Risk estimation of CSI 300 index spot and futures in China from a new perspective

Economic Modelling 49 (2015) 344–353

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Economic Modelling journal homepage: www.elsevier.com/locate/ecmod

Risk estimation of CSI 300 index spot and futures in China from a new perspective Yuan-Yuan Suo a, Dong-Hua Wang a,b,⁎, Sai-Ping Li c a b c

School of Business, East China University of Science and Technology, Shanghai 200237, China Research Center for Econophysics, East China University of Science and Technology, Shanghai 200237, China Institute of Physics, Academia Sinica, Nankang, Taipei 115, Taiwan

a r t i c l e

i n f o

Article history: Accepted 31 May 2015 Available online 12 June 2015 Keywords: CSI 300 index Recurrence interval Probability distribution Memory effect Risk estimation

a b s t r a c t We investigate the statistical behavior and application in risk estimation of recurrence intervals between highfrequency returns that are either larger than a given positive threshold or smaller than a negative threshold for the stock index and stock index futures markets in China. By studying the probability density function of recurrence intervals, we find symmetric profiles for both the positive and negative occurrence thresholds, which can be fitted with stretched exponential functions. The probability density function further scales with the mean interval as the unified functional form for different thresholds. We further study the dependence of the conditional probability density function and the scaled mean condition recurrence interval on the previous recurrence interval, and demonstrate the existence of short memory in recurrence intervals. The result from detrended fluctuation analysis exhibits long-term correlations, where the detrended fluctuation function decays as an exponential function, with an exponent between 0.5 and 1. Based on the results of the analysis of recurrence intervals, we construct a hazard function and define a loss probability in order to evaluate risk in financial markets. To our surprise, a crossover is found in the loss probability plot of the stock index and its futures market, which sheds light on the issue of value at risk (VaR) overestimation (underestimation) based on recurrence interval analysis of complex financial markets. The study would enable one to improve risk estimation and is useful for management of risks in financial markets. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Since the introduction of stock index futures trading, extensive research has been devoted to the question of whether index futures trading contributes to the risk management of the underlying stock markets in terms of hedging (Lafuente and Novales, 2003; Lee, 2010; Nguyen, 2014; Park and S. L. N., 1995; Wang et al., 2013). However, due to the extreme events that occur more frequent than Gaussian (Bunde et al., 2002; Derrida et al., 1996; Sornette, 2003) fluctuations, revealing the characteristics of big losses, it is impossible for the investors to be in a full-hedging state all the time. Therefore, quantifying the regime of large and unpredictable fluctuations in the financial markets is important both for understanding the present and predicting the possibilities of future disasters. The importance of understanding financial market behaviors from new perspectives urges many researchers from the field of economics, econometrics and mathematics to seek new ideas from other fields such as physics, which recently created a new interdisciplinary field of econophysics (Chen and Li, 2012; Mantegna and ⁎ Corresponding author at: 130 Meilong Road, P.O. Box 114, School of Business, East China University of Science and Technology, Shanghai 200237, China. Tel.: + 86 21 64253507. E-mail address: [email protected] (D.-H. Wang).

http://dx.doi.org/10.1016/j.econmod.2015.05.011 0264-9993/© 2015 Elsevier B.V. All rights reserved.

Stanley, 2000). In this study, we adopt the recurrence interval analysis to discuss the behaviors of the spot and futures returns of the CSI 300 stock index in China. Recurrence interval, also known as return interval or interspike interval, is the time interval between two consecutive events above (or below) some threshold q (Mcfadden, 1956). This is related to a well-established technique in statistical mechanics popularly referred to as “first passage statistics” or “persistence”(Majumdar, 1999). The method developed to analyze recurrence intervals in time series has been applied to many different areas, such as earthquakes (Livina et al., 2005), climate (Bunde et al., 2004, 2005), statistical models (Derrida et al., 1994) and heartbeat in medicine science (Bogachev et al., 2009). The idea of recurrence interval analysis is to predict catastrophic events like floods, droughts or stock crisis by using the laws of small fluctuations for the reason that we can find some scaling properties in different levels of the entire data set, which effectively solves the problem of insufficient data from extreme fluctuations (Chattopadhyay and Burroughs, 2007).The recurrence intervals between volatilities have been carefully studied (Cont, 2001; Engle and Patton, 2001; Kaizoji and Kaizoji, 2004; Lee et al., 2006; Qiu et al., 2008; Ren et al., 2009a,b; Tseng and Li, 2011; Wang et al., 2006, 2007; Xie et al., 2014; Yamasaki et al., 2005). The volatility v(t) can be defined as the absolute value of the logarithm yields at time t, namely, v(t) =

