Jumps and stock market variance during the COVID-19 pandemic: Evidence from international stock markets

Abstract

Based on the work of Buncic and Gisler (2017), this paper investigates whether the roles of jump components will change in forecasting the volatility of international equity markets during the COVID-19 pandemic. Interestingly, in contrast to the conclusions of Buncic and Gisler (2017), we find jump components of the international equity indices are useful to predict the international stock markets’ volatility during the COVID-19 pandemic. Our study tries to provide new evidence of jump components in stock markets.

1. Introduction

Drastic shocks, also known as jumps, can often lead to sharp fluctuations in volatility and contain contents of huge movements of asset markets (Aït-Sahalia et al., 2015; Bandi and Renò, 2016). Commonly considered for volatility forecasting, researchers find that jumps contain predictive information for volatility forecasting (Eraker et al., 2003; Becker et al., 2009; Clements and Liao, 2017; Ma et al., 2019; Maneesoonthorn et al., 2020). However, Buncic and Gisler (2017) find jump components are not effective in forecasting the realized volatility for international equity markets. We are inspired to investigate the roles of jumps in predicting the realized volatility of the stock market.

It is noteworthy that the COVID-19 pandemic has brought a fierce strike to the whole world, and this pandemic continues to provide its destructive influence. Thus, based on the work of Buncic and Gisler (2017), we investigate whether the roles of jump components will change in forecasting the volatility of international equity markets during this pandemic. In other words, are jump components helpful for forecasting the realized volatility of international equity markets? This paper contributes to the existing literature by checking the predictability of jumps for international equity markets during the COVID-19 pandemic.

2. Methodology

2.1. Realized volatility, bipower variation and jumps

The log-price p_t is assumed to follow a continuous-time diffusion process driven by Brownian motion, which can be:

$$d{p}_t=u_t{d}_t+{\sigma }_{t}d{W}_t+k_{t}d{q}_t,$$ (1)

where u_t is a locally bounded drift term, σ_t is a volatility process that is bounded away from zero, W_t is standard Brownian motion, and q_t is a counting process with (possibly) time-varying intensity.

The quadratic variation (QV) of the log-price process is:

$$\mathrm{Q}{\mathrm{V}}_t\to \underset{0}{\overset{t}{\int }}{\sigma }^2\left(s\right)\text{ds}+\sum _{0<s\le t}{k}^2\left(s\right),$$ (2)

where ${\int }_0^t{\sigma }^2\left(s\right)\text{ds}$ is the integrated variance (IV) of the process and it is the continuous part of the quadratic variation, while ${\sum }_{0<s\le t}{k}^2\left(s\right)$ is the squared jump component between 0 and t and is the discontinuous part of the quadratic variation.

Following Andersen and Bollerslev (1998), the realized variance (RV) can be:

$$\mathrm{R}{\mathrm{V}}_t=\sum _{j=1}^M{r}_{t,i}^2,$$ (3)

where r _t,i represents the ith intraday return of day t, and M is the interval. When the intraday sampling frequency increases,

$$\text{pli}{\mathrm{m}}_{M\to \infty }\mathrm{R}{\mathrm{V}}_t=\mathrm{Q}{\mathrm{V}}_t,$$ (4)

According to Barndorff-Nielsen and Shephard (2004), the bi-power variance (BPV) is:

$$\text{BP}{\mathrm{V}}_t=\frac{\pi }{2}\left[\frac{M}{M-2}\right]\sum _{i=2}^M\left|r_{t,i}\right|\left|r_{t,i-1}\right|,$$ (5)

where $\left[\frac{M}{M-2}\right]$ is a finite sample bias correction term.

