Table 4 Descriptive statistics of...

Table 4

Descriptive statistics of cumulants of the S&P 500 returns

	Sample	Imp	Prac-Real	Conv-Real
Panel A. The mean and standard deviation of the cumulant estimates based on 30-day returns
2nd cumulant	0.24	0.40	0.32	0.32
( × 100)		(0.37)	(0.50)	(0.50)
3rd cumulant	−0.11	−0.34	−0.22	−0.01
( × 1,000)		(0.46)	(0.57)	(0.09)
4th cumulant	0.18	0.80	0.25	−1.00
( × 10,000)		(1.27)	(0.97)	(5.08)
Skewness	−0.90	−1.37	−1.11	0.03
Skewness		(0.50)	(0.72)	(0.17)
Kurtosis	3.04	5.94	4.33	−2.86
Kurtosis		(4.02)	(5.87)	(0.04)
Panel B. The mean and standard deviation of the cumulant estimates based on 90-day returns
2nd cumulant	0.75	1.18	0.89	0.89
( × 100)		(0.80)	(0.97)	(0.97)
3rd cumulant	−0.11	−1.42	−0.92	−0.01
( × 1,000)		(1.13)	(1.49)	(0.12)
4th cumulant	0.62	3.76	1.28	−5.05
( × 10,000)		(3.30)	(2.33)	(17.68)
Skewness	−0.16	−1.17	−1.11	0.00
Skewness		(0.38)	(0.49)	(0.07)
Kurtosis	1.11	3.35	2.93	−2.94
Kurtosis		(1.95)	(2.51)	(0.02)

	Sample	Imp	Prac-Real	Conv-Real
Panel A. The mean and standard deviation of the cumulant estimates based on 30-day returns
2nd cumulant	0.24	0.40	0.32	0.32
( × 100)		(0.37)	(0.50)	(0.50)
3rd cumulant	−0.11	−0.34	−0.22	−0.01
( × 1,000)		(0.46)	(0.57)	(0.09)
4th cumulant	0.18	0.80	0.25	−1.00
( × 10,000)		(1.27)	(0.97)	(5.08)
Skewness	−0.90	−1.37	−1.11	0.03
Skewness		(0.50)	(0.72)	(0.17)
Kurtosis	3.04	5.94	4.33	−2.86
Kurtosis		(4.02)	(5.87)	(0.04)
Panel B. The mean and standard deviation of the cumulant estimates based on 90-day returns
2nd cumulant	0.75	1.18	0.89	0.89
( × 100)		(0.80)	(0.97)	(0.97)
3rd cumulant	−0.11	−1.42	−0.92	−0.01
( × 1,000)		(1.13)	(1.49)	(0.12)
4th cumulant	0.62	3.76	1.28	−5.05
( × 10,000)		(3.30)	(2.33)	(17.68)
Skewness	−0.16	−1.17	−1.11	0.00
Skewness		(0.38)	(0.49)	(0.07)
Kurtosis	1.11	3.35	2.93	−2.94
Kurtosis		(1.95)	(2.51)	(0.02)

Note(s): This table represents descriptive statistics of cumulants of S&P 500 returns from January 1996 to August 2014. Skewness and Kurtosis are cumulants divided by the third and fourth powers of the standard deviation, respectively. In Panel A, Sample denotes the sample estimates (cumulant, skewness, and kurtosis) computed from the 30-day returns to maturity for each of the 223 options with expiry dates from February 1996 to August 2014. Imp, Prac-Real, and Conv-Real report the averages of these 223 estimates, with the corresponding standard deviations shown in parentheses. Specifically, Imp is computed at the beginning of each 30-day period, while Prac-Real and Conv-Real are obtained from the realized returns and option prices over each period. Panel B is constructed in the same way using 90-day returns to maturity for 73 quarterly options with expiry dates from June 1996 to June 2014

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