*2.2. Jump Detection Scheme*

The timing of jumps has an essential meaning for examining anomalous behavior around jumps. To identify overnight gaps via jump tests, the precise time must be known. For this purpose, we rely on the jump detection scheme introduced by Andersen et al. (2010). This jump identification procedure is designed on the premise that jumps are rare events. If it is assumed that *t* equals one day and at most one jump can emerge during the corresponding period, the only intraday jump can be determined with:

$$RV\_t - BPV\_t \stackrel{p}{\rightarrow} c\_{t,\*}^2 \tag{20}$$

where *c*2*t* represents the jump variation in period *t*. The intuitive idea is that the jump must be incorporated in the highest absolute return on that specific day. Hence, the timing of the jump can be determined by seeking the highest absolute return of the period. Furthermore, the precise jump size can be calculated in the following way:

$$\widetilde{\mathbf{c}}\_{t} = \text{sgn}\left( \left\{ r\_{t,\varepsilon} : |r\_{t,\varepsilon}| = \max\_{j \in \{1, \ldots, M\}} |r\_{t,j}| \right\} \right) c\_{t'}^{2} \tag{21}$$

where *rt*,*<sup>c</sup>* denotes the intraday return that contains the jump contribution, while *sgn*(·) is equal to 1 or −1, depending on the sign of the argument.

### **3. Event Study of the S&P 500 Index**

This section uses the outlined methodology of Section 2 to identify and analyze overnight price gaps in the S&P 500 index. Following the approaches of Fung et al. (2000) and Grant et al. (2005), we conducted the following four steps.

At first, the data were filtered according to the event of interest, the presence of overnight gaps. To identify overnight gaps, we conducted daily the BNS jump tests, as introduced in Section 2.1. For the test, we used high-frequency intraday returns of the previous day and the overnight return and a significance level of 0.1 percent. The timing of jumps was determined by the jump detection procedure of Andersen et al. (2010) (see Section 2.2). If the timing of the jump corresponded with the overnight return, the day was marked as an event day and included in our study.

Second, for every event day, the cumulative return of the S&P 500 index at minute *t* after the market opening was computed by:

$$CR\_{i,t} = \frac{P\_{i,t}}{P\_{i,0}} - 1,\tag{22}$$

where *Pi*,*<sup>t</sup>* denotes the index price on event day *i* at minute *t* after the beginning of the trading day. Respectively, *t* = 0 represents the market opening.

Third, the average cumulative return (*ACR*) at time *t*:

$$ACR\_t = \frac{1}{N} \sum\_{i=1}^{N} CR\_{i,t\prime} \tag{23}$$

was computed for all event days. This figure is available for any minute *t* after the start of the trading day. *N* is defined as the total number of days fulfilling the event day properties.

Fourth, *t*-tests were conducted to determine whether a given price movement after a specified event was significant. Specifically, we calculated the corresponding test statistic to examine if the *ACRt* at time *t* was significantly distinct from zero. The test statistic had the following form:

$$t\_{ACR\_l} = \sqrt{N} \frac{(\overline{ACR\_l} - 0)}{S\_{ACR\_l}} \sim t(N - 1),\tag{24}$$

where 0 < *t* ≤ *T* and *ACRt* denotes the mean of the sample. Furthermore, *SACRt* represents its standard deviation, and *N* defines the total numbers of days in the filtered dataset. Under the null hypothesis of no distinction from zero, the test statistic follows a *t*-distribution with *N* − 1 degrees of freedom.

Table 1 shows the characteristics of the overnight price gaps detected by our jump test procedure. In total, we observed 2128 overnight gaps during the sample period: 1154 of those gaps were positive, while 974 were negative. On average, the S&P 500 index faced positive (negative) overnight gaps of 0.60 percent (−0.67 percent). The largest overnight gaps occurred during the global financial crisis with 6.02 percent and −7.64 percent. The fact that both the range and the standard deviation of negative gaps were higher than those of positive overnight movements confirms the existing literature: market participants tend to react stronger to bad news rather than to good headlines (Suleman 2012). Concluding, Table 1 shows that there was a sufficient number of overnight price gaps leading to temporary market inefficiencies. As a result, this jump behavior generated high-frequency stock price dynamics that created major trading opportunities. In stark contrast to the approach of Fung et al. (2000) and Grant et al. (2005), the gaps identified by our jump-test scheme were both flexible and data-driven.

