Abstract: The analysis of co-jump facilitates the further research on systemic risk and transmission of market risk. Based on the three main theoretical methods to test for co-jumps, it is found that LM identifies 709 co-jumps, accounting for 2.03% percent of the data while BLT and FHLL methods identify 130 and 79 co-jumps, using high frequency data on 50 selected stocks from CSI 300 Index from January 21^st 2013 to January 21^st 2016. All the three methods suggest around 15% of the co-jump happens within half an hour after stock market opens even though intradaily seasonality is controlled. LM method shows that only when market crashed more than 20 individual stocks co-jump and the other 2 methods also capture the market crash feature. The three methods identify 5 common time, which are in market turbulent periods. Moreover, the moment when the circuit breaker is first triggered in 2016 is also identified as co-jump by all the three methods, which we refer as "circuit breaker co-jump". In general, all the three methods are effective to identify co-jump. LM method identifies more market movement using individual stock jump while BLT and FULL methods are more conservative to identify co-jumps. On the other hand, BLT and LM methods share most identification due to utilization of return data only while the identification of FULL is different by using more data. Despite the difference, the common identifications are usually referred as market turbulence, indicating the sensitivity of co-jump identification to market turbulence.
Due to the aggregation of jump and co-jump, the Hawkes process is introduced into the analysis of jump and co-jumps, constructing factor models based on Hawkes process. It is also shown that Hawkes process can better describe the aggregation of jumps than Poisson process by calculating the MJ and CJ statistics, representing the aggregation of jumps and co-jumps of individual stock respectively. Under the Hawkes assumption, the MJ statistics coincide with the empirical results however the CJ statistics indicates that independent Hawkes process cannot describe the aggregation of co-jumps. Due to the estimation difficulty of multi-dimensional Hawkes process, a factor model based on Hawkes process is proposed to describe individual stock jump with different probability when the market factor jumps under the circumstances when special idiosyncratic Jump happens. The jump of three individual stocks is treated as market factor event and delete the data when probability of the market factor event is high. The estimation results show that the market factor parameter converges significantly and the MJ and CJ statistics based on the model fits data well, indicating the fact that factor models can better describe aggregation of jump and co-jumps.

Key words: stock price, Hawkes process, factor model, systemic co-jump, high frequency data