Discrete Dynamics in Nature and Society

Volume 2019, Article ID 8904162, 15 pages

https://doi.org/10.1155/2019/8904162

## 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

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

Correspondence should be addressed to Liang Wang; nc.ude.tuax@gnailgnaw

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.