −88.6862  
Vasicek  0.013835  0.243282  3.313312  —  —  —  −56.6224  
CIR  0.080097  0.105921  3.354525  —  —  —  −37.7186  
MRJD  0.001618  0.078245  2.000189  0.038285  0.023883  0.714316  −27.5073 
As shown, the models vary in their explanatory power for interest rate changes. These estimations reveal some important stylized facts of SHIBOR. There exists significant mean reversion in SHIBOR. For example, the estimates of the drift parameters in Vasicek, CIR, and MRJD models in Table
Comparing the log likelihoods from Table
Results of the Vuong test.
Merton  GBM  Vasicek  CIR  MRJD  

Merton  —  4.180489  2.241832  4.799925  4.317839 
GBM  —  —  3.370632  3.110825  3.799735 
Vasicek  —  —  —  2.012456  2.482186 
CIR  —  —  —  —  3.302152 
MRJD  —  —  —  —  — 
Continuoustime models of the term structure of interest rates developed so far use a stochastic process in order to model the dynamics of the shortterm interest rate. While there is a great deal of interest in using stochastic differential equations to model financial time series data, it has been difficult to find effective ways to estimate these models. An extensive collection of continuoustime models of the shortterm interest rate is evaluated over data sets of mature markets that have appeared previously in the literature. However, due to the relatively short history of the Chinese bond markets and the strict regulation of Chinese interest rates, there has been little study of spot interest rates in China.
Despite the numerous empirical studies of spot interest rate models, litter effort has been denoted to examining the behavior of SHIBOR. In this paper, we have contributed to literature by providing the comprehensive empirical analysis of SHIBOR. We propose the maximum likelihood methodology to estimate the parameters of five singlefactor interest rate models and compare these five different models of shortterm interest rate dynamics in order to determine which model best fits the SHIBOR data. In addition to the wellknown optimality properties of MLE, the availability of the likelihood provides a convenient tool for specification analysis. The results of the Vuong test show that the popular models, including Merton, GBM, Vasicek, CIR, and MRJD, perform poorly relative to continuous models. The MRJD should be preferred other five models.
There are a couple of issues that hold potential for future research in this area. First, we must acknowledge that the single factor model used here, though empirically tractable, is also subject to some criticism. The dependence on a single factor greatly limits the possible shapes of the yield curve and often leads into situations, where the theoretical yield curve does not correspond to the market yield curve. Consequently, the first suggestion is to introduce the twofactor model or the threefactor model. Although regime switching and jumps are important for modeling Chinese spot rate dynamics, they are still grossly misspecified. There is a long way to go before we reach a correct specification for Chinese spot rate dynamics. Hence, the second suggestion is to explore possible sources of model misidentification by examining the marginal distribution and model dynamics separately. Finally, one of the main assumptions of Vuong’s model selection criterion is that the data are independent and identically distributed, which normally does not hold for time series. Thus the third direction is to present some modified tests of the classical Vuong test.
The author declares that he has no financial and personal relationships with other people or organizations that can inappropriately influence his work, there is no professional or other personal interest of any nature or kind in any product, service, and/or company that could be construed as influencing the position presented in, or the review of, the paper.
This research was supported by the National Natural Science Foundation of China (nos. 71101056 and 71301144), the Major Program of National Social Science Foundation of China (11&ZD156), the Humanity and Social Science Youth foundation of Ministry of Education of China (no. 13YJC630227), the Zhejiang Provincial Natural Science Foundation of China (no. LQ13G010001), and the Educational Commission of Zhejiang Province of China (no. Y201329832).