Discrete Dynamics in Nature and Society
Volume 2019 (2019), Article ID 8904162, 15 pages
https://doi.org/10.1155/2019/8904162

Research Article

Research on the Value at Risk of Basis for Stock Index Futures Hedging in China Based on Two-State Markov Process and Semiparametric RS-GARCH Model

Liang Wang, Tingjia Xu, Longhao Qin, and Chenge Liu

School of Economics and Business Administration, Xi’an University of Technology, Xi’an 710048, China

Correspondence should be addressed to Liang Wang; wangliang@xaut.edu.cn

Received 8 February 2019; Revised 30 April 2019; Accepted 9 May 2019; Published 2 June 2019

Academic Editor: Luca Pancioni

Copyright © 2019 Liang Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This article aims to investigate the Value at Risk of basis for stock index futures hedging in China. Since the RS-GARCH model can effectively describe the state transition of variance in VaR and the two-state Markov process can significantly reduce the dimension, this paper constructs the parameter and semiparametric RS-GARCH models based on two-state Markov process. Furthermore, the logarithm likelihood function method and the kernel estimation with invariable bandwidth method are used for VaR estimation and empirical analysis. It is found that the three fitting errors (MSE, MAD, and QLIKE) of conditional variance calculated by semiparametric model are significantly smaller than that of the parametric model. The results of Kupiec backtesting on VaR obtained by the two models show that the failure days of the former are less than or equal to that of the latter, so it can be inferred that the semiparametric RS-GARCH model constructed in this paper is more effective in estimating the Value at Risk of the basis for Chinese stock index futures. In addition, the mean value and standard deviation of VaR obtained by the semiparametric RS-GARCH model are smaller than that of the parametric method, which can prove that the former model is more conservative in risk estimation.

1. Introduction

As the economic globalization and financial innovation are intensifying the income fluctuation of financial markets, Value at Risk (Morgan, 1996) [1] has become one of the important tools to invest and operate for financial institutions and to conduct market supervision by regulators. At the same time, with the launch of stock index futures of CSI 300, SSE 50, and CSI 500 in China, hedging has increasingly become a hot issue for practitioners and regulators. The basic principle of hedging is to utilize the high correlation between the spot market and the futures market, establishing opposite positions in the two markets, and use one market’s profit to offset the losses in the other market, thereby achieving the purpose of assets hedge. However, in this process, the risk caused by the basis volatility is the key factor for the hedging effect (Working, 1953) [2], which can be measured by the VaR method.

The traditional theory holds that hedging can reduce the risk because the basis will not change greatly during this process, but, in the actual transaction, the nonsynchronization of the volatility in the futures market and the spot market will also cause basis volatility and generate risks. Hence, many foreign scholars have explored this issue. Working (1953) [2] first defined the concept of basis, which is the spot price minus the futures price, and he believed that the best hedging is to keep the basis remain unchanged. Fraser & Mckaig (2000) [3] conducted a research by using data of Financial Times Stock Index Futures, the basis of U.S. Treasury futures, the difference between three-month Treasury yields in the United States and the United Kingdom. It is found that macroeconomic factors and the investors’ expectations to the market will significantly cause basis volatility. Gulley & Tilton (2014) [4] found that when the basis is positive, the demand of investors in futures market will also affect spot and futures prices, but the basis volatility is smaller. Zheng & Huang (2013) [5] constructed a macroeconomic factor model for the basis of commodity futures, and the empirical results showed that factors such as market interest rate, equity risk premium, and futures price change will have significant impacts on the basis, and the impact in the bull market is stronger than the bear market. Fama & French (2015) [6] found that the interest rate change, storage costs, and opportunity costs have different effects on the basis volatility for commodity futures. Broll, Welzel & Wong (2015) [7] pointed out that if the random price is negatively correlated with the expected value of the basis risk, then partial hedging is optimal; if there is a positive correlation, then excessive hedging is optimal, and the optimal position will be uncertain. Zhuang et al. (2016) [8] explored the basis risk for the hedging in the steel futures market and analyzed the impact of macroeconomic factors and micromarket factors on the basis risk empirically, and it was found that the VaR of basis provides a foundation for hedging the risks with parametric, semiparametric, and nonparametric GARCH methods.

Structural state transition is common in the immature financial market. The implementation of major economic policies and the change of financial supervision system may induce the economic state transition. Therefore, it is necessary to apply the volatility model with state transition to estimate and predict the volatility of China financial market and improve the fitting and estimating accuracy of volatility, while the Regime-Switch GARCH (RS-GARCH) model can better estimate the structural transition (Hamilton & Susmel, 1994; Francq & Zakoan, 2005) [9, 10]. Although the GARCH model has prominent effect in measuring the risk of financial market (Engle, 1982; Bollerslev, 1986) [11, 12], it cannot describe the state transition of financial sequences (Hamilton & Susmel, 1994; Francq & Zakoan, 2005) [9, 10]. Many scholars have introduced state transition GARCH model (RS-GARCH) for research. Lamoureux & Lastrapes (1990) [13] found that, due to state changes of return series, the GARCH model will overestimate the volatility. Hamilton & Susmel (1994) [9] and Gray (1996) [14] combined the Markov state transition process with the GARCH model and developed the maximum likelihood estimation method for parameter estimation. Sajjad, Coakley & Nankervis (2008) [15] examined whether the Bayesian MS-GARCH models with two regimes improve the forecasting volatility of VaR model by comparing with their single-regime counterpart. The empirical results showed that Bayesian two-regime MS-GJR-GARCH model with a GED distribution has the best fitting effect to the data based on DIC. Elenjical et al. (2016) [16] assessed the forecasting performance of popular GARCH-based volatility models in the context of VaR estimation and conducted a cross-regime analysis between time periods whereby market conditions experience a shift. Yang & Zhang (2013) [17] applied RS-GARCH and RS-APGARCH models to estimate and forecast the volatility of return series for Shanghai Composite Index and Shenzhen Component Index with Markov regime switching. Peng & Chen (2015) [18] combined the Markov state transition process with the DCC-GARCH model and found that the introduction of Markov state transition process and the range rate of return can effectively improve the estimation accuracy of the hedge ratio.

