The Short-term Swap Rate Models in China

International Journal of Business Marketing and Management (IJBMM)
Volume 4 Issue 10 October 2019, P.P. 26-34
ISSN: 2456-4559
www.ijbmm.com
International Journal of Business Marketing and Management (IJBMM) Page 26
The Short-term Swap Rate Models in China
Hanhan Guo, Wanxiao Tang, Peibiao Zhao
Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu,
P. R. China
Abstract: This paper analyzes the swap rates issued by the China Inter-bank Offered Rate(CHIBOR) and
selects the one-year FR007 daily data from January 1st, 2019 to June 30th, 2019 as a sample. To fit the data,
we conduct Monte Carlo simulation with several typical continuous short-term swap rate models such as the
Merton model, the Vasicek model, the CIR model, etc. These models contain both linear forms and nonlinear
forms and each has both drift terms and diffusion terms. After empirical analysis, we obtain the parameter
values in Euler-Maruyama scheme and relevant statistical characteristics of each model. The results show that
most of the short-term swap rate models can fit the swap rates and reflect the change of trend, while the CKLSO
model performs best.
Keywords: swap rate models; Euler-Maruyama scheme; Monte Carlo Simulation; swap rates.
I. Introduction
Short-term interest rate is one of the most important and fundamental economic variables in finance, which
determines the structure of interest rates and the price of assets. In Chinese swap market, we have short-term
swap rates as a kind of short-term interest rates. Expressed as dr(t) = f(r(t), t)dt + (r(t),t)dWt, a short-term swap
rate model starts from the stochastic differential equation of interest rate and describes the continuous changes
during continuous time and continuous state. Classic models like Hull-White model[1], Vasicek model[2], CIR
model[3] has been widely used in theoretical modeling and empirical analysis to determine the price and the risk
of fixed income assets[4, 5, 6]. Contributing to the flexible form and the advancing statistical methods, short-
term interest rates are not limited to the fixed-income market, but widely used in stock prices [7] and stock
options[8], insurance[9, 10] and other fields. Can these current models describe the changes in Chinese swap
market? Which model performs best in the empirical analysis? They are the problems to be solved in this article.
While in fitting the swap market with the short-term swap rate models, we must make proper parameter
estimation. However, the conditional density functions of these stochastic volatility models cannot be explicitly
expressed as well as the likelihood functions and unconditional moments. As a result, the parameter estimation
is still complicated. Recent studies suggest three methods in parameter estimation. The first method is based on
the traditional parameter estimation way by using the approximate methods or the simulated methods to
construct the likelihood function and unconditional moment. It mainly include simulating maximum likelihood
estimation (like SML), simulating moment estimation (like SMM)[11], etc. The second method indirectly
estimates stochastic volatility models by introducing an auxiliary model (such as ARMA, GARCH model)[12,
13] or semi-parametric method[14], which includes indirect inference and effective moment estimation
(EMM)[15, 16], etc. The third method is the parameter posterior distribution analysis based on the Bayesian
principle[17]. Because of some high dimensional integration, we can hardly calculate the posterior mean and
standard deviation. Although scholars overcome the posterior distribution calculation by using the Markov chain

Monte Carlo (MCMC) method, more difficulties appear such as requirements for computing ability and
calculation software.
So far, the academic research of short-term interest rate models has achieved fruitful results in both
Chinese stock market and bond market interest rate. In this paper, we applies these models to the Chinese swap
market. We wonder whether the short-term swap rate models can describe the statistical characteristics in
Chinese swap market. As for parameter estimation, since the short-term swap rate models are all stochastic
differential equations, we adopt the method issued by Cai Xinrui [18]. Based on the discrete data, this method
can derive the distribution of the correlation operation of the numerical solutions in the Euler-Maruyama
scheme. Let the data obey this distribution, we can compute the unknown parameters in the drift coefficient and
the diffusion coefficient. Based on the algorithm above, we make Monte Carlo simulation with Matlab. As a
result, we obtain the parameter values and relevant statistical characteristics along with the most efficient model.
II. Preliminaries
There are many short-term swap rate models with )()),(()),(()( tdWttrdtttrftdr  format. The
functions )(f and )( are different in specific models. This paper cites nine typical classic short-term swap
rate models and improved short-term swap rate models, which is shown in Table 1.
They are widely used in bond pricing, option pricing, and swap pricing. The drift function and the diffusion
function are versatile with linear and nonlinear forms.
Table 1: Short-term swap rate models
Number Model Expression
1 Merton dWadtdr 
2 O-U dWardtdr 
3 Vasicek dWdtbardr  )(
4 CIR dWrdtrbadr  )(
5 GBM brdWardtdr 
6 CKLSO dWdbrdtcardr )()( 
7 dWdtadr   )2/( 2
8 dWradtradr 22
1)5.0( 
9 dWradtrradr )1()]1([ 222