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|lnp(t) − lnp(t − Δt)| where p(t) is the price at time t. Yamasaki et al. (2005) studied the probability distribution of the recurrence interval between the daily volatilities of the stocks and currencies price changes in the USA, and indeed find a scaling behavior and long-term correlations. Wang et al. (2007, 2006)further tested this newly discovered statistical property using daily and intraday samples of stock indices, stock prices, exchange rates, crude oil and gold price in the U.S. financial market, and the scaling function f(x) can be well-approximated by the γ

stretched exponential function f ðxÞ ¼ ατe−ðβτxÞ . Similar scaling and memory properties of the recurrence interval between volatilities are also found in the Chinese stock market, and the correlation exponents γ are different for different thresholds q (Liu et al., 2009; Ren and Zhou, 2008; Ren et al., 2009a; Xie et al., 2014). The recurrence interval analysis is also applied to the Japanese and Korean stock markets (Kaizoji and Kaizoji, 2004; Lee et al., 2006), and their volatility recurrence interval distributions obey power law decays (Bush and Chattopadhyay, 2014) of the form f(x) = αx− β (Majumdar et al., 1996). For the volatility recurrence intervals of both daily and high frequency data sets from different financial markets, one arrives at similar conclusions: (1) the distribution of the scaled recurrence interval τ=τ can be approximated by a stretched exponential function or power–law function, and (2) the sequence of the recurrence interval has a long-term memory related to the original volatility series. Subsequently, some researchers also focus on the trading volume, which is well known as an important variable reflecting the liquidity of the financial market, based on the recurrence interval analysis. Podobnik et al. (2009) used the recurrence interval analysis method to calculate the time intervals between the logarithmic change in trading volume of the world-wide financial indices and stock prices, and eventually demonstrated that the average intervals obey a power law distribution which follows an inverse cubic law. Ren and Zhou (2010b) investigated the recurrence intervals of trading volume for the 20 liquid Chinese stocks, in which a power–law scaling was found in the tail of the recurrence interval distribution. To better understand these scaling features and correlations, the recurrence interval analysis was further carried out to study the price returns of different financial products (Bogachev and Bunde, 2009; Bogachev et al., 2007, 2008; Meng et al., 2012; Ren and Zhou, 2010a; Yamasaki et al., 2006). Bogachev et al. (2007, 2008) investigated the distribution of recurrence intervals between large fluctuations exceeding some thresholds in artificial multifractal data sets with nonlinear correlations, and observed power–law behaviors which depend on the threshold value and system size. They further introduced a new risk measurement approach by using a probability function Wq(Δt|t). In the recurrence interval analysis (RIA), Wq(Δt|t) is the central quantity for risk estimation, which is defined as the probability of reoccurrence of certain type of events at least once in the next Δt units of time, if the last time such type of events occurred t time units ago. More recently, Bogachev and Bunde (2009) used the RIA method to study how to improve the estimation of VaR, which is probably the best-known risk measure of the risk of loss in financial markets. Similar to findings in (Bogachev and Bunde, 2009), power–law tails were found in the distribution of recurrence intervals in the high-frequency data sets of (Ren and Zhou, 2010a), which were tested by three goodness-of-fit measures. They further applied the RIA to the risk estimation for the Chinese stock markets. The availability of high-quality financial data leads to significant advancement in modeling and estimating risk. In this paper, we investigate the recurrence intervals that occur in the high-frequency data of the CSI 300 index spot and futures markets. There are several reasons to consider in both the spot and futures indices in this study. First, from the aspect of policymakers and regulators of China Financial Futures Exchange (CFFEX), the volatility of the spot market tends to play an important role in futures trading. Since the CSI 300 index is the underlying asset of the index futures, many trading rules and contract specifications are designed which are based on the spot

