In this paper, we consider the leverage effect on the CSI 300 Index yield and Hong Kong Hang Seng Index yield. It is modeled by the SV model with leverage. In this model, we compare the mainland and the Hong Kong stock market with stock market long-term effect, the degree on fluctuation reply and leverage effect so on. The analysis results show that the leverage stochastic volatility model can well fitting rate of return on the CSI300 index and the Hang Seng index in Hong Kong; The Shanghai and Shenzhen stock market volatility and leverage effect obviously stronger than the Hong Kong stock market.
In financial applications, a very important topic is to grab statistical properties of the yield of assets through the model. In the past few decades, the stock yield model has made great progress. From the initial Black-Scholes-Merton model’s constant volatility, more complex and finer models are presented to capture the characteristics of stock prices. These features include the significant events of the stock market impact, fluctuations in the price of agglomeration, leverage effect, etc. There are two basic models for describing volatility: the ARCH conditional heteroscedasticity model proposed by [
The article [
Also, domestic scholars have done a lot of research on the SV model of stock price. The article [
For stocks, it can often be observed a phenomenon is: if it changes in the market up or down, its volatility in the downward sliding process is higher than the volatility in the upward movement, which is called the leverage effect. Leverage is important for the stock market, nevertheless the leverage effect of the exchange rate market is much lower. Compared with the basic SV model, the leverage effect SV model has an additional parameter, that is, the correlation coefficient. The model is as follows:
y t | θ t , ρ = exp ( θ t 2 ) ε t (1)
θ t + 1 | θ t , μ , φ , τ , ρ = μ + φ × ( θ t − μ ) + τ × η t + 1 (2)
( ε t η t + 1 ) ~ i . i . d N ( ( 0 0 ) , ( 1 ρ ρ 1 ) ) (3)
y t represents the yield at time t, while θ t represents the logarithmic fluctuation at time t. μ indicates the long-term effects of logarithmic fluctuations, depicting the extent of long-term logarithmic volatility. φ is said the reversion the logarithmic fluctuations, which reflects the extent to which the current volatility is affected by future volatility. ρ indicates the degree of leverage. When ρ is zero, the model above is the basic stochastic volatility model, which depicts the characteristics such as time-varying and agglomeration of fluctuation, but for the financial market, especially the common volatility in the stock market cannot depict asymmetric phenomenon. When less than 0, the above model called SVL model, which has the leverage stochastic volatility model, this model by income shocks ε t and volatility impact η t + 1 is negatively related to the asymmetry of volatility.
There are the methods of estimating SV model parameters as follows: MCMC simulation, quasi-maximum likelihood (QML), generalized moment estimation (GMM), etc. The MCMC method is better and more flexible among them, since that on the one hand, MCMC simulation parameter estimation accuracy is superior to other methods; on the other hand, in the estimation parameters it can be predicted the forecast value at the same time. In this paper, MCMC simulation is used to infer the above model and predict the fluctuation.
The basic idea of the MCMC method is to establish a stable distribution π ( θ ) of the Markov chain, samples π ( θ ) were obtained by random sampling, based on these samples to do a variety of statistical inference. The key is to construct a stationary distribution π ( θ ) with a specified value. In general, it is followed three steps: (1) select a suitable Markov chain, the transfer probability of p ( * , * ) , so that the corresponding smooth distribution in π ( θ ) ; (2) by observing a certain point θ ( 0 ) from the sample, using the Markov chain in (1) that produce point sequence θ ( 1 ) , θ ( 2 ) , ⋯ , θ ( n ) ; (3), for an m and a sufficiently large n,
E ( θ ) = 1 n − m ∑ t = m + 1 n θ ( t ) . (4)
At this point, E ( θ ) performs as the point of the estimation of θ .
This paper selects the Hong Kong’s Hang Seng Index (110000) and the CSI 300 Index (000300) from January 2014 to July 2016, with a total of 587 trading days per index. Take the return on assets
r t = p t − p t − 1 p t − 1 * 100 % , (5)
where p t is the closing price of t.
The following
what’s get in the observation on the chart, the yield on the CSI 300 Index is greater than the Hong Kong ones. The minimum and maximum returns of the CSI 300 stock index yield are almost 2 times of Hong Kong’s Hang Seng index. Compared with the median return on Hang Seng index, CSI 300 stock index returns the median is far greater than zero. In terms of variance, the CSI 300 stock index is bigger than Hong Kong’s.