Due to the short history of stock index futures in China, there are relatively few literatures on the basis risk. In view of this, this paper takes the CSI 300, SSE 50, and CSI 500 stock index futures as the research object, analyzes the formation process of the basis for stock index futures hedging, and further proposes the method to measure the VaR of the basis for long and short hedges. Considering that the RS-GARCH model can effectively solve the structural transition problem of variance in VaR estimation and the Markov process can describe the characteristics of state transition with periodicity for financial variables, this paper intends to calculate the conditional variance of the basis based on Markov process and parametric RS-GARCH model.

Although the multivariate nonparametric regression model can effectively avoid the incorrect setting of the variable distribution in the parametric model, the former also has certain limitations. When there are more explanatory variables, the situation of “dimensional disaster” is prone to occur, such as the sharp increase of variance, the rapid decline of convergence rate of kernel estimation, and local linear estimation. However, the semiparametric model can effectively solve these problems and flexibly deal with the problem of unknown probability density function of variables and disobedient parameter distribution of samples. Furthermore, this paper develops a semiparametric RS-GARCH model based on two-state Markov process to obtain the conditional variance of the basis and uses the log-likelihood function along with the kernel estimation with invariable bandwidth approach to solve the model. Finally, an empirical study is carried out on the VaR of the basis for long and short hedges in three stock index futures.

This paper is organized as follows. Section 1 puts forward the issues concerned in this paper with literature review. Section 2 constructs the parametric and semiparametric RS-GARCH models based on two-state Markov process respectively, and the solution methods of the two models are given. Section 3 analyzes the empirical findings and Section 4 concludes. The innovations of this paper are as follows: It constructs the parametric and semiparametric RS-GARCH models based on two-state Markov process, proposes the measurement method of VaR for long and short hedges, and adopts both the log-likelihood function and kernel estimation with invariable bandwidth approach for model solution; it selects the CSI 300, SSE 50, and CSI 500 stock index futures in China for empirical analysis, and the result shows that the fitting errors (MSE, MAD, and QLIKE) of conditional variance obtained by semiparametric RS-GARCH model are significantly lower than that of the parametric model. Through the Kupiec backtesting on VaR, it is further found that the semiparametric model established in this paper is better in estimating the VaR of the basis.

2. Methodology

This section first analyzes the basis of hedging for stock index futures and then proposes the method of measuring VaR for long and short hedges. Next, the parametric and semiparametric RS-GARCH models are constructed based on two-state Markov process, respectively, and the solution methods of the two models are presented.

2.1. Analysis of the Basis for Stock Index Futures Hedging

The basis is the spread between the spot and the futures price during the hedging process, which plays an important role in the price discovery and information transmission of the futures market. The most perfect hedge is that the basis remains changeless. However, in the real market, the price fluctuations of spot and futures are not synchronized, and the optimal hedge is to ensure the minimization of the basis risk. For the hedge with a hedge ratio of 1, the basis is defined as Eq. (1).

where is the basis and and denote the spot price and the futures price, respectively. In general, hedge can be divided into short and long hedges. The former refers to buying spot while selling futures at the time of opening a position, and the latter refers to selling spot and buying futures when opening a position. The return of short hedge is

The return of long hedge is

In Eq. (2) and Eq. (3), and denote the price of futures and spot when opening a position, respectively. For short hedgers, the increase of the basis when closing a position means that the hedge is successful and profitable, while the long hedge is just the opposite.

2.2. The VaR Model for the Basis in Stock Index Futures Hedging

2.2.1. The VaR Model for Long and Short Hedges

Risk management has received much attention from practitioners and regulators in the last few years, with Value at Risk (VaR) emerging as one of the most popular tools. Morgan (1996) [1] proposed the parametric method for VaR, and the zero-value of VaR can be expressed as Eq. (4).

where is the final value of the asset at a certain confidence level, is the minimum yield, and denote the mean value and standard deviation of the yield, and is the quantile. For a long hedge, when the basis becomes larger, the return may be negative, so the VaR of the basis can be expressed as Eq. (5) (Jorion, 2010) [19].

For a short hedge, when the basis becomes smaller, the return may be negative, so the VaR of the basis can be expressed as Eq. (6).

In Eq. (5) and Eq. (6), is the basis of the quantile at a certain confidence level, and and denote the mean value and standard deviation, respectively. It can be seen that how to obtain the standard deviation in Eq. (5) and Eq. (6) is the key step to get VaR. However, for the financial time series, the standard deviation changes with time and has the phenomenon of structural transition. Hence, this paper will develop the parametric and semiparametric RS-GARCH models for investigation.

2.2.2. The Kupiec Backtesting of VaR

The backtesting of VaR is the coverage degree of the model results to the actual loss, and one feasible method is the failure rate test introduced by Kupiec (1995) [20]. Assuming that the estimation of VaR is time-independent, if the actual loss exceeds the VaR, this situation is recorded as failure; otherwise, it is recorded as success. Therefore, the binomial results of failure observation represent a series of independent Bernoulli experiment, and the expected probability of failure is . In addition, it is assumed that the total number of backtesting days is , and the number of failure days is ; then the failure probability is . The null hypothesis is , and the statistic of the likelihood ratio test for null hypothesis can be expressed as Eq. (7):

where LR obeys the -distribution when the degree of freedom is 1.