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market. Owing to the large fluctuations of the present stock market, the margin requirement has increased from the initial 8% to 12% of the contract value, where 6% is indeed our target margin. Understanding the behavior of price fluctuations in the spot market will help us set the proper margin of the index futures market. Second, for arbitrageurs who attempt to profit from price inefficiencies in the market by making simultaneous trades that offset each other and capturing risk-free profits, understanding the similarities and differences of price changing between the spot and futures markets can help them get the risk-free profits from basis changes. Finally, from the perspective of risk management, previous studies verified that there exists risk contagion effect in financial markets. When making assets location in the spot market, one cannot just consider the spot market risk but neglect the futures market. Therefore, it is meaningful to reveal and compare the similarities and differences of the behavioral properties between the CSI 300 index and futures markets. In our study here, 1-min high-frequency datasets are used to gain better statistics. Our study therefore contributes to the literature in the following perspectives. First, to the best of our knowledge, this is the first study to explore and compare the behavior of recurrence intervals between the stock index and its futures market in China, even though global currency markets (Yamasaki et al., 2005), U.S. stock market (Wang et al., 2006, 2007) and Chinese stock markets (Ren and Zhou, 2010a,b; Ren et al., 2009b) have been much explored. Second, most of the previous works focus on studying the statistical properties of recurrence intervals series between extreme events. In this paper, we attempt to apply recurrence interval analysis to risk estimation, and to provide a relation between mean recurrence interval and VaR, in which case empirical evidence is given why traditional methods overestimate or underestimate value at risk in the complex financial system. Finally, we test whether behaviors of CSI 300 index and futures are consistent with results of previous studies by using loss probability distribution. To our surprise, a crossover is found in the loss probability plot, which is different from the power–law properties found in (Ren and Zhou, 2010a). This difference has its own economic explanations which will be given below. The rest of this paper is organized as follows. Section 2 describes data and sample selection, and presents basic statistics. Section 3 examines the empirical distributions of the recurrence intervals of the spot and futures returns. Section 4 studies the short and long term memory effect based on the conditional probability distribution and detrended fluctuation analysis (DFA) method. Section 5 explores the risk improvement for the stock index futures and spot markets in China. Section 6 is the conclusion. 2. Data description Since the inception of the Shanghai Securities Exchange (SHSEX) and the Shenzhen Securities Exchange (SZSEX) in the early 1990s, the Chinese financial markets have experienced tremendous growth and attracted increasing attention. By the end of 2006, there were already 1434 listed companies in China, out of which 109 companies issued B shares. A total of 1265.5 billion shares were issued with an aggregate market capitalization of RMB 8940.4 billion, and 78.49 million securities investment accounts were opened, mostly by retail investment. However, there was no tool for them to hedge their position against downturns in China's notoriously volatile markets. On September 8, 2006, China Financial Futures Exchange (CFFEX) was founded in Shanghai. After more than three years of practice trading, the CSI 300 index futures contract was launched on April 16, 2010 on the CFFEX, providing investors with a tool to hedge risks in stock markets. The CSI 300, which stands for the China Securities Index 300 stock index, was created by the China Securities and Index Company Ltd. on April 8, 2005. The index comprises 300 stocks listed on the SHSEX or SZSEX, and accounts for approximately 70% of the total market capitalization of both stock exchanges. Therefore, the index is widely perceived to comprehensively reflect both the price fluctuations and performance of China A-share markets. Also note that the aim of designing the stock

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Table 1 Contract specifications of the CSI 300 index futures. Underlying index Contract multiplier Unit Tick size Contract months Trading hours Trading hour on last trading day Limit up/down Margin requirement Last trading day Delivery day Settlement method Transaction code Exchange

CSI 300 index CNY 300 Index point 0.2 point Monthly: current month, next month, next two calendar quarters (four total) 09:15 am–11:30 am, 01:00 pm–03:15 pm 09:15 am–11:30 am, 01:00 pm–03:00 pm +/−10% of settlement price on the previous trading day 12% of the contract value Third friday of the contract month, postponed to the next business day if it falls on a public holiday Third friday, same as last trading day Cash settlement IF China Financial Futures Exchange

index is for use as the basis for derivatives innovation. A few years later, for the purpose of managing risk, maintaining market stability and improving capital allocation in the Chinese economy, index futures based on the CSI 300 was launched as a historic milestone on the path to a market-driven economy of the mainland stock markets. In order to prevent drastic fluctuations of futures market, the CSI 300 index futures are strictly monitored by regulators. Eligible individual investors must have a minimum of RMB 500,000 yuan (approximately US $73,000) to open an account for stock index futures trading. Experience in futures trading and simulated trading of index futures are needed. They are also required to complete a training course and pass a test before they can do trading in the market. In contrast, the minimum deposit of institutional investors is RMB 1 million (approximately US $145,000). In addition, margin requirements for stock index futures is set at 12 percent to control trading risk, and index futures are subject to a +/−10% of settlement price on the previous trading day. Details of the contract specifications of the CSI 300 index futures are shown in Table 1. The CSI 300 index futures contract prices and underlying asset market prices are obtained from the RESSET Financial Research Database (http://www.cffex.com.cn/). The sample period is from April 16th, 2010 through December 31st, 2012. We construct a continuous time series for the futures prices by using the price of the main contract according to the industry standard. The main contract is used because it is highly liquid and most active. We specify the contract as the main contract based on the trading volume on a trading day. Table 1 lists the four CSI 300 index futures contracts being traded simultaneously. To construct the continuous futures price series, we choose the intraday prices of the main contracts, which is defined as the contract that has the largest trading volume among the four contracts on each day.

