Abstract: The price discovery of stock index futures is a core function for futures market, and is also regarded as an important indicator for measuring the level of market development. An effective price discovery function can be helpful for stabilizing the financial markets and improving the asset pricing mechanism, and optimizing the allocation of financial resources. Hence, it is of great significance to explore the relationship between Chinese stock index futures and spot markets, and evaluate the effect of price discovery function. In current literatures, the studies of price discovery are usually focused on the leading role of futures market in the aspect of trend and wave, while there is rarely a study of the jump. In theory, the new information flows from futures market to spot market, which means that the occurrences of a jump in futures market can, to some extent, herald a jump in spot market. Since the futures and spot jumps can cause enormous impact on trading system and disturb the normal order of financial market, and severely affect the investment decisions of investors, enough attention should be paid to the jump spillover from futures market to spot market by policymakers, practitioners and investors. Hence, it is significant in making reasonable investment strategy and managing jump risk effectively to explore the price discovery of Chinese stock index futures in jumps.
To study the price discovery of Chinese stock index futures in jumps, this paper uses the 2-dimension Hawkes processes with self-excitement and cross-excitement features to model the jumps of futures and spot returns. Hawkes processes allow jumps of returns to cluster across the time and spread between futures and spot markets, and can describe the interaction mechanism between the jumps of futures and spot returns effectively. Base on Hawkes processes, this paper uses daily data to perform some empirical analyses of the jumps of futures and spot returns of CSI 300, SSE 50 and CSI 500, respectively. The sample period of CSI 300 is from January 1, 2011, to December 31, 2020, and the sample periods of SSE 50 and CSI 500 are both from July 1, 2015, to December 31, 2020. This paper sets the quantile of 98.5% of sample data as the threshold of the upward jumps and the quantile of 1.5% of sample data as the threshold of the downward jumps, and estimates the parameters in Hawkes processes by using maximum likelihood estimation. Besides, this paper tests the robustness of parameter estimators by changing the threshold of the jumps, and obtains the fitting degree of Hawkes process for the occurrences of jumps by constructing residual processes and using Kolmogorov-Smirnov test. Finally, this paper tests the price discovery of three kinds of stock index futures in jumps by using Z test.
The empirical results show that the empirical distribution of residual processes related with Hawkes processes are consistent with the exponential distribution whose mean is equal to 1, which means that Hawkes processes can fit the occurrences of the jumps well for three stock index futures and spot. It is obvious that there exists codirectional self-excitement of the jumps of stock index futures and spots, and except for the CSI 500, the self-excitement of downward jump is stronger than that of upward jump. The occurrence of jumps in stock index futures will significantly stimulate the extreme jumps to occur in stock index, but the occurrence of jumps in stock index generally cannot stimulate the extreme jumps to occur in stock index futures. In summary, Chinese stock index futures have the function of price discovery in jumps. The jumps of futures price can lead to the jumps of spot price, but the leading function of spot price in jumps is not significant. The main contributions of this paper are as follows. First, this paper provides a theoretical model and some empirical methods to measure the price discovery effect in jumps of stock index futures, which can enrich the relevant theory. Second, this paper applies the above model and methods to Chinese stock index futures and spot markets, and explores the price discovery in the jumps of futures returns, which can fill the existing gap in the empirical study. Third, this paper provides useful insights for investors to hedge the jump risk by using stock index futures, and can also help regulators take effective measures to monitor and control the jump risk in futures and spot markets.

Key words: stock index futures; price jumps; price discovery; Hawkes processes

摘要： 为了揭示中国股指期货在跳跃上的价格发现功能,本文采用具有自刺激和交叉刺激特征的二维Hawkes过程,对股指期货和现货的跳跃进行建模,并分别以我国沪深300、上证50和中证500股指期货和现货收益率作为样本,探究我国股指期货和现货的跳跃自刺激行为以及相互之间的跳跃交叉刺激作用。实证结果表明:Hakwes过程对三类股指期货和现货的跳跃均拟合得较好;股指期货和现货的跳跃存在着明显的同向自刺激,且除了中证500股指以外,下跳的自刺激均强于上跳;股指期货的跳跃会显著刺激股指现货发生跳跃,但股指现货的跳跃对股指期货的刺激并不显著。总而言之,中国股指期货在跳跃上具有价格发现功能,期货价格的跳跃能够引导现货价格发生跳跃,而现货价格跳跃对期货价格跳跃的引导作用则不明显。

关键词: 股指期货, 价格跳跃, 价格发现, Hawkes过程