Basic statistics | y_hs | y_hk |
---|---|---|
Minimum Median Maximum Mean Standard deviation Variance kurtosis Skewnessa | −7.8680 0.0549 6.7150 0.0800 1.5676 2.4574 3.9465 −0.5390 | −3.1760 0.0021 3.8030 0.0151 0.9522 0.9067 0.8539 0.0091 |
either the CSI 300 index, or Hong Kong’s Hang Seng index, while the distribution of the yield is not a normal distribution. Besides, according to
In conclusion, we can draw the following conclusions:
・ The CSI 300 Index is more volatile and more volatile than the Hong Kong Hang Seng Index, indicating that China’s stock market is more unstable.
・ CSI 300 Index and Hong Kong Hang Seng are obviously fluctuation phenomenon of agglomeration, the yield distribution do not obey the normal distribution, there are the characteristics of peak tailing.
As shown in
As shown in
Variables | W | p-value |
---|---|---|
y_hs y_hk | 0.9419 0.9898 | 2.11(*10−14) 0.0003712 |
Variables | Dickey-Fuller | Lag order | p-value |
---|---|---|---|
y_hs y_hk | −6.1976 −7.0423 | 8 8 | 0.01 0.01 |
Variables | Chi-squared | df | p-value |
---|---|---|---|
y_hs y_hk | 105.3 31.27 | 12 12 | 2.19(*10−16) 0.001799 |
test of the chi-square statistic Chi-squared = 105, the degree of freedom df = 12, corresponding to p-value = 2.20E−16 < 0.05, indicating a significant level of 0.05, reject the original hypothesis, after that the CSI 300 index yield sequence exists ARCH effect. Similarly, at the 0.05 significance level, the Hong Kong Hang Seng Index yield sequence also exists in ARCH effect.
Through the MCMC method, the parameters of the SVL model are estimated, and the iterations are 20000 times, and the previous 10000 times are removed and the parameters are estimated by the iteration value of 10000 times. With reference to the setting of the prior distribution of [
μ ~ N ( 0 , 100 ) . (6)
φ = 2 ∗ φ ∗ − 1 , φ ∗ ∼ beta ( 20 , 1.5 ) . (7)
τ = 1 τ ∗ , τ ∗ ~ Ga ( 2.5 , 0.025 ) . (8)
ρ ~ U ( − 1 , 1 ) . (9)
Such as
Rhat is the statistic for the MCMC convergence diagnosis based on the normal approximation proposed by [
R ^ = ( n − 1 n + m + 1 m n * B W ) d f d f − 2 . (10)
Series | Variable | Mean | sd | 2.50% | 97.50% | Rhat |
---|---|---|---|---|---|---|
y_hk | μ φ τ ρ | −0.1962 0.9460 0.1521 −0.1828 | 0.1495 0.0284 0.0442 0.1359 | −0.5155 0.8795 0.0890 −0.3432 | 0.0795 0.9892 0.2524 −0.0894 | 1.0350 1.1280 1.1450 1.0430 |
y_hs | μ φ τ ρ | 0.7150 0.9764 0.1782 −0.3598 | 0.3224 0.0156 0.0402 0.1598 | 0.0880 0.9374 0.1159 −0.6795 | 1.3950 0.9964 0.2725 −0.0570 | 1.0010 1.1160 1.1400 1.0320 |
where m is the number of initial values, 2n represents the total number of iterations, the first half of the n iteration values are removed, the nth iteration values are used for the second half, B is the variance between groups, W is the intra-group variance, df is the approximate T the degree of freedom of distribution. Conclusion in [
According to the characteristics of
φ stands for the return nature of logarithmic fluctuations, indicating that the current volatility will have a long-term impact on future volatility. The point estimate of the CSI 300 index yield model is 0.9764 and the confidence interval is 95%. [0.9374, 0.9964]. The Hong Kong Hang Seng Index yield rate is estimated at 0.9460 and the confidence interval of 95% is estimated to be [0.8795, 0.9892]. The point estimation φ of the y_hs model is slightly larger than that of the y_hk model, which is close to 1, and the interval estimates of the two are overlapped with a large part. Therefore, the Shanghai and Shenzhen stock markets and the Hong Kong stock market have almost the same degree of fluctuation. The nature of the return, which shows that the phenomenon of fluctuations in income agglomeration is obvious. The agglomeration of volatility is the persistence of risk, which reflects the long-term impact of current risks on future risks, which investors, especially long-term investors, do not want to see. For investors, both in Shanghai and Shenzhen stock markets or Hong Kong stock market, we must consider the sustainability of market risk.