Compared to the trading hours of the CSI 300 index futures contract, the spot index contract is traded from 9:30 a.m. to 11:30 a.m. and from 1:00 p.m. to 3:00 p.m. (Beijing Time), which is 15 minutes after the opening hours and 15 minutes before the closing hours of the futures index. Excluding weekends and holidays for which trading is closed, and also the mismatched data, we end up with 156,000 price observations for each time series. The returns of each series are calculated by taking differences of the logarithms of price as r(t) = |lnp(t) − lnp(t − Δt)|, where p(t) is the closing price of the tth minute. The logarithmic returns of the CSI 300 index and the futures index are shown in Fig. 1(a) and (b) respectively and their statistics are summarized in Table 2. Table 2 shows that the fluctuations of the spot and futures range between − 0.03 and 0.05 while the mean and standard deviation of two returns series are similar. On the other hand, both the spot and futures indices display fat excessive kurtosis with large kurtosis coefficients (384.88 and 160.90, respectively). We note that the main difference is their skewness behavior: the CSI 300 index is negatively skewed while the futures is positively skewed (−0.086 and 2.020 respectively), in agreement with the result of (Hou and Li, 2013). To the best of our knowledge, most researchers are more interested in using the returns of asset price to study the properties and dynamical mechanisms of the financial markets, e.g., price discovery, volatility transmission and risk estimation (Ekin and Arda, 2010; Ghosh, 1993; Helena and Hipolit, 2008; Tse, 1999; Yang et al., 2001, 2012). Referring to Fig. 1, it is easy to see that there are clusters (namely volatility clustering phenomenon) in the time series, which is an indication of long term memory (Lin and Fei, 2013; Liu and Chen, 2013; Yalama and Celik, 2013), i.e. small fluctuations tend to follow small fluctuations, and large fluctuations tend to follow large fluctuations. This phenomenon has been investigated by many researchers (Cunado et al., 2010; Xie et al., 2014; Yamasaki et al., 2005). A question of interest in this study is whether there is a new method we can use to describe this behavior and to estimate the time intervals between large fluctuations, which can help to improve the risk estimation in the complex market of financial derivatives? Specifically, when will the next large fluctuation occur after a large fluctuation is being observed? 3. Empirical probability density function 3.1. Recurrence interval To examine the memory behaviors and to improve risk estimation of the CSI 300 index futures and spot market in China, we follow the work of (Bogachev and Bunde, 2009; Bogachev et al., 2007; Ren and Zhou, 2010a; Yamasaki et al., 2005) and take the probability distribution of recurrence intervals between large fluctuations into consideration first. In

Fig. 1. Logarithmic returns of CSI 300 index and futures prices.

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Table 2 Statistics of the logarithmic returns of the CSI 300 and its futures.

Spot Futures

Minimum

Maximum

Mean

Standard deviation

Skewness

Kurtosis

Nobs

−0.031 −0.027

0.046 0.051

−1.88e−6 −1.97e−6

6.83e−4 8.44e−4

−0.086 2.020

384.88 160.90

156,000 156,000

order to apply the recursive interval analysis, we need to normalize the price returns of each series by dividing its standard deviation Rðt Þ ¼ h

r ðt Þ 2

b r ðt Þ N − b rðtÞ N 2

i1=2 :

ð1Þ

has occurred, the probability that the next extreme event would occur will be higher for larger thresholds. The probability density function of the recurrence interval between large returns of the CSI index and its futures are now fitted with the γ stretched exponential function f ðxÞ ¼ ατe−ðβτxÞ (Xie et al., 2014).

Using the normalized sequences R(t), we measure the recurrence intervals by introducing a threshold q. A recurrence interval τ is defined as the time interval from the time when the returns of the normalized prices drops below the threshold q N 0 or rises above the threshold q b 0 to the next time point that it passes the threshold again as illustrated in Fig. 2. The return interval τq = 1 and τq = −1 for the thresholds q = 1 and q = −1 are indicated in the figure for the part of normalized return series R(t) of the CSI 300 index return, and the patches in Fig. 2 are highly obvious as the results of (Yamasaki et al., 2005), which are the indication of memory, as short recurrence intervals tend to follow short intervals, and long intervals tend to follow long intervals.

The parameters of the function based on the maximum likelihood estimation are shown in Table 3. The coefficient of the exponent γ of the stretched exponential function is the correlation exponent characterizing the long term memory of the recurrence intervals. Note that the correlation exponents of futures are in the range between 0.26 and 0.39, while the corresponding exponents of the spot index are much smaller within the interval (0.20, 0.24). It is obvious that the long memory effect in the recurrence intervals of futures dies out faster than the spot, meaning that high frequency trading should be preferred for futures speculators. Although the results above seem to be useful to the investors and regulators, the more relevant question is: Are there any scaling relations between these empirical probability density functions?

