Cross-sectional volatility index as a proxy for the VIX in an Asian market

Figures & data

Table 1. Descriptive statistics on volatility return index measures sample period (2004–2014)

Measure	CBOE VIX	S&P 500 CSV	VXJ	NIKKEI-225 CSV
Mean	0.0023	0.0258	0.0021	0.0209
Median	$-0.0052$	$-0.0015$	$-0.0038$	$-0.0063$
Maximum	0.6421	1.6997	0.7826	4.3323
Minimum	$-0.2957$	$-0.5779$	$-0.3089$	$-0.5502$
Std. deviation	0.0705	0.2412	0.0670	0.2257
Skewness	0.6862	0.1750	1.5262	0.3392
Kurtosis	7.2681	3.7917	14.8883	5.7385
Jarque-Bera	5,584.2981	2,208.9521	58,242.1700	323,686.1000
Observation	2429	2429	2519	2519

Table 2. Descriptive statistics of implied volatility index and CSV index

Series	Mean	Standard deviation	Correlation	Daily returns (annualised)
CSV NIKKEI-225 index-levels	26.72	9.79	0.6293	2429
VXJ levels	25.28	10.91	0.6293	2429
CSV S&P 500 index-levels	26.25	13.01	0.7693	2519
VIX levels	19.97	9.99	0.7693	2519

Table 3. Correlation matrix analysis of volatility return index

Correlation	CBOE VIX	S&P 500 CSV	VXJ	NIKKEI-225 CSV
CBOE VIX	1	0.7662	0.5203	0.3607
S&P 500 CSV		1	0.53	0.4016
VXJ			1	0.6294
NIKKEI-225 CSV				1

Figure 1. The volatility of CBOE VIX and the cross-sectional volatility index for the sample period of 2004–2014.

Figure 2. The volatility between the Japan CBOE VIX and the NIKKEI-225 Cross-Sectional Volatility Index for the sample period of 2004–2014.

Figure 3. Two hundred fifty two days rolling window correlation of VIX and CSV S&P 500.

Figure 4. Two hundred fifty two days rolling window correlation of VXJ and CSV NIKKEI-225.

Table 4. The estimation of conditional volatility measures of GARCH/EGARCH

	GARCH(1,1)				EGARCH(1,1)
	CBOE VIX	CSV S&P 500	Japan VXJ	CSV NIKKEI-225	CBOE VIX	CSV S&P 500	Japan VXJ	CSV NIKKEI-225
Constant	0.0060*	0.0083*	0.0003*	0.0095*	$-0.3334$*	$-0.6771$*	$-0.4964$*	$-1.0700$*
$\mathit{\alpha }$	$-0.4710$	0.06501*	0.1465*	0.0920*	0.0625*	0.1330*	0.1258*	0.1829*
$\mathit{\beta }$	0.7293*	0.7432*	0.7870*	0.6298*	0.1784*	0.03060	0.2031*	0.03312
$\mathit{\gamma }$	–	–	–	–	0.9467*	0.8181*	0.9281*	0.7259*

*Statistically significant level of 0.05 percent.

Table 5. The volatility forecast error with different approach of GARCH model

	GARCH(1,1)				EGARCH(1,1)
	CBOE VIX	CSV S&P 500	Japan VXJ	CSV NIKKEI-225	CBOE VIX	CSV S&P 500	Japan VXJ	CSV NIKKEI-225
Root mean square error (RMSE)	0.0872	0.1693	0.0714	0.1600	0.1005	0.1689	0.0714	0.1602
Mean absolute error	0.0723	0.1410	0.0409	0.1322	0.0780	0.1406	0.0415	0.1323
Theil inequality coefficient	0.5958	0.6553	0.9178	0.6621	0.9676	0.6551	0.9296	0.6613
Bias proportion	0.0000	0.0425	0.0006	0.0038	0.0313	0.0374	0.0008	0.0056
Variance proportion	0.4563	0.4611	0.8522	0.3722	0.9586	0.4655	0.8822	0.3689
Covariance proportion	0.5437	0.4694	0.1472	0.6241	0.0101	0.4970	0.1170	0.6255

Figure 5. Volatility forecast of VIX and cross-sectional volatility index (CSV) using GARCH in US market.

Figure 6. Volatility forecast of VXJ and cross-sectional volatility index (CSV) using EGARCH in the US market.

Figure 7. Volatility forecast of VXJ and cross-sectional volatility index (CSV) using GARCH in the Japanese market.

Figure 8. Volatility forecast of VXJ and cross-sectional volatility index (CSV) using EGARCH in the Japanese market.