0
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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!

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  • $\begingroup$ Check the following link for various measures of skewness and kurtosis itl.nist.gov/div898/handbook/eda/section3/eda35b.htm $\endgroup$ – Kevin Aug 2 '16 at 8:26
  • $\begingroup$ Do you know if a postive skewed curve can look negative? Using these methods I get positive skew. $\endgroup$ – Ptru Aug 2 '16 at 9:08

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