3.2. Empirical probability density function

3.3. Scaled probability density function

We begin by studying the behavior of the empirical probability density function Pq(τ) of the recurrence intervals between returns with different values of threshold q for the spot and futures, and show its dependence on q. The results are shown in Fig. 3(a) and (b). Fig. 3 shows that Pq(τ) decays slower for larger thresholds, in agreement with the fact that the recurrence intervals between returns that exceed the thresholds should be longer than that for smaller thresholds. Furthermore, Pq(τ) is symmetrical between positive threshold q N 0 and negative threshold q b 0, having the same magnitude for both the CSI 300 index and futures. When focus on the same threshold q, the probability Pq(τ) decreases with increasing recurrence interval τ. This implies that when the last extreme event above a threshold q occurred t units of time ago, the probability Pq(τ) of another extreme event above q to occur Δt1 later is larger than Δt2, if Δt1 b Δt2. This result suggests that if the investors suffered heavy loss when experienced the large fluctuation in the stock market just some time units ago, they should be aware that the probability that the next strong fluctuation would come within a shorter time interval is larger than that of a longer time interval. On the other hand, with a fixed recurrence interval γ, the probability Pq(τ) increases with increasing threshold values. If an extreme event

To explore the universal properties of the recurrence intervals, we scale the recurrence intervals by the mean interval τ≡τ with threshold q, and the scaled probability density function P q ðτ Þτ, as a function of the scaled return intervals τ=τ is shown in Fig. 4. One can see that the curves for different thresholds q approximately collapse onto a single curve as shown for the spot and futures in Fig. 4(a) and 4(b). Thus the probability density functions take the following scaling relation (Yamasaki et al., 2005) P q ðτÞ ¼

1 f ðτ=τÞ : τ

ð2Þ

Note that the universal scaling function f(x) does not depend directly on the mean return interval τ, but only through the threshold q. Therefore, if the probability density function Pq(τ) for one threshold q is known, the probability density function for other thresholds can be deduced from Eq. (2). One can therefore predict the large fluctuations or extreme events in the financial market by first estimating the probability distribution of small fluctuations with rich data, and then extrapolate the probability distribution of extreme events by using the scaling function. To better understand the origin of the scaling behavior, we reshuffle the time series and study the probability density function of the recurrence intervals of the reshuffled returns of the CSI 300 index and futures time series. This would then remove correlations in the returns sequences. It is easy to see that the original time series has higher probability for both small and large intervals when compared to the reshuffled series as shown in Fig. 4. 4. Memory effect

Fig. 2. Schematic illustration of recurrence intervals between returns.

Although Pq(τ) provides us with a scaling behavior, it may still not fully characterize the recurrence interval series between returns. Indeed, if the recurrence intervals in the series are independent of each other without correlation, they will totally be determined by Pq(τ). On the other hand, if the recurrence intervals are correlated, they may be a result of the memory effect. The clustering phenomenon above

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Fig. 3. Empirical probability distributions Pq(τ) of the recurrence intervals between returns with different threshold q for CSI 300 index and futures.

Table 3 Estimates of α, β and γ of the stretched exponential function.

Future

q

α

β

γ

α

β

γ

1.0 1.2 1.4 1.6 1.8 −1.0 −1.2 −1.4 −1.6 −1.8

0.05 0.07 0.05 0.03 0.03 0.10 0.04 0.03 0.02 0.01

0.13 0.33 0.35 0.29 0.40 0.24 0.11 0.13 0.07 0.11

0.40 0.32 0.30 0.28 0.26 0.38 0.38 0.33 0.33 0.29

0.79 0.49 0.27 0.11 0.07 0.60 0.27 0.16 0.07 0.04

16.5 17.8 11.7 4.0 3.4 11.1 5.95 5.11 3.4 3.0

0.23 0.22 0.21 0.22 0.21 0.24 0.23 0.22 0.21 0.20

Spot

suggests that memory exists in these sequences, which is another important characteristic feature of financial markets. To demonstrate the memory effect in the recurrence intervals between large returns of the CSI 300 index and futures, we use the conditional probability density function Pq(τ|τ0), the mean conditional interval τ ðτ0 Þ and carry out the detrended fluctuation analysis (DFA).