2.3. The Estimation of Conditional Variance Based on Two-State Markov Process and Parametric RS-GARCH Model

2.3.1. Analysis of State Transition for Conditional Variance Based on Two-State Markov Process

For Markov process (1907), the future change of the variable only depends on its current state, not on its past situation. According to this characteristic, the problem of the state transition path for variance in RS-GARCH model can be solved. Assuming that is the conditional variance of the basis at time t, we can obtain

Without using Markov process, the variance change is shown in the left diagram of Figure 1, where denotes the conditional variance when , and indicates the conditional variance when with State 2, is the State 1 when , and so on, indicates the conditional variance that the State 2 when converts to the State 1 when . The right diagram in Figure 1 shows the path changes of variance by two-state Markov process. It can be seen that the variance change is converted from the original paths to only two paths (), which avoids the geometrical growth of the paths and reduces the computational dimension of the model.

Figure 1: The path change of conditional variance.

In this section, we use two-state Markov process to analyze the probability of the state transition. The first column and the first row in Table 1 give the state of variance at time t and t+1, respectively. If the probability that the variance remains at State 1 from time t to time t+1 is P, then the probability of converting to State 2 at time t+1 is 1-P. Similarly, the probability of the variance remaining at State 2 from time t to time t+1 is Q, and the probability of transforming to State 1 at time t+1 is 1-Q.

Table 1: Two-state Markov process.

State 1
State 2

State 1
P
1-P

State 2
1-Q
Q

After multiple transitions, Markov process will tend to be stable with the steady-state probability. Assuming that the steady-state probability is and in State 1 and State 2, respectively, then we can obtain Eq. (9) (Markov, 1907):

From Eq. (9), and can be formulated as

Let denote the basis of the time series from time 1 to time t, and let and represent State 1 and State 2 at time t, respectively. Since the transition of state is affected by , we have

Further, there is

With Figure 1, we can get at time t-1. By substituting it into Eq. (12), we can develop Eq. (13) according to Bayesian formula:

Hence, the probability of State 1 at time t is

where and are the probability densities of State 1 and State 2 at time t-1, respectively.

2.3.2. The Parametric RS-GARCH Model Based on Two-State Markov Process

According to the study of Hamilton & Susmel (1994) [9], we estimate the conditional variance of basis sequence with two-state Markov process and parametric RS-GARCH model and obtain Eq. (15):

In Eq. (15), at time t, is the basis; and represent the conditional mean and standard deviation, where indicates State 1 and State 2; and are the probability of the variance transforming to State 1 and State 2, respectively; is a random variable. According to Eq. (15), the equations of the mean value and conditional variance are

In Eq. (16) and Eq. (17), is the parameter to be estimated, and and are the conditional variance of State 1 and State 2 at time t, where is the random error at time t-1. Moreover,

and the variance of the basis bt can be written as Eq. (19):

Assuming that the probability densities and obey the normal distribution, we have

Let denote the daily volatility at time t, that is, the percentage of the increase or decrease of the price at time t compared with the time t-1, and is the mean value of .

Furthermore, we derive the parameters in Eq. (16) and Eq. (17) with the following steps:

Step 1. Use the basis sequence bt in the sample period to obtain the daily volatility and the mean value . When , let the state at time t be ; otherwise, . Hence, the homogeneous Markov chain is derived.

Step 2. Let .

Step 3. According to equations (16)-(20) and the basis sequence in the sample period, we get the estimated value of the parameters: , , , , , , , and .

Step 4. Substitute the estimated value in Step 3 into the log likelihood function

Step 5. Use the “fminunc” function in Matlab; when reaches the minimum value at time t, the estimated value of the parameter vector can be obtained.

Step 6. Substitute the estimated value of in Step 5 into equations (16)-(20), and recalculate the estimated value of the parameters: , , , , , , , and .

Step 7. Let .

Step 8. Substitute the estimated value in Step 6 into the log likelihood function in Step 4; repeat Steps 5–8.

Step 9. When , stop the loop, and get the final estimated value of the parameter vector .

Step 10. According to the estimated value of the parameters at time , the conditional variance sequence can be obtained from Eq. (19), and the sequence of Value at Risk (VaR) of the basis can be further confirmed by Eq. (5) and Eq. (6).

2.3.3. The Semiparametric RS-GARCH Model Based on Two-State Markov Process

The relationship of the economic variables is unknown in reality, and there are specification errors of the traditional econometric model in practical applications, which cannot satisfy the needs of application research in economy and management. The nonparametric regression model assumes that the relationship of economic variables is unknown, so the model is more realistic, compared with linear and nonlinear regression models. The nonparametric regression model has the following characteristics: the form of regression function can be arbitrary, without any constraints, and the distribution of explanatory as well as explained variables is rarely limited. Hence, the model has greater adaptability. Although the nonparametric regression model has a better fitting effect than the classical regression model, in reality, there are several influencing factors for economic phenomena, and when the number of explanatory variables increases, the convergence rate of kernel estimation and local linear estimation of the multivariate nonparametric regression model will slow down. Since the semiparametric model has the characteristics of parametric and nonparametric models, it can improve the convergence rate of model estimation. On the basis of the previous research, this section will construct a semiparametric RS-GARCH model and propose the estimation method.