Consider stochastic differential equation )()),,(()),,(()( tdWtrdttrftdr   . We denote r(t0) =
r0, r(ti) = ri, where Ttt 0
. i is an element of the underlying probability space  . Function )(f and )(
are known for given dr(t). W(t) is a standard Brownian motion. In addition, )(f and )( are unknown
parameters to be estimated. To simplify the exposition, we assume the parameters  and  are one-dimensional.
Moreover, we divide the time interval ],[ 0 Tt into n pieces of equal length h, such that Ttn  ，
htt nn  1
，…， htt  12
， htt  01
.
Discretizing the differential equation above, we obtain the Euler-Maruyama scheme
)]()()[,(),( 11 iiiiii tWtWrrhfrr    (2.1)
Suppose we have m observations ),(,),,(),,( 11211
101010
trtrtr tt
m
tttt 
  , each appearing after evolution of a
time period h from t0 to t1 after giving the initial value r0.
Notice that )),(),,((~ 2
1  iiii rhrhfNrr 
and we have
),())(),((
1
1
),(),(
1
0
22
0
1
1
00
1
1
10
10


rhrhftr
m
rrhftr
m
m
j
tt
j
m
j
tt
j









(2.2)
According to (2.2), we can estimate 1
ˆ and 1
ˆ .
Similarly, we can define m observations ),(,),,(),,( 22221
212121
trtrtr tt
m
tttt 
  , each appearing after evolution
of a time period h from the given initial value r1. Then, we have
),())(),((
1
1
),(),(
1
1
22
1
1
2
11
1
2
21
21


rhrhftr
m
rrhftr
m
m
j
tt
j
m
j
tt
j









(2.3)
According to (2.3), we can estimate 2
ˆ and 2
ˆ .
Repeatedly, we can define m observations ),(,),,(),,( 11211
111





 
 i
tt
mi
tt
i
tt
trtrtr iiiiii
 , each appearing after
evolution of a time period h from the given initial value ri. Then we can estimate n ˆ,,ˆ,ˆ 43  and n ˆ,,ˆ,ˆ
43  .
Finally, we take the average to get the value of the parameters  and  , such that
 

n
i
i
n
i
i
nn 11
ˆ1
ˆ
1
 ， (2.4)