Thus,

$$\text{pli}{\mathrm{m}}_{M\to \infty }\left(\mathrm{R}{\mathrm{V}}_t-\text{BP}{\mathrm{V}}_t\right)=\sum _{0<s\le t}{k}^2\left(s\right),$$ (6)

To make the jumps to be non-negative, following Barndorff-Nielsen and Shephard (2004) and Andersen et al. (2007), the jump component is:

$$J_t=\text{max}\left\{\mathrm{R}{\mathrm{V}}_t-\text{BP}{\mathrm{V}}_t,0\right\},$$ (7)

The continuous component then can be:

$$C_t=\mathrm{R}{\mathrm{V}}_t-J_t=\mathrm{R}{\mathrm{V}}_t-\text{max}\left\{\mathrm{R}{\mathrm{V}}_t-\text{BP}{\mathrm{V}}_t,0\right\},$$ (8)

Moreover, following Barndorff-Nielsen and Shephard (2006), the “remarkable” jumps can be:

$$Z_t={\Delta }^{-1/2}\frac{\left(\mathrm{R}{\mathrm{V}}_t-\text{BP}{\mathrm{V}}_t\right){\text{RV}}_t^{-1}}{\sqrt{\left(\frac{{\pi }^2}{4}+\pi -5\right)\text{max}\left(1,\frac{\mathrm{T}{\mathrm{Q}}_t}{{\left(\text{BP}{\mathrm{V}}_t\right)}^2}\right)}},$$ (9)

where TQ_t is the realized tri-power quarticity.

Additionally, following Andersen et al. (2007), we set α as the significance level and the critical value, which is denoted as ∅_α. The jump component can be denoted as follows:

$$\mathrm{C}{\mathrm{J}}_t=I\left(Z_t>{\Phi }_{t,\alpha }\right)·\text{max}(\mathrm{R}{\mathrm{V}}_t-\text{BP}{\mathrm{V}}_t,0),$$ (10)

where I( • ) is an indication function.

2.2. HAR-RV-type models

Model 1: HAR-RV

$$\text{logR}{\mathrm{V}}_{t+1}={\beta }_0+{\beta }_1\text{logR}{\mathrm{V}}_t+{\beta }_2\text{logRV}{\mathrm{W}}_t+{\beta }_3\text{logRV}{\mathrm{M}}_t+{\epsilon }_{t+1},$$ (11)

where RV_t, RVW_t and RVM_t represent daily, weekly and monthly RV, respectively. Moreover, $\text{RV}{\mathrm{W}}_t=\frac{1}{5}{\sum }_{i=1}^5\mathrm{R}{\mathrm{V}}_i$, $\text{RV}{\mathrm{M}}_t=\frac{1}{22}{\sum }_{i=1}^{22}\mathrm{R}{\mathrm{V}}_i$, and ε_t + 1 represents the disturbance term.

Based on Buncic and Gisler (2017), jump components are not useful to predict the stock market volatility. In this paper, we try to check whether the contents of jumps are helpful for predicting stock market volatility during the COVID-19 pandemic based on the HAR-CRV-CJ model.

Model 2: HAR-CRV-CJ

$$\text{logR}{\mathrm{V}}_{t+1}={\beta }_0+{\beta }_1\text{logCR}{\mathrm{V}}_t+{\beta }_2\text{logCRV}{\mathrm{W}}_t+{\beta }_3\text{logCRV}{\mathrm{M}}_t+{\beta }_4\text{logC}{\mathrm{J}}_t+{\beta }_5\text{logCJ}{\mathrm{W}}_t+{\beta }_6\text{logCJ}{\mathrm{M}}_t+{\epsilon }_{t+1},$$ (12)

where CRV_t, CRVW_t and CRVM_t represent the continuous parts of daily, weekly and monthly RV, respectively and CJ_t, CJW_t, and CJM_t each separately reflect averages of the daily, weekly and monthly jump components, respectively. Moreover,$\text{CJ}{\mathrm{W}}_t=\frac{1}{5}{\sum }_{i=1}^5\mathrm{C}{\mathrm{J}}_t$, $\text{CJ}{\mathrm{M}}_t=\frac{1}{22}{\sum }_{i=1}^{22}\mathrm{C}{\mathrm{J}}_t$and ε_t + 1 presents the disturbance term. In this paper, we consider 18 international stock indices.