(1) The Semiparametric RS-GARCH Model Based on Two-State Markov Process. The conditional variance with two states in Eq. (17) is rewritten as the nonparametric form, while the other equations in Eq. (16) to Eq. (20) are unchanged; then the semiparametric RS-GARCH model is obtained:

Here, Eq. (22) is the nonparametric form of conditional variance equation, where () and () are unknown functions. In this paper, the kernel estimation with invariable bandwidth approach in multivariate parametric regression model is used to estimate the semiparametric RS-GARCH model (Ye, 2003) [21]. Let Y be the explained variable, which is a random variable; X is the d-dimensional explanatory variable vector, and the elements of the vector can be either deterministic or random. Given the samples and assuming that is independent and identically distributed, then a multivariate nonparametric regression model can be established:

where is an unknown function; is a random error term, which reflects the influence of other factors on the explained variable as well as the specification error of the model. The estimation of can be expressed as

In Eq. (24), H is the bandwidth, , and is the d-dimensional density function, where and . According to Ye (2003) [21], this paper uses the following kernel function:

where . Substitute and in Eq. (22) into in Eq. (23), respectively. X is the two-dimensional vector as the form of in Eq. (22). Furthermore, we can use Eq. (23)-Eq. (25) to finish the nonparametric estimation of conditional variance.

(2) The Kernel Estimation with Invariable Bandwidth of Semiparametric RS-GARCH Model. The specific steps for estimating the semiparametric RS-GARCH model by using the method of kernel estimation with invariable bandwidth are as follows.

Step 1. Follow Step 1 to Step 9 in Section 2.3.2; we can obtain the estimated value of the vector .

Step 2. Substitute into Eq. (16) and Eq. (17), and get the parameter estimated value , , , , , and .

Step 3. Take , , , and into Eq. (23)-Eq. (25), programming with Matlab (the code is shown in the Appendix) for the kernel estimation with invariable bandwidth, and further have the new parameter estimated value and .

Step 4. Substitute , in Step 3 and , , , in Step 2 into Eq. (19); the conditional variance of the basis can be obtained from the semiparametric RS-GARCH model.

3. Empirical Analysis

3.1. Data Selection

At present, China Financial Futures Exchange (CFFEX) has launched three kinds of stock index futures contracts: CSI 300, SSE 50, and CSI 500, which are taken as the research samples in this paper. Furthermore, due to the limited trading time of futures contracts (seasonal contracts have a relatively long trading time), investors usually choose continuous dominant contract according to the volume of trading. This paper selects the futures contract with the largest daily trading volume as the continuous dominant contract, so that the continuity of the sample can be maintained. The sample range of the CSI 300 stock index futures contract is from April 16, 2010, to May 31, 2018, with a total of 1975 samples, while SSE 50 and CSI 500 are from April 16, 2015, to May 31, 2018, totaling 764 trading days.

3.2. The Descriptive Statistical Analysis

Assume that the closing price of the index is on day t, the closing price of the dominant index contract is , and the basis on day t is . Further, we obtain the basis sequence diagrams of CSI 300, SSE 50, and CSI 500, as shown in Figures 2–4. It can be seen that, in Figure 2, before June 2015, the CSI 300 stock index futures contract was basically in the “reverse” market (the basis is less than zero), while it was in the “normal” market (the basis is greater than zero) after June. In Figures 3 and 4, the SSE 50 and CSI 500 stock index futures contracts were basically in a “normal” market situation after June 2015. This is probably because the stock index futures trading was affected by Chinese stock market crash in 2015. In addition, the basis sequences have an obvious cluster effect in Figures 2–4; that is, a large (small) fluctuation is accompanied by a large (small) fluctuation. Particularly, in the early stage of futures listing and before June 2015, the basis sequences showed strong volatility agglomeration. In addition, the trend of basis sequences for stock index futures has a strong consistency, which showed a tendency of increasing first and then decreasing in 2015. For the convenience of the following study, this paper divides the volatility of basis into two states, high and low volatility, respectively. Specifically, the mean values of the daily volatility for the basis sequence of three futures are obtained separately, where refers to the percentage of the increase or decrease of the price at time t compared with the time t-1. When on day , we record the state as high volatility; otherwise we set it as low volatility.

Figure 2: The basis sequence of CSI 300 stock index futures.

Figure 3: The basis sequence of SSE 50 stock index futures.

Figure 4: The basis sequence of CSI 500 stock index futures.

Table 2 shows the descriptive statistical results of the basis for the three futures. It can be seen that the mean value of the basis is greater than zero, and the standard deviation is between 0.9 and 2. The JB statistic is larger and the P-value is zero, indicating that the basis sequences disobey the normal distribution. Moreover, the skewness is greater than zero, and the kurtosis is larger, which suggests that the basis sequences have the characteristics of leptokurtosis and fat-tail with right skew. In Table 3, the ADF statistic is lower than the critical value at each significance level, and the P-value is zero, illustrating that the original hypothesis is rejected and the sequences have no unit root. Therefore, we can deduce that the three basis sequences are stationary. Table 3 also gives the ARCH-LM test results of the basis sequences, and the P-value is zero, which means that there are significant heteroscedasticity effects in basis sequences, so the GARCH model can be used to measure the variance of basis sequences.

Table 2: The descriptive statistics of basis sequences for stock index futures.

Stock index futures
Mean value
Standard deviation
Skewness
Kurtosis
JB statistics
P-value

CSI 300
0.205
1.061
2.717
22.22
32816.660
0.000

SSE 50
0.471
0.964
3.135
22.03
12778.254
0.000

CSI 500
1.414
1.727
2.439
12.849
3845.504
0.000

Table 3: The ADF and ARCH-LM test of basis sequences for stock index futures.