III. Selection of models and empirical analysis
In this section, we analyze the swap rates compiled and issued by the China Inter-bank Offered Rate
(CHIBOR). On the one hand, we fit the swap rates with short-term swap rate models mentioned in section 2. On
the other hand, we obtain the parameter values in Euler-Maruyama scheme and relevant statistical
characteristics with Monte Carlo Simulation. The sample data comes from the China Money Network. It covers
the FR007 fixing curve for 122 working days from January 1st, 2019 to June 30th, 2019. The recording time is
12:00. The price type is average. And the period is one year. First, we divide and serialize the equals time of the
data. Generally, the 122 working days in the first half of 2019 will be approximated to 0.5, while the whole year
is regarded as unit 1. As a result, in the first working day we have t0 = 0 , in the second working day we have t1
= 0.004,…, in the 122nd working day we have t121 = T = 0.484. Daily swap rate corresponds to its time.
We realize the algorithm with Matlab2016b. During the experiment, we should store the sample data as a
‘.dat’ file, set the trajectories as 10,000 and adjust a smaller step-size of time as 0.001. Before the simulation, we
should enter a set of free values for the parameters as ‘free parameters’ beforehand. Only when the free
parameter is reasonable, can the parameter estimation be returned. After running the program, we obtain three
figures in total. Figure (a) is a map of 10,000 simulated trajectories. Figure (b) plots original data (  ) vs the
empirical mean (green line), the 95% confidence bands (dashed lines) and the first-third quartile (dotted lines) of
10,000 simulated trajectories. Figure (c) is a histogram to show the distribution of 10,000 trajectories.
Note that the parameter estimation will not be returned if the free parameters is unreasonable or the sample
data fluctuates greatly. If the free parameters is reasonably, the mean line and the distribution will be similar to
the sample data. To illustrate this phenomenon, we exemplify the Merton model (Model 1) and set different free
parameters to distinguish ‘reasonable free parameters’.
(a) (b) (c)
Figure 1: Contrast between different parameters
In Figure 1(a), we set 1.01.0  ，a . In Figure 1(b), we set 8.18.1  ，a . In Figure 1(c), we set
8.02.0  ，a .
We can observe the original data (  ) vs the empirical mean (green line), the 95% confidence bands
(dashed lines) and the first-third quartile (dotted lines) of 10,000 simulated trajectories. Obviously, nearly half
of the sample points ran out of the 95% confidence interval in Figure 1(a) while almost all sample points were
below the mean value in Figure 1(b). In comparison, the sample points are quite similar to the mean and
confidence domains in Figure 1(c). As a result, the free parameters of Figure 1(c) are ‘reasonable free
parameters’

Table 2: Free parameters of short-term interest rate models
Number Expression Free parameters
1 dWadtdr  1,1.0  a
2 dWardtdr  1,1.0  a
3 dWdtbardr  )( 1,1.0,1.0  ba
4 dWrdtrbadr  )( 8.0,1.0,1.0  ba
5 brdWardtdr  2.1,2.0  a
6 dWdbrdtcardr )()(  1,5.0,2,0.15  cba
7 dWdtadr   )2/( 2
5.1,25.0  a
8 dWradtradr 22
1)5.0(  4.0a
 4.0a
We set the free parameters of short-term swap rate models, which is shown in Table 2. Then, fit the data
10,000 times and we obtain the parameter mean and 95% confidence interval, which is shown in Table 3. The
mean of the simulated values of the parameters is equivalent to the estimated value of the parameters. And the
last two columns are the 2.5% quantile and the 97.5% quantile, respectively.
Table 3: Parameter Estimation
NO. Expression Param
eter
Mean 2.50% 97.5%
1 dWadtdr 
a 0.115441 -1.2248 1.4557
 0.479524 0.4212 0.5378
2 dWardtdr 
a 0.027171 -0.5318 0.4775
 0.479587 0.4213 0.5379
3 dWdtbardr  )(
a 0.110633 -10.431 10.652
b 0.115675 -27.891 28.122
 0.479775 0.4211 0.5384
4 dWrdtrbadr  )( a -0.042784 -0.7772 0.6916

b -0.010413 -54.832 54.812
 0.291128 0.2556 0.3267
5 brdWardtdr 
a 0.059523 -0.4358 0.5549
b 0.176920 0.1552 0.1987
6 dWdbrdtcardr )()( 
a -3.000000 -3.0000 -3.0000
b 0.817881 0.6153 1.0205
c 8.327462 7.2973 9.3577
d -1.709720 -2.2337 -1.1857
7
a 0.230301 -1.1095 1.5701
 0.479527 0.4212 0.5379
8 dWradtradr 22
1)5.0(  a 0.419982 0.4188 0.4212
 a 0.057446 0.0504 0.0645
In addition, we also get the statistical characteristics of the swap rate fixing curves with MC simulation
shown in Table 4.
Table 4: Statistics characteristics of the swap rate fixing curves
NO. Expression Mean Var. Std. Skewness Kurtosis
1 dWadtdr  2.6522 0.1128 0.3359 -1.3522 13-
10 2.9553
2 dWardtdr  2.6307 0.1144 0.3382 -8.5086 13-
10 2.9543
3 dWdtbardr  )( 2.7966 0.1192 0.3453 -2.5583 14-
10 2.9513
4 dWrdtrbadr  )( 2.6510 0.1114 0.3337 0.1842 2.9992
5 brdWardtdr  2.6725 0.1105 0.3324 0.3658 3.1924
6 dWdbrdtcardr )()(  2.7345 0.0446 0.2112 1.4085 7.0328
7 dWdtadr   )2/( 2
2.6521 0.1128 0.3359 1.1948 13-
10 2.9553
8 dWradtradr 22
1)5.0(  — — — — —
 2.6501 0.1066 0.3265 0.6729 3.9116

