I have the data set
Ticker Bid Ask Last Volm \
Strike
1775 SPXW 8/31/16 C1775 323.299805 326.100098 0.000000 0
1800 SPXW 8/31/16 C1800 299.000000 301.799805 260.500000 0
1825 SPXW 8/31/16 C1825 274.899902 277.599854 0.000000 0
1850 SPXW 8/31/16 C1850 251.000000 253.600006 244.250000 0
1875 SPXW 8/31/16 C1875 227.300003 229.800003 220.750000 0
1900 SPXW 8/31/16 C1900 203.800003 206.300003 178.000000 0
1925 SPXW 8/31/16 C1925 180.699997 183.100006 109.849998 0
1950 SPXW 8/31/16 C1950 158.100006 160.399994 152.300003 0
1975 SPXW 8/31/16 C1975 136.100006 138.199997 130.600006 0
2000 SPXW 8/31/16 C2000 114.800003 116.800003 114.900002 0
2025 SPXW 8/31/16 C2025 94.500000 96.300003 69.000000 0
2050 SPXW 8/31/16 C2050 75.199997 76.800003 69.250000 0
2075 SPXW 8/31/16 C2075 57.099991 58.600006 54.799999 0
2100 SPXW 8/31/16 C2100 41.000000 42.299988 38.099998 0
2125 SPXW 8/31/16 C2125 26.899994 28.100006 25.049999 0
2150 SPXW 8/31/16 C2150 15.800000 16.800003 15.600000 0
2175 SPXW 8/31/16 C2175 7.800000 8.600000 7.830000 0
2200 SPXW 8/31/16 C2200 3.100000 3.700000 3.500000 0
2225 SPXW 8/31/16 C2225 1.200000 1.650000 1.400000 0
2250 SPXW 8/31/16 C2250 0.550000 0.950000 0.750000 0
2275 SPXW 8/31/16 C2275 0.350000 0.750000 0.350000 0
2300 SPXW 8/31/16 C2300 0.250000 0.650000 0.350000 0
2350 SPXW 8/31/16 C2350 0.150000 0.500000 0.200000 0
2400 SPXW 8/31/16 C2400 0.050000 0.400000 0.150000 0
2500 SPXW 8/31/16 C2500 0.000000 0.300000 0.050000 0
Avg. Price Implied Vol IV(Bid) IV(Ask)
Strike
1775 324.699951 0.2521 0.2318 0.2687
1800 300.399902 0.2419 0.2243 0.2567
1825 276.249878 0.2315 0.2175 0.2446
1850 252.300003 0.2225 0.2106 0.2333
1875 228.550003 0.2134 0.2031 0.2221
1900 205.050003 0.2038 0.1949 0.2115
1925 181.900002 0.1941 0.1870 0.2009
1950 159.250000 0.1849 0.1792 0.1909
1975 137.150002 0.1764 0.1714 0.1808
2000 115.800003 0.1673 0.1633 0.1712
2025 95.400002 0.1585 0.1553 0.1617
2050 76.000000 0.1492 0.1466 0.1518
2075 57.849998 0.1390 0.1369 0.1415
2100 41.649994 0.1294 0.1276 0.1315
2125 27.500000 0.1191 0.1173 0.1210
2150 16.300002 0.1093 0.1075 0.1110
2175 8.200000 0.0995 0.0976 0.1012
2200 3.400000 0.0908 0.0887 0.0929
2225 1.425000 0.0872 0.0846 0.0896
2250 0.750000 0.0904 0.0862 0.0923
2275 0.550000 0.0946 0.0875 0.1021
2300 0.450000 0.1019 0.0919 0.1084
2350 0.325000 0.1171 0.1106 0.1261
2400 0.225000 0.1285 0.1155 0.1414
2500 0.150000 0.1624 NaN 0.1714
I've calculated the pdf and cdf using the formulas:
$F(x) = e^{rT}(\frac{P_{n+1} - P_{n-1}}{X_{n+1}-X_{n-1}})$
$f(x) = e^{rT}\frac{P_{n+1} - 2 P_{n} + P_{n-1}}{(\delta X)^2}$
where $r$ is the risk-free rate, $P$ is the price (Bid or Ask) and $X$ is the Strike.
Using this i get a curve with slight negative skew and high kurtosis or at least it seems to be so by visual inspection. Does anyone know how to calculate these values for the mean, median, standard deviation, skewness and kurtosis for such a curve?
Thanks!