Stock index futures
ADF test
ARCH-LM test

t-Statistic
Prob
ObsR-squared
Prob

CSI 300
-7.866
0.000
110.228
0.000

SSE 50
-6.299
0.000
64.925
0.000

CSI 500
-8.503
0.000
50.588
0.000

3.3. The Estimation of Parametric and Semiparametric RS-GARCH Models Based on Two-State Markov Process

In this section, we divide the basis sequences into two states and establish a two-state Markov chain. Table 4 shows the transition times between various states of basis sequences during the sample period.

Table 4: The state transition statistics of basis sequences for stock index futures.

CSI 300
State 1
State 2
SSE 50
State 1
State 2
CSI 500
State 1
State 2

State 1
801 times
307 times
State 1
136 times
70 times
State 1
174 times
63 times

State 2
306 times
200 times
State 2
69 times
127 times
State 2
63 times
102 times

Calculating the number of four state transitions in Table 4, we divide each transition time by the total transition times to obtain transition probability P and Q. Further, the steady-state probability and can be derived according to Eq. (9). The results are shown in Table 5. It can be seen that are all greater than , which indicates that the basis is likely to be at a low volatility state. In addition, as the study in Section 2.3.1, P are all greater than Q for the three basis sequences in Table 5; it indicates that the basis at the low volatility state remains unchangeable with the characteristic of stronger continuity. This is also in line with the fact that the bull market in Chinese securities market has a short duration with high volatility and the bear market lasts for a long time with small price fluctuation. In addition, the steady-state probability and state transition probability shown in Table 5 will be taken as the initial parameter values for solving the RS-GARCH model later.

Table 5: The steady-state probability and state transition probability of basis sequences for stock index futures.

Stock index futures
Probability

P
Q

CSI 300
0.766
0.234
0.827
0.435

SSE 50
0.835
0.165
0.907
0.532

CSI 500
0.765
0.235
0.788
0.307

Further, combined with the estimation methods of Steps 1–10 in Section 2.3.2, we program with Matlab to derive the parameters of RS-GARCH model (detailed code in Appendix), and the results are shown in Table 6.

Table 6: The estimation of parameters of RS-GARCH model.

The parameters of RS-GARCH model
CSI 300 stock
SSE 50 stock
CSI 500 stock

index futures
index futures
index futures

(19.059)
(12.343)
(13.307)

Note: , , and mean that the parameters are significant at significance levels of 10%, 5%, and 1%, respectively, and it is t-statistic in parentheses.

In Table 6, , , , are all greater than zero, , which ensures the stationarity of the conditional variance (Bollerslev, 1986) [12]. and represent the high volatility state; and indicate the low volatility state. shows that the high volatility state is more susceptible to the previous state than the low volatility state (Bollerslev, 1986) [12]. According to Table 6, we estimate the conditional variance of the basis by using Steps 1–4 of the semiparametric RS-GARCH model in Section 2.3.3 (2). Due to space limitations, we only give the descriptive statistical results in Table 7. It can be seen that the conditional variance of SSE 50 is smaller, while that of CSI 300 is larger.

Table 7: The estimation of conditional variance for parametric and semiparametric RS-GARCH models.

Stock index futures
Model
Mean value
Standard deviation
Maximum
Minimum

SSE 50
Parametric RS-GARCH
0.556
0.841
9.970
0.235

Semi-parametric RS-GARCH
0.522
0.668
7.353
0.279

CSI 500
Parametric RS-GARCH
1.023
1.715
28.663
0.488

Semi-parametric RS-GARCH
0.9705
1.001
18.758
0.729

CSI 300
Parametric RS-GARCH
2.461
1.419
26.839
0.934

Semi-parametric RS-GARCH
2.409
1.225
27.455
0.933

Furthermore, we adopt three error measure methods, which are mean square error (MSE), mean absolute error (MAD), and error of Gaussian quasi-maximum likelihood loss function (QLIKE), to assess the fitting effect of models. And we introduce the GARCH model for comparison, where . It is found that and are significant, and indicates that the conditional variance sequences are stationary and predictable, so GARCH (1,1) can be used for regression (Bollerslev, 1986) [11]. The parameter estimation results are shown in Table 8. Where , , , is the square value of the difference between the basis on day t and the mean value of basis from day 1 to day t, and is the estimated value of the conditional variance by using the method in Section 2.3.2.

Table 8: The parameter estimation results of GARCH model.

Stock index futures
GARCH parameters

SSE 50
0.062(0.000)
0.404(0.000)
0.556(0.000)

CSI 500
0.687(0.018)
0.600(0.000)
0.347(0.000)

CSI 300
0.303(0.000)
0.542(0.000)
0.388(0.000)

In addition, we use the parametric and semiparametric RS-GARCH models to estimate the conditional variance of basis sequences for the three futures, calculate the fitting error, and use the GARCH model for comparison, which are shown in Figures 5–7. In the figures, the ordinate is the result of the fitting error, and the abscissa is the fitting error method. For the convenience of observation, the errors of different models are connected by different curves, and the three curves are the fitting errors of GARCH model and parametric and semiparametric RS-GARCH, respectively. It can be seen that the semiparametric model has the best fitting effect, followed by the parametric model, while GARCH model has the worst fitting effect. For the three futures, the fitting errors (MSE, MAD, and QLIKE) of the semiparametric RS-GARCH model are lower than the parametric method. Further, we select the transaction data from June 1, 2018, to September 28, 2018, as out-of-sample data and compare the fitting effect of the two RS-GARCH models (Figures 8–10); the results show that the semiparametric model is still better. In summary, it can be found that the semiparametric RS-GARCH model constructed in this paper improves the estimation accuracy of conditional variance of the basis, so this model can better estimate the daily volatility of stock index futures in China.