3. Data

Following the Buncic and Gisler (2017), we apply daily realized measures data from the Oxford-Man Institute's Quantitative Finance Realized Library. The 18 international stock indices are the S&P 500 (SPX, United States), the FTSE 100 (FTSE United Kingdom), the Nikkei 225 (N225, Japan), the DAX 30 (GDAXI, Germany), the All Ordinaries (AORD, Australia), the CAC 40 (FCHI, France), the Hang Seng (HIS, Hong Kong), the KOSPI (KS11, South Korea), the AEX (AEX, The Netherlands), the Swiss Market Index (SSMI, Switzerland), the IBEX 35 (IBEX, Spain), the S&P CNX Nifty (NSEI, India), the IPC Mexico (MXX, Mexico), the Bovespa (BVSP, Brazil), the S&P TSX (GSPTSE, Canada), the Euro STOXX 50 (STOXX50E, Euro area), the FT Straits Times (STI, Singapore), and the FTSE MIB (FTMIB, Italy). The sample period ranges from January 1, 2000 to December 30, 2021 except for the S&P TSX and the FTSE MIB indices, which start from May 2, 2002 and June 1, 2009, respectively. Table 1 shows the descriptive statistics. From Table 1, we find that all the international stock indices’ RVs have right skew and high kurtosis. The Jarque-Bera (JB) test shows that no Gaussian distributions exist in all the RV at the 1% significance level. The Ljung-Box test shows that all the RVs have serial auto-correlations up to the 20th order at the 1% significance level. The Augmented Dickey-Fuller (ADF) test shows that all the RVs have no unit root at the 1% significance level, further showing that the data series are stationary.

Table 1.

Descriptive statistics.

Equity index RV	Country	Observations	Mean	Std.dev	Skewness	Kurtosis	Jarque-Bera	Q(20)	ADF
SPX	United States	5515	0.0001	0.0003	10.9804	201.4261	9,415,270.6964***	25,810.5580***	−31.5832***
FTSE	United Kingdom	5547	0.0001	0.0003	15.8377	413.769	39,722,802.2189***	12,316.8599***	−43.4482***
N225	Japan	5347	0.0001	0.0002	9.1627	127.6124	3,695,384.4501***	16,632.6800***	−34.3812***
GDAXI	Germany	5574	0.0002	0.0003	7.7867	100.0298	2,375,539.0198***	26,745.9680***	−30.5722***
AORD	Australia	5556	0.0001	0.0001	18.0814	491.5454	56,125,920.1610***	16,746.1869***	−36.3125***
FCHI	France	5612	0.0001	0.0002	9.1087	129.3657	3,983,130.2228***	21,998.1593***	−32.1119***
HSI	Hong Kong	5389	0.0001	0.0002	10.7017	187.3056	7,964,309.2765***	21,330.0847***	−33.9895***
KS11	South Korea	5413	0.0001	0.0002	9.3285	159.079	5,774,367.8597***	26,917.8647***	−29.4234***
AEX	The Netherlands	5610	0.0001	0.0002	7.8819	94.3215	2,133,490.8420***	26,138.4720***	−28.9730***
SSMI	Switzerland	5512	0.0001	0.0002	12.2359	220.9085	11,322,846.6330***	18,506.3164***	−33.7436***
IBEX	Spain	5575	0.0001	0.0002	9.3113	147.7301	5,140,005.2531***	14,536.9034***	−34.6935***
NSEI	India	5443	0.0001	0.0004	25.549	885.0314	177,873,998.8841***	3466.9611***	−48.0011***
MXX	Mexico	5516	0.0001	0.0002	13.2484	290.0526	19,458,562.2393***	8093.5236***	−48.9158***
BVSP	Brazil	5409	0.0002	0.0003	8.878	111.734	2,878,921.1460***	27,475.9789***	−29.1306***
GSPTSE	Canada	4916	0.0001	0.0005	47.7843	2838.882	1,648,980,318.8636***	3319.0682***	−49.4592***
TOXX50E	Euro Area	5595	0.0002	0.0003	12.3419	271.7286	17,321,101.7723***	16,649.2033***	−37.4944***
STI	Singapore	3571	0.0001	0.0001	11.645	231.5952	8,036,586.2429***	5836.4154***	−36.8374***
FTMIB	Italy	3190	0.0001	0.0002	6.6495	66.6972	612,690.0159***	10,309.7749***	−24.2406***