As we can see, except for Model 8, the rest models can describe the statistical characteristics of the swap
rates. The reason why Model 8 failed is the actual value of swap rate r. In our sample, the swap rate r is always
larger than 2, making the fluctuation term an imaginary number which exceeds our calculation range. Hence, its
trajectories cannot be plotted on the real number field.
We focus on the parameter estimations among the short-term swap rate models that fit the swap rates
successfully. Considering the standard deviation we find Model 6 has the best fitting effect because its standard
deviation is the smallest. In Model 6, both the drift function and the diffusion function are linear, which implies
the linear form may be good enough to fit the swap rates. In fact, Model 6 also fits better in the bond market
than any other models, perhaps more links are worth exploring. As for skewness, Model 1, Model 2, Model 3,
and Model 7 exhibit a normal distribution with almost unbiasedness. Model 4, Model 5, Model 9 and Model 6
exhibit a slightly right-biased distribution and the degree became higher. As for kurtosis, model 6 is slightly
steep while others are relatively flat.
(a) (b) (c)
Figure 2: Results of Monte Carlo simulation(MC)
We take Model 6 as an example to display more statistical characteristics from Monte Carlo simulation.
Figure 2(a) is a map of 10,000 simulated trajectories. Figure 2(b) plots original data ( ) vs the empirical mean
(green line), the 95% confidence bands (dashed lines) and the first-third quartile (dotted lines) of 10,000
simulated trajectories. Figure 2(c) is a histogram to show the distribution. We can see the swap rates are
growing gradually and fluctuating around 2.6. Besides, shown as Figure 2(c), most trajectories are distributed
between 2.6 and 2.7 and the distribution is approximately normal. According to the the empirical mean, we find
the overall trend of swap rates is increasing, though the increase is tiny.
In fact, except for Model 8, the rest models can describe the statistical characteristics of the swap rates
well. Further more, the fitting results are quite similar. The approximate results may attribute to the stability of
Chinese swap rate trend (compared with bond and stock). As we can see, the data has fluctuation but not large,
more complicated drift functions and the diffusion functions may not be required. Therefore, in the financial
stochastic analysis, especially in swap market, we can select a short-term swap rate model with a simple form.
In the way, we are more likely to get a precise empirical result even an analytic solution.

IV. Conclusions
We analyze the swap rates issued by the China Inter-bank Offered Rate(CHIBOR) and select the one-
year FR007 daily data from January 1st, 2019 to June 30th, 2019 as a sample. Basing on the data, we conduct
Monte Carlo simulation with several typical continuous short-term swap rate models to select the most efficient
one. Results are as follows.
(1) Among the tested models, except for model 8, the remaining models can not only fit our sample, but also
lead to parameter estimation and statistical characteristics of the swap rates. During the experiment,
’reasonable free parameters’ seems the key to parameter estimation.
(2) The empirical mean of Chinese swap rates is relatively horizontal with normally fluctuations.
The overall trend is increasing though the increase is tiny.
(3) Among the tested models that fit the swap rates, Model 6 has the best effect. However, the other models also
perform well and the results are quite similar. The approximate results maybe mostly attribute to the stability
of Chinese swap rate trend (compared with bond and stock).
Competing interests
The authors declare that they have no competing interests.
Author’s contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final
manuscript.
Acknowledgements
This research was supported NNSF of China(No.11371194; No.11501292) and by a Grant-in-Aid for Science
Research from Nanjing University of Science and Technology (30920140132035).
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