Figure 5: The model fitting error results in sample period (SSE 50).

Figure 6: The model fitting error results in sample period (CSI 500).

Figure 7: The model fitting error results in sample period (CSI 300).

Figure 8: The model fitting error results out of sample period (SSE 50).

Figure 9: The model fitting error results out of sample period (CSI 500).

Figure 10: The model fitting error results out of sample period (CSI 300).

3.4. Estimation and Test of VaR of the Basis for Stock Futures Hedging

According to the conditional variance of basis sequences obtained in Section 3.2, we use Eq. (5) and Eq. (6) in Section 2.2 to obtain the VaR sequences for long and short hedges on day t. Table 9 gives the descriptive statistical results of VaR sequences for stock index futures in China. It can be seen that VaR calculated by the semiparametric method is smaller than that of the parametric method, which indicates that the semiparametric method is more conservative in risk estimation. Moreover, as can be seen from Table 9, the standard deviation of VaR calculated by semiparametric method is smaller than that of parametric model as a whole, which also shows that the result by adopting semiparametric method is more stable.

Table 9: The estimation results of VaR of the basis for stock index futures.

Stock index futures
Long/Short
Model
Confidence level
Mean value
Standard deviation
Maximum
Minimum

SSE 50

RS-GARCH
95%
1.537
1.179
10.908
0.039

97.5%
1.747
1.265
11.759
0.243

99%
1.991
1.368
12.748
0.479

Semi-parametric RS-GARCH
95%
1.533
1.107
11.825
0.082

97.5%
1.742
1.177
12.680
0.091

99%
1.985
1.262
13.673
0.332

RS-GARCH
95%
0.654
0.735
5.303
0.003

97.5%
0.863
0.766
6.017
0.021

99%
1.107
0.817
6.846
0.023

Semi-parametric RS-GARCH
95%
0.651
0.729
5.760
0.008

97.5%
0.860
0.747
6.525
0.012

99%
1.103
0.780
7.413
0.015

CSI 500

RS-GARCH
95%
2.889
2.012
18.163
0.075

97.5%
3.451
2.117
19.850
0.248

99%
3.535
2.244
21.809
0.662

Semi-parametric RS-GARCH
95%
2.830
1.791
15.028
0.121

97.5%
3.188
1.846
15.684
0.405

99%
3.117
1.913
16.446
0.066

RS-GARCH
95%
0.225
1.397
7.901
0.005

97.5%
0.523
1.404
9.015
0.009

99%
0.871
1.415
10.310
0.024

Semi-parametric RS-GARCH
95%
0.166
1.343
5.117
0.004

97.5%
0.453
1.345
5.653
0.005

99%
0.787
1.365
6.564
0.032

CSI 300

RS-GARCH
95%
2.693
0.949
15.107
1.803

97.5%
3.179
1.031
16.739
2.314

99%
3.742
1.129
18.635
2.668

Semi-parametric RS-GARCH
95%
2.677
0.902
15.536
1.509

97.5%
3.159
0.973
17.187
2.150

99%
3.720
1.0576
19.105
2.667

RS-GARCH
95%
2.374
0.503
13.633
1.169

97.5%
2.859
0.540
15.210
1.474

99%
3.423
0.599
17.041
1.828

Semi-parametric RS-GARCH
95%
2.358
0.4596
7.375
0.293

97.5%
2.840
0.4783
8.306
1.431

99%
3.401
0.5153
9.387
1.827

Note: denotes the VaR of the basis for the long hedge, and is the VaR of the basis for the short hedge.

Furthermore, we adopt the Kupiec backtesting method to estimate the VaR of the basis for the three stock index futures, and the results are shown in Table 10. It can be found that, in the acceptance domain, the lower the number of failures, the better the prediction effect of the model. However, if the failure rate is excessively low, it means that the model is too conservative and investors may miss investment opportunities. In Table 10, the acceptance domain of LR statistic is (0.00016,6.635) at the significance level of 1%, (0.001,5.024) at the significance level of 2.5%, and (0.004,3.841) at the significance level of 5%. It can be seen that, at 95% confidence level, a few LR statistics fall into the rejection domain, indicating that both the parametric and semiparametric RS-GARCH models can reasonably estimate the VaR of the basis. Further, Table 10 shows that the higher the confidence level, the fewer the times of failing the test. At three different confidence levels, the failure times of the semiparametric RS-GARCH model are less than or equal to that of the parametric RS-GARCH model, which shows that the former model can effectively improve the estimation accuracy of VaR of the basis for stock index futures in China. In addition, at 95% confidence level, some LR statistics are larger, falling into the rejection domain, and the failure rate is far lower than 5%, which indicates that the model at this confidence level overestimates the risk to a certain extent. Moreover, we find that the VaR of the basis for Chinese stock index futures can better characterize the leptokurtosis and fat-tail of the basis sequences. Therefore, it is recommended to use the confidence level of 97.5% and 99% for VaR estimation. It can be further deduced that the semiparametric RS-GARCH model can effectively improve the measurement accuracy of the basis risk for Chinese stock index futures.

Table 10: The Kupiec backtesting of VaR of the basis for stock index futures.