Note: Descriptive statistics of RVs of equity indices are exhibited. In line with Jarque and Bera (1987), we set the null hypothesis of a normal distribution for each variable. Ljung and Box (1978) propose the Ljung-Box statistic called Q(n); in our study, the 20th order serial correlation is tested. The Augmented Dickey-Fuller test is used to test whether the time series is stationary. Asterisks ^{***, **}and * denote rejections of null hypothesis at 1%, 5% and 10% levels.

4. Empirical results

4.1. Out-of-sample analysis

In line with Paye (2012) and Liang et al. (2020), the out-of-sample R ²($R_{\text{OOS}}^2$) method is efficient in capturing the distinction among the predictability models. The out-of-sample R ² statistic is:

$$R_{\text{OOS}}^2=1-\frac{{\sum }_{t=1}^M{(\mathrm{R}{\mathrm{V}}_t-{\text{RV}}_t^j)}^2}{{\sum }_{t=1}^M{(\mathrm{R}{\mathrm{V}}_t-{\text{RV}}_t^0)}^2},j=\text{Model}\left(2\right),$$ (13)

where RV_t is the actual realized volatility, ${\text{RV}}_t^j$ is the prediction from model j, where j ∈ Model (2), and ${\text{RV}}_t^0$ is the volatility forecasting from the benchmark model. A positive $R_{\text{OOS}}^2$ of a model shows that this model is superior to the benchmark model. Following Clark and West (2007), the MSPE metric is applied to check the difference among the models for oil futures market volatility.

Table 2 shows the results of the out-of-sample R ² test for 18 equity markets during the COVID-19 pandemic. For all the international equity indices, the first out-of-sample observation starts on March 11, 2020.1 We can observe some remarkable findings from Table 2. During the COVID-19 pandemic, the values of $R_{\text{oos}}^2$ are significantly positive for 11 of 18 equity markets, including FTSE, N225, GDAXI, AORD, FCHI, KS11, IBEX, MXX, BVSP, GSPTSE, and STOXX50E, implying that jump components of the international equity indices are useful to improve forecasting performance for most of the observed indices during the COVID-19 pandemic. However, Buncic and Gisler (2017) find that jumps are not useful for predicting the volatility of international equity indices.

Table 2.

Results of the out-of-sample R² test.

Forecasting models	Tos	$R_{\text{OOS}}^2$(%)	MSPE-Adj.	pvalue
HAR-RV-JSPX	451	0.6312	1.0163	0.1547
HAR-RV-JFTSE	456	16.6876**	2.1613	0.0153
HAR-RV-JN225	437	1.3794*	1.4031	0.0803
HAR-RV-JGDAXI	455	6.1198**	1.2846	0.0995
HAR-RV-JAORD	459	1.7383***	2.3304	0.0099
HAR-RV-JFCHI	465	2.1905*	1.5148	0.0649
HAR-RV-JHSI	443	−0.6181	−0.4218	0.6634
HAR-RV-JKS11	447	0.9491**	1.8856	0.0297
HAR-RV-JAEX	464	−14.5774	−0.3758	0.6465
HAR-RV-JSSMI	456	−102.4253	1.5731	0.0578
HAR-RV-JIBEX	463	4.4091**	1.6832	0.0462
HAR-RV-JNSEI	442	−1.9548	−0.6991	0.7577
HAR-RV-JMXX	452	4.6145***	2.9519	0.0016
HAR-RV-JBVSP	436	4.9235**	1.8756	0.0304
HAR-RV-JGSPTSE	449	5.6046***	2.3765	0.0087
HAR-RV-JSTOXX50E	449	4.8872**	1.783	0.0373
HAR-RV-JSTI	451	−0.9095	0.4801	0.3156
HAR-RV-JFTMIB	456	−0.4061	0.1548	0.4385

Notes: Columns display forecasting models, the effective number of out-of-sample observations T_os, the $R_{\text{oos}}^2$(%), MSPE-adjusted statistic, p-value, respectively. If the $R_{\text{oos}}^2$ (%) is larger than zero, implying that forecasting model outperform the benchmark model. Asterisk ⁎⁎⁎, ⁎⁎ and ⁎ denote rejections of null hypothesis at 1%, 5% and 10% level.