4.1. Short memory effect To quantify the memory effect in the recurrence interval, we first investigate the conditional probability density function Pq(τ|τ0), which is the probability of finding a recurrence interval τ immediately after the recurrence interval τ0. To get better statistics, we study the conditional probability density function Pq(τ|τ0) not for a specific value of τ0, but for values of τ0 in certain intervals. Specifically, we have sorted the recurrence interval records in increasing order and partitioned it into

four bins Q1, Q2, Q3 and Q4 with the same length. By definition, the N/4 smallest recurrence intervals are put in the first bin Q1 while Q4 contains the largest quarter of N. We show the scaled conditional probability density function P q ðτjτ 0 Þτ as a function of τ=τ for τ0 in the largest (open symbols) and smallest (filled symbols) subsets in Fig. 5. For τ0 in Q1, P q ðτjτÞ is larger for small τ=τ while for τ0 in Q4, P q ðτjτ Þ is larger for large τ=τ. The fact that small τ tends to follow small τ0 and large τ follows large τ0 indicates that short memory exists in the recurrence intervals. Furthermore, P q ðτjτ Þ collapses into a single scaling function for both Q1 and Q4 for all thresholds q. This suggests that the sequences of the recurrence intervals cannot be characterized by Pq(τ) and memory is indeed in the sequences. The short term memory can also be observed in τ0 =τ, which is the mean recurrence intervals when the two extreme events occurred some time ago were separated by an interval τ0. To calculate τðτ 0 Þ using the divided-bin method of Pq(τ|τ0), we put the recurrence intervals into eight consecutive bins for each threshold q in increasing order. Each bin contains N/8 intervals, and we calculate the mean value in every sub-interval. Fig. 6 gives the scaled mean condition recurrence interval τðτ0 Þ=τ as a function of scaled recurrence interval τðτ0 Þ of the CSI 300 index and futures with open color symbols, which shows clearly the effect of memory as the Pq(τ|τ0), i.e., small recurrence intervals are more likely to follow small intervals, and large ones tend to follow large ones, similar to the phenomenon in Pq(τ|τ0). On the other hand, the reshuffled sequences with black symbols are almost constant (approaches 1) as the expected equation τ=τ≡1, which demonstrates that recurrence intervals are independent of the preceding return interval in the reshuffled records.

Fig. 4. Scaled probability distributions of recurrence intervals of original and reshuffled (inset) data for different thresholds of the CSI 300 spot index and futures.

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Fig. 5. Scaled conditional probability density function P q ðτjτ 0 Þτ of scaled recurrence intervalsτ=τ with τ0 in Q1 (solid symbols) and Q4 (open symbols) for the CSI 300 index and futures. The solid curves are fitted curves.

The quantities of conditional probability density function Pq(τ|τ0) and the mean conditional intervals τðτ 0 Þ indicate that short memory exists in the recurrence intervals τ between large returns of the CSI index and futures. These results imply that if two extreme events occurred some time ago were separated by a small interval τ0, another extreme event with the same threshold q would occur with a larger probability within a smaller interval τ than a longer interval. In other words, when there is a sharp decline in the stock market, there is a high probability that it will continue to fall. A scenario which could probably be due to fear across the market. 4.2. Long memory effect In the previous sections, scaling behavior and short-term memory are shown in the recurrence interval sequence, which are related to the long-term memory presented in (Yamasaki et al., 2005) and (Wang et al., 2006). To further investigate the long-term memory of the recurrence intervals between high-frequency returns of the Chinese stock index and futures market, detrended fluctuation analysis (DFA) is adopted here, which is a powerful method to characterize the long-term correlation in time series (Chen et al., 2002, 2005; Hu et al., 2001; Kantelhardt et al., 2001) DFA computes the detrended fluctuation function F(s) of a time sequence within a window of s time units and determines the exponent H from the scaling form F(s) ∼ s H, where H is the Hurst exponent. If H N 0.5, the sequence is long-term correlated while the sequence is uncorrelated for H = 0.5. Fig. 7 shows the detrended fluctuation function F(s) of the recurrence intervals for negative thresholds q between returns of the CSI 300 index and futures.

The fitted lines in the log–log plot reveal the fact that the detrended fluctuation function F(s) satisfies a scaling relation as F(s) ∼ s H. The exponents H can therefore be obtained from the slope of the fitted lines. To further understand the long memory effect, Fig. 8 is a plot of the estimated Hurst exponents H of the recurrence intervals between returns. For both the CSI 300 index and futures, the exponents H are greater than 1/2, suggesting that long term memory exists in the recurrence interval series. One also notices that the Hurst exponents of futures are larger than the Hurst exponents of the spot for the same thresholds, meaning that futures have stronger long-term memory effect. For comparison, we also obtain the Hurst exponents of the reshuffled time series which are shown in Fig. 8. Their Hurst exponents are close to 0.5 as expected. This is in agreement with previous studies (Ren and Zhou, 2010b; Ren et al., 2009b; Wang et al., 2006, 2007) on the recurrence intervals in financial markets and confirms the fact that long-term memory effect on recurrence intervals exists in the Chinese stock index and futures high-frequency returns time series. 5. Risk estimation In the previous section, we have carried out a thorough statistical analysis of the return intervals of the CSI 300 index and futures. In this section, we will apply the scaling memory behavior and memory properties of recurrence intervals between returns to estimate risk. We will use the hazard function Wq(Δt|t) (Bogachev and Bunde, 2009; Bogachev et al., 2007; Ren and Zhou, 2010a; Xie et al., 2014) and loss probability (Wang et al., 2007) to estimate the risk in the CSI 300 index and futures markets.