Stock index futures
Long/Short
Model
Confidence level
Failure days
Failure rate
LR statistics

SSE 50

RS-GARCH
95%
22
2.88%
8.480

97.5%
14
1.83%
1.537

99%
9
1.18%
0.231

semi-parametric RS-GARCH
95%
20
2.62%
10.969

97.5%
13
1.70%
2.246

99%
5
0.65%
1.050

RS-GARCH
95%
13
1.70%
23.24

97.5%
9
1.18%
6.792

99%
7
0.92%
0.056

semi-parametric RS-GARCH
95%
12
1.57 %
25.544

97.5%
6
0.79%
12.533

99%
3
0.39%
3.6997

CSI 500

RS-GARCH
95%
38
4.97%
0.001

97.5%
27
3.53%
2.976

99%
15
1.96%
5.591

semi-parametric RS-GARCH
95%
36
4.71%
0.136

97.5%
26
3.40%
2.301

99%
15
1.96%
5.591

RS-GARCH
95%
18
2.36%
13.868

97.5%
12
1.57%
3.112

99%
7
0.92%
0.055

semi-parametric RS-GARCH
95%
14
1.83%
21.093

97.5%
8
1.05%
8.441

99%
5
0.65%
1.049

CSI 300

RS-GARCH
95%
89
4.51%
1.0466

97.5%
44
2.23%
0.622

99%
24
1.22%
0.865

semi-parametric RS-GARCH
95%
85
4.30%
2.111

97.5%
42
2.13%
1.1891

99%
22
1.11%
0.249

RS-GARCH
95%
77
3.9%
5.434

97.5%
36
1.82%
4.096

99%
11
0.56%
4.633

semi-parametric RS-GARCH
95%
74
3.75%
7.123

97.5%
35
1.77%
4.770

99%
11
0.56%
4.633

Note: indicates that the LR statistic falls within the rejection domain.

4. Conclusion

This paper aims to investigate the VaR measurement of the basis for CSI 300, SSE 50, and CSI 500 stock index futures in China. Since the RS-GARCH model can effectively deal with the structural transition of variance and the two-state Markov process can transform the original 2N state transition paths into two paths, thus effectively reducing the computational dimension and complexity, this paper firstly constructs a parametric RS-GARCH model based on two-state Markov process to obtain the conditional variance of the basis. If there are more explanatory variables, the convergence rate of kernel estimation and local linear estimation will decrease sharply when using the multivariate nonparametric regression method. In view of this, this paper constructs the semiparametric RS-GARCH model based on two-state Markov process to estimate the conditional variance of the basis and adopts the log-likelihood function and kernel estimation with invariable bandwidth method for model solution. On this basis, this paper calculates the VaR of long and short hedges for Chinese stock index futures and carries out the Kupiec backtesting.

Through empirical research, the conclusions are as follows.

First, the basis volatility of hedging has obvious agglomeration for stock index futures in China, especially in the early stage of futures listing and before June 2015, while the basis is basically in the “normal” market after June 2015 (the basis is greater than zero).

Second, this paper divides the basis of stock index futures hedging into two states of high and low volatility and establishes a two-state Markov process, which finds that the duration of the basis sequence in a low volatility state is longer than that of the high volatility state in China.

Third, the parametric and semiparametric RS-GARCH models based on two-state Markov process can better describe the characteristics of leptokurtosis and fat-tail in Chinese stock index futures market; comparing the three fitting errors (MSE, MAD, and QLIKE) of the two models, we can find that the semiparametric RS-GARCH model is better.

Fourth, the mean value and standard deviation of VaR calculated by semiparametric RS-GARCH model are smaller than that of the parametric model, which indicates that the former is more conservative in risk estimation.

Fifth, the Kupiec backtesting shows that the number of failure days of VaR based on semiparametric RS-GARCH model is less than or equal to that of the parametric model, which demonstrates that the semiparametric RS-GARCH model is better, and it can avoid the incorrect setting of the model, thus making the estimation result of VaR closer to the actual loss value.

Appendix

 %%%%%%rs-garch parametric estimation %%%% function x= test1 (b,P,Q) h1(1)=18969.3325; h2(1)=20765.5881; u1(1)=173.8939; u2(1)=105.0945; p(1)=0.5896; h(1)=p(1) (u1(1)∧2 + h1(1)) + (1-p(1)) (u2(1)∧2 + h2(1)) - (p(1) u1(1) + (1-p(1)) u2(1))∧2; e(1)=b(1) - (p(1) u1(1) + (1-p(1)) u2(1)); f1(1)=exp((-(b(1) - u1(1))∧2)/(2h1(1)))/sqrt(2pih1(1)); f2(1)=exp((-(b(1) - u2(1))∧2)/(2h2(1)))/sqrt(2pih2(1)); s = @(x) 0; for i=2:length(b)

 p(i)=(1-Q) f2(i-1) (1-p(i-1))/(f1(i-1) p(i-1) + f2(i-1) (1-p(i-1))) + P f1(i-1) p(i-1)/(f1(i-1) p(i-1) + f2(i-1) (1-p(i-1))); h1i=@(x) x(1) + x(2) e(i-1)∧2 + x(3) h(i-1); h2i=@(x) x(4) + x(5) e(i-1)∧2 + x(6) h(i-1); u1i=@(x) x(7) + x(8) b(i-1); u2i=@(x) x(9) + x(10) b(i-1); f1i=@(x) exp ((-(b(i) - u1i(x))∧2)/(2h1i(x)))/sqrt(2pih1i(x)); f2i=@(x) exp ((-(b(i) - u2i(x))∧2)/(2h2i(x)))/sqrt(2pih2i(x)); s=@(x) s(x) + log(f1i(x) p(i) + f2i(x) (1-p(i))); fun= @(x) -s(x); x0 = [28.3240    0.1874    0.1023    168.5506    0.2418 1.2899    -4.4329    0.9775    47.9077    0.6006]; options = optimset(‘MaxFunEvals’,4000); x = fminunc (fun,x0,options); h1(i)=x(1) + x(2) e(i-1)∧2 + x(3) h(i-1); h2(i)=x(4) + x(5) e(i-1)∧2 + x(6) h(i-1); u1(i)=x(7) + x(8) b(i-1); u2(i)=x(9) + x(10) b(i-1); h(i)=p(i) (u1(i)∧2 + h1(i)) + (1-p(i))(u2(i)∧2 + h2(i))-(p(i) u1(i) + (1-p(i)) u2(i))∧2; e(i)=b(i) - (p(i) u1(i) + (1-p(i)) u2(i)); f1(i)=exp((-(b(i) - u1(i))∧2)/(2h1(i)))/sqrt(2pih1(i)); f2(i)=exp((-(b(i) - u2(i))∧2)/(2h2(i)))/sqrt(2pih2(i));