Why do the roles of jumps change during the COVID-19 pandemic? A possible reason can be that jumps are able to predict stock market volatility based on the channel of investor sentiment. Jumps often contain valuable information that is connected to extreme conditions (Ma et al., 2019). More specifically, people are more sensitive to sharp fluctuations (jumps) in the stock market during the COVID-19 pandemic or the crisis because of the increase of investor fear increase (Smales and Kininmonth, 2016; Ergun and Durukan, 2017; Goldstein et al., 2017; Chang et al., 2020; Ftiti et al., 2021). In addition, existing studies find that models tend to have better performance during recessions (Rapach et al., 2010; Neely et al., 2014).

4.2. Robustness check

To ensure that our results are robust, we consider a different out-of-sample period in this subsection. This period ranges from 23 January 2020 to the end of the sample period, which begins with the lockdown of the city of Wuhan in China because of the outbreak of COVID-19. This may affect China's stock market volatility and have linkage effects on international equity markets. The empirical results are shown in Table 3 . We find that out-of-sample R ² are significantly positive for 10 of 18 equity markets in this period. The results are consistent with the previous conclusion except for the AORD index. These results are consistent with the out-of-sample results.

Table 3.

Results of the out-of-sample R² test based on city sealing of Wuhan.

Forecasting models	Tos	$R_{\text{OOS}}^2$(%)	MSPE-Adj.	p-value
HAR-RV-JSPX	484	1.8465**	1.2446	0.1066
HAR-RV-JFTSE	490	5.9328**	2.2895	0.011
HAR-RV-JN225	469	2.3115*	1.9275	0.027
HAR-RV-JGDAXI	489	6.2267*	1.4449	0.0742
HAR-RV-JAORD	492	−1.9765	−0.359	0.6402
HAR-RV-JFCHI	499	1.8996**	1.453	0.0731
HAR-RV-JHSI	475	−0.5310	−0.3124	0.6226
HAR-RV-JKS11	479	1.0285**	2.0025	0.0226
HAR-RV-JAEX	498	−14.6608	−0.3932	0.6529
HAR-RV-JSSMI	490	−40.7366	1.6969	0.0449
HAR-RV-JIBEX	497	4.6273**	1.8138	0.0349
HAR-RV-JNSEI	475	−2.0407	−0.63	0.7356
HAR-RV-JMXX	485	4.2967**	3.0599	0.0011
HAR-RV-JBVSP	468	4.5394**	1.8931	0.0292
HAR-RV-JGSPTSE	482	6.0653***	2.6071	0.0046
HAR-RV-JSTOXX50E	482	3.0928*	1.5087	0.0657
HAR-RV-JSTI	484	−0.4518	0.5265	0.2993
HAR-RV-JFTMIB	489	−0.2879	0.1813	0.4281

Notes: Columns display forecasting models, the effective number of out-of-sample observations T_os, the $R_{\text{oos}}^2$(%), MSPE-adjusted statistic, p-value, respectively. If the $R_{\text{oos}}^2$ (%) is larger than zero, implying that forecasting model outperform the benchmark model. Asterisk ⁎⁎⁎, ⁎⁎ and ⁎ denote rejections of null hypothesis at 1%, 5% and 10% level.

5. Conclusion

Extending the work of Buncic and Gisler (2017), this paper checks the roles of jump components for predicting the volatility of international equity markets during the COVID-19 pandemic. We find that jump components of the international equity indices are useful to predict the international stock markets’ volatility during the COVID-19 pandemic, which is inconsistent with the results of Buncic and Gisler (2017). Our study emphasizes the importance of jump components in stock market volatility during the COVID-19 pandemic. As the COVID-19 pandemic continuously affects the world economy, understanding the information of jumps is essential for market participants and policy makers.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.