Fig. 6. Scaled mean condition recurrence interval τ ðτ 0 Þ=τ as a function of scaled recurrence intervals τ 0 =τ between returns of the CSI 300 index and futures.

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Fig. 7. Detrended fluctuation function F(s) of the recurrence intervals between returns of the CSI 300 index and futures.

5.1. Hazard function The hazard function Wq(Δt|t) is the central quantity for risk estimation in the recurrence interval analysis. It is defined as the probability that within a short time interval Δt at least one extreme event (below negative thresholds) would occur, given that the last extreme event occurred at t time units ago (Bogachev and Bunde, 2009; Bogachev et al., 2007). By definition, the hazard function can be expressed as follows Z

tþΔt

W q ðΔtjt Þ ¼ tZ t

∞

P q ðτÞdτ

:

ð3Þ

P q ðτÞdτ

5.2. Loss probability

We here use the parameters given in Table 3 to compute the theoretical value of Wq(Δt|t) for a given short time interval. To numerically evaluate Wq(Δt|t) from the empirical time series, we use the following discretized form 

count t b τ q b t þ Δt   W q ðΔtjt Þ ¼ count τ q N t

Color symbols shown in Fig. 9 are the hazard function Wq(Δt = 1|t) for different negative thresholds obtained numerically by using Eq. (4) for CSI 300 index and futures. Wq(Δt = 1|t) shows a slow decreasing trend with respect to t in the range between 1 and 60 time units, in agreement with the long term correlation in the previous section. Given a threshold q, one can compute the probability of the occurrence of the next extreme event. The black solid lines in Fig. 9 are the theoretical results obtained by using Eq. (3). One can see that the theoretical and numerical results of Wq(Δt|t) agree with each other for large t. The difference increases when t gets smaller, suggesting that the hazard function Wq(Δt|t) evaluated by using Eq. (3) overestimates risk for short time period.

 ð4Þ

where “count(τq N t)” is the number of recurrence intervals longer than t time units and “count(t b τq b t + Δt)” is the number of recurrence intervals between t and t + Δt for a given q.

In this section, we will use the loss probability density function in RIA to evaluate the value at risk (VaR), a well-known indicator of risk in financial markets. VaR is defined as the risk at a level of loss q (q b 0, same as the negative thresholds above) as follows: Z

−∞

q

ð5Þ

where P(R) is the probability density function for the normalized returns R(t) and P * is the probability of loss. The mean recurrence interN

q val is defined as τ q ≡ N1q ∑i¼1 τq;i , where Nq is the number of intervals

N

q τq;i is which fall below the negative threshold q. Furthermore, ∑i¼1 approximately equal to the total number of returns and Nq + 1 is the number of returns below the negative threshold q. We then get the relation between mean recurrence interval τq and VaR via Eq. (5) as follows:

1 ¼ τq

Fig. 8. Hurst exponent H of the CSI 300 index (red open symbols) and futures (blue open symbols) and the reshuffled series (solid symbols).

P ðRÞdR ¼ P 

Z

q

−∞

dR ¼

number of returns below negative threshold q : total number of returns

ð6Þ

It was pointed out by (Ren and Zhou, 2010a) that the reciprocal of the mean recurrence interval 1=τ and the absolute threshold |q| exhibits power–law relationship. We here instead find crossover points in the log–log plot for the CSI 300 index (at q0 = − 4) and futures (at q0 = − 8),as shown in Fig. 10. This becomes clear if one performs a semi-log plot as shown in Fig. 11. It is easy to see that the distributions are significantly different in the two regimes. By definition, we note 1=τ as loss probability and threshold q the loss. In the small loss regime (q N q0), we use a power law function 1=τ∼jqj−γ1 to fit the curve. The values of the exponent γ1 are 0.23 and 2.6 for the CSI 300 index and

Y.-Y. Suo et al. / Economic Modelling 49 (2015) 344–353

351

Fig. 9. The theoretical (solid curve) and numerical (color symbols) results of Wq(Δt = 1|t) for the CSI 300 index and futures.