ρ represents the degree of leverage on the rate of return on volatility. The point estimate of the CSI300 index yield model is what the interval of 95% confidence is estimated to be [−0.6795, −0.0570]. The estimation of the Hong Kong Hang Seng Index yield model is −0.1828 and the confidence interval of 95% is estimated to be [−0.3432, −0.0894]. Whether the CSI300 Index yield or the Hong Kong Hang Seng Index yield, the 95% confidence interval does not include 0, and the points are estimated to be less than 0, indicating that the Shanghai and Shenzhen stock markets and the Hong Kong stock market there are obvious leverage effect. The absolute value of the point estimation ρ of the CSI300 index yield model is significantly larger than the absolute value of the Hong Kong Hang Seng Index yield model, indicating that the leverage effect of the Shanghai and Shenzhen stock markets is stronger than that of the Hong Kong stock market. In other words, when the market shows negative returns, the volatility impact in the Shanghai and Shenzhen stock markets is greater than which caused by the Hong Kong stock market. Negative correlation not only shows that stock market participants are often risk hobby, but also implies that the stock market speculation exists, which also evidence from a side of the mainland stock market is emerging stock market.
To sum up, we can draw the following conclusions:
・ stock market volatility exist in both Shanghai and Shenzhen stock market and the Hong Kong concentration, the current market fluctuations, greatly affecting the market fluctuations would in the next issue, the instability yields..
・ The volatility of the Shanghai and Shenzhen stock markets is larger than that of the Hong Kong stock market, partly because of the long-term characteristics of different markets and partly because the market leverage is not the same. The leverage effect of Shanghai and Shenzhen stock markets is greater, and the current volatility is affected by the previous period. It is obvious that the impact of the negative income in the previous period is greater than that of the positive income.
This paper constructs the stochastic volatility model with leverage effect on the yield of Shanghai and Shenzhen 300 Index and the Hong Kong Hang Seng Index yield respectively, and compares the characteristics of Shanghai and Shenzhen stock market and Hong Kong stock market. Empirical results show that the mainland stock market long-term fluctuations in the larger leverage more. This shows that China’s stock market is immature, and the market speculators occupy a large part. The market, some tiny events or news, is enough to make the market a strong volatility. In addition, China’s stock market system and Hong Kong or Europe and the United States stock market system there are differences. In the transaction mode, the mainland market stock trading is taken by T + 1 trading, and Hong Kong stock market trading is taken by the T + 0 trading; in the trading varieties, the mainland market only stocks and funds, Hong Kong stock market, including Stocks, Hang Seng futures, options and other complex varieties of transactions. The Hong Kong stock market is more similar to the European and American stock markets. Therefore, some Hong Kong or Europe and the United States market mechanism does not necessarily apply to China’s stock market. Not long ago short-lived fuse system, just four days, there have been two closed. Each down to 5% of the suspension of trading points, after the resumption of trading, immediately fell to 7% of the day to stop trading points. This shows that the proportion of speculators in China’s stock market is very large, a considerable number of people follow suit “investment”. When you see the stock fell, a considerable part of the investors did not rational analysis, have to sell, resulting in further stock price drop. Therefore, the improvement of China’s financial system, deepening the financial system reform, is still long way to go.
Thanks for my teacher JL Yin, who has walked me through all the stages of the writing of this article. Without his consistent and illuminating instruction, this paper could not have reached its present form. Secondly, this article would not have been possible without the consistent and valuable reference materials that I received from my good friend Mum Kin Lee, whose insightful guidance and enthusiastic encouragement in the course of my shaping this article definitely gain my deepest gratitude.
Lin, J.H. (2017) Stock Exchanges Comparison between Mainland China and H.K. Based on the SVL Model. Open Journal of Statistics, 7, 383-393. https://doi.org/10.4236/ojs.2017.73027