**Table 1.** Characteristics of positive and negative overnight gaps, which are identified by the Barndorff–Nielsen and Shephard (BNS) jump test, from January 1998–December 2015.


Figure 1 illustrates the detected jumps in a more detailed way. We observe a higher variation of negative overnight gaps, which is not surprising since financial data possess an asymmetric distribution (Cont 2001). Interestingly, the interval with the highest number of observations for both positive and negative overnight gaps was about ±0.15.

**Figure 1.** Histogram of positive and negative overnight gaps, which were identified by the BNS jump test, from January 1998–December 2015.

Figure 2 presents the number of detected overnight gaps over time. With rising volatility in financial markets, the number of overnight gaps also increased; fluctuations in the market imply jumps. Thus, it is not surprising that we observed almost no jumps in the first years of our sample period. In stark contrast, the number of overnight price gaps increased in times of high market turmoil. In general, more positive than negative gaps affect the S&P 500 index. As expected, this pattern changes during crises such as the dot-com crash in the early 2000s and the financial crisis in 2008. This also demonstrates the flexibility of the approach used to identify overnight gaps.

**Figure 2.** Development of positive and negative overnight gaps, which were identified by the BNS jump test, from 1998–2015.

Figure 3 depicts the average cumulative returns after overnight gaps identified by the BNS jump test. The detailed development of the *ACR* for positive and negative price gaps is reported in Table A1. The typical price pattern after overnight gaps is still persistent in modern financial markets, despite that markets should become more efficient in the course of digitalization and improved information flow (see Fung et al. (2000) and Grant et al. (2005)). In the case of a positive overnight gap, the average cumulative returns rose for a brief period before reverting to the minimum at −0.0316 percent. After reaching the lowest *ACR* 105 min after market opening, it began to rise until it crossed the zero percent line. From this point, the returns almost fell close to the minimum before increasing again. The upswing accelerated towards market closing, reaching 0.0236 percent at the end of the trading day. Following a negative overnight gap, the *ACR* move inverted. Starting with a brief continuation of the initial overnight movement, which marked the minimum of −0.0093 percent two

minutes after the stock exchange opens, the *ACR* began to reverse to its maximum of 0.0463 percent after approximately one and a half hours. The *ACR* remained relatively stable between 0.0200 and 0.0400 percent subsequent to hitting the upper limit. During the last ten minutes, the *ACR* rapidly decreased until the end of the trading day. Noticeable is that the magnitude of the variation of the *ACR* was stronger after negative price gaps. This is in line with stronger expected reactions of market participants to bad information that was also observable in the represented gap characteristics (Table 1). The *p*-values for both *ACR* realizations indicated that the returns were statistically different from zero on a 10 percent significance level for most of the time before the 115-min mark. After that threshold has passed, *p*-values well exceeded 10 percent; this fact is not surprising since many professional day traders stop trading after two trading hours because volatility and volume tend to decrease (see Balance (2019)). Furthermore, we recognized that the *ACR* for positive overnight gaps were not significant for a target time of 5, 35, 65, and 95 min based on a 10% significance level; it seems that the pattern is systematically repeated at 30-min intervals. This statement is confirmed by Business Insider (2015), which shows that the trading volume increases in the first minutes of every trading hour. Furthermore, Bedowska-Sojka (2013) demonstrated that this volatility is influenced by macroeconomic releases, which are typically published at 9:30, 10:00, 10:30, and 11:00. As a result, the test-statistic decreased, leading to non-significant *p*-values.

Concluding, our event study confirms the overreaction hypothesis and supports the results of Fung et al. (2000) and Grant et al. (2005). The findings of the event study further sugges<sup>t</sup> that we are in a position to develop a statistical arbitrage strategy that exploits the mean-reversion characteristic of stocks after statistically-significant overnight price gaps (see Poterba and Summers (1988), Leung and Li (2015), Lubnau and Todorova (2015)). Specifically, it seems profitable to open trades after overnight gaps and close them after 2 h, i.e., we should set a target time of 120 min.

**Figure 3.** Average cumulative returns (%) after positive and negative overnight gaps, which were identified by the BNS jump test, from January 1998–December 2015.