Fig. 10. Reciprocal of mean recurrence interval 1=τ q as a function of absolute threshold |q| for the CSI 300 index and futures, where q is from − 16 to − 1. Green circles correspond to the whole range of thresholds, blue diamonds and red squares in the insets correspond to the small loss and large loss regimes respectively.

futures respectively. In the large loss regime (q b q0), the curve follows an exponential decay 1=τ∼expð−γ 2 jqjÞ as illustrated in Fig. 11:  1 jqj−γ1 ∼ : expð−γ 2 jqjÞ τq

ð7Þ

The results are shown in column 4 of Table 4 and the insets of Fig. 10. Therefore, if one wants to know the risk level corresponding to 1% probability of loss within the time interval of unit time, one will look at the loss probability 1=τ ¼ 1%, and get the corresponding risk level q, which is the VaR one is looking for. It is worth to point out the mechanism of loss probability functional changes at the crossover point. The loss probability initially decreases sharply as a power–law function, and turns into an exponential decay at the crossover point. The result implies that it is inevitable to overestimate or underestimate the value at risk (Dimitrios et al., 2014; Su, 2014; Su and Hung, 2011) in the complex financial market if one only uses a single functional form of distribution for the fitting. The crossover behavior in Fig. 11 strongly suggests that the two regimes are governed by different mechanisms. One further notices that the crossover points for the CSI 300 index and futures market are different, as illustrated in Fig. 11. Theoretically speaking, from the experience of developed economies (Bohl et al., 2001; Kavussanos et al., 2008;

Stoll and Whaley, 1990; Yang et al., 2001) it is widely perceived that the index futures market leads over the spot index market and plays a dominant role in price recovery, which is confirmed by the study of (Hou and Li, 2013) on the financial market of China. Therefore, the price

Fig. 11. Semi-log plot of the reciprocal of mean recurrence interval 1=τ q as a function of absolute threshold |q|,where q is from − 16 to − 1 for the CSI 300 index and futures.

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Table 4 Estimated parameters γ1 and γ2 of loss probability.

Future Spot

q0

γ1

γ2

−4 −8

0.23 2.6

0.16 2.6

change in index futures market is more volatile, and the corresponding probability is bigger at the same loss level q, which contribute to the larger crossover position. The above result implies that there is higher (lower) risk in the spot market than the futures markets during volatile (normal) periods. On the other hand, in Chinese markets, there are more institutional investors for the higher barriers in the stock index futures market, and empirical evidences suggest that institutional investors are more rational and better informed than individuals (Ahn et al., 2008; Boehmer et al., 2008; Chakravarty, 2001). The risk in the futures market during volatile periods would then be lowered than the spot market which mainly consists of individuals who mostly follow the capital flows of institutions especially during market crashes. The above findings therefore reveal the fact that for the same threshold q, the probability of loss in the futures market is larger than the spot market when the financial system is normal (|q| is small) but is smaller than the spot market when the market is more volatile (|q| is large). This in turn contributes to the different crossover positions in the CSI 300 index and futures markets. 6. Conclusion Using intraday high frequency data, we investigate the scaling behavior, memory effects and risk estimation by the recurrence interval analysis method for the Chinese stock index and the newly introduced stock index futures markets in China. We find that the probability density functions of the recurrence intervals with different thresholds collapse onto a single curve which only depends on the scaled interval τ=τ for both the CSI 300 index and futures. This is consistent with recent findings on the global currency markets (Yamasaki et al., 2005), U.S. stock market (Wang et al., 2006, 2007) and Chinese stock markets (Ren and Zhou, 2010a,b; Ren et al., 2009b). Short memory effect is also found in recurrence intervals between returns by investigating the conditional probability density function Pq(τ|τ0) and the mean conditional recurrence interval τðτ0 Þ, meaning that a small recurrence interval is more likely to be followed by a small interval, and large intervals are more likely to be followed by large intervals. In addition, long-term correlation exists in the recurrence intervals, which is tested by the detrended fluctuation analysis (DFA) method. The findings on memory effect are also in line with that of (Ren and Zhou, 2010a), who first demonstrated the existence of memory effect in recurrence intervals between financial returns. Finally, we apply the recurrence interval analysis (RIA) to evaluate the risk for the Chinese stock index and stock index futures markets, which provide relatively accurate estimates of the hazard function Wq(Δt|t) and forge a link between loss probability and VaR. The crossover in the profile of loss probability indicates that there are different mechanisms in the small and large loss regimes. This in turn sheds light on the issue of either overestimating or underestimating the value at risk in complex financial markets with a single functional form of distribution of loss. These results would enable us to improve risk estimation and are useful for management of risks in financial markets. Acknowledgment The authors would like to thank an anonymous reviewer for his constructive suggestions and comments. This work was supported by

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