 end %%%%%%%%%%%%%%%%%%%%%%%% function [h1,h2,h,e,u1]= testrs (b,P,Q,x) h1(1)=1.536164415241; h2(1)=1.336363744144; u1(1)=0.940125; u2(1)=0.589625; p(1)=0.5088; h(1)=p(1) (u1(1)∧2 + h1(1)) + (1-p(1)) (u2(1)∧2 + h2(1)) - (p(1) u1(1) + (1-p(1)) u2(1))∧2; e(1)=b(1) - (p(1) u1(1) + (1-p(1)) u2(1)); f1(1)=exp((-(b(1) - u1(1))∧2)/(2h1(1)))/sqrt(2pih1(1)); f2(1)=exp((-(b(1) - u2(1))∧2)/(2h2(1)))/sqrt(2pih2(1)); for i=2:length(b)

 p(i)=(1-Q) f2(i-1) (1-p(i-1))/(f1(i-1) p(i-1) + f2(i-1) (1-p(i-1))) + P f1(i-1) p(i-1)/(f1(i-1) p(i-1) + f2(i-1) (1-p(i-1))); h1(i)=x(1) + x(2) e(i-1)∧2 + x(3) h(i-1); h2(i)=x(4) + x(5) e(i-1)∧2 + x(6) h(i-1); u1(i)=x(7) + x(8) b(i-1); u2(i)=x(9) + x(10) b(i-1); f1(i)=exp((-(b(i) - u1(i))∧2)/(2h1(i)))/sqrt(2pih1(i)); f2(i)=exp((-(b(i) - u2(i))∧2)/(2h2(i)))/sqrt(2pih2(i)); e(i)=b(i) - (p(i) u1(i) + (1-p(i)) u2(i)); h(i)=p(i) (u1(i)∧2 + h1(i)) + (1-p(i))(u2(i)∧2 + h2(i))-(p(i) u1(i) + (1-p(i)) u2(i))∧2;

 end h1 h2 h e %%%%%%%%%%条件方差非参数估计%%%%%%%%%%% function m = feicanshuguji(y,x,h) n = length(y); m = zeros(n,1); p = ones(n,1); for i =1:n

 w = zeros(n); for j =1:n

 e = ((x(j,1) - x(i,1))/h)∧2 +((x(j,2)-x(i,2))/h)∧2; w(i,j) = (2/pi)(1-e)(e<1);

 end m(i) = (p’wy)/(p’wp);

 end %%%%%%%%%%%%%半参数RS-GARCH 计算条件方差%%%%%%%%%%%%% function [h,e]= testrss (b,P,Q,x,h1,h2) h1(1)=1.536164415241; h2(1)=1.336363744144; u1(1)=0.940125; u2(1)=0.589625; p(1)=0.5088; h(1)=p(1) (u1(1)∧2 + h1(1)) + (1-p(1)) (u2(1)∧2 + h2(1)) - (p(1) u1(1) + (1-p(1)) u2(1))∧2; e(1)=b(1) - (p(1) u1(1) + (1-p(1)) u2(1)); f1(1)=exp((-(b(1) - u1(1))∧2)/(2h1(1)))/sqrt(2pih1(1)); f2(1)=exp((-(b(1) - u2(1))∧2)/(2h2(1)))/sqrt(2pih2(1)); for i=2:length(b)

 p(i)=(1-Q) f2(i-1) (1-p(i-1))/(f1(i-1) p(i-1) + f2(i-1) (1-p(i-1))) + P f1(i-1) p(i-1)/(f1(i-1) p(i-1) + f2(i-1) (1-p(i-1))); u1(i)=x(7) + x(8) b(i-1); u2(i)=x(9) + x(10) b(i-1); f1(i)=exp((-(b(i) - u1(i))∧2)/(2h1(i)))/sqrt(2pih1(i)); f2(i)=exp((-(b(i) - u2(i))∧2)/(2h2(i)))/sqrt(2pih2(i)); e(i)=b(i) - (p(i) u1(i) + (1-p(i)) u2(i)); h(i)=p(i) (u1(i)∧2 + h1(i)) + (1-p(i))(u2(i)∧2 + h2(i))-(p(i) u1(i) + (1-p(i)) u2(i))∧2;

 end h e

Data Availability

The authors declare that the data will be available upon acceptance of this article. The data used to support the findings of this study are included within the article; it is true and reliable. If readers need related data, they can contact the authors.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant number 71171155; Xi’an Social Sciences Planning Fund Major Projects under Grant number 17J92; Special Scientific Research Project of Education Department of Shaanxi Province under Grant numbers 18JK0535 and 16JK1527; Xi’an Development and Reform Commission Regional Economic Issues under Grant number SXTY2018-08-15; and Humanities and Social Sciences Research Planning Fund of Ministry of Education under Grant number 19YJA630080.

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