# How to estimate option implied skewness and kurtosis in R

Suppose that i have data that for each day i have more than one option, either put or call. I.E. I have more than 20 put options and 20 call options for each specific day.

What is the way to estimate the option implied skewness and kurtosis for that specific dataset? Is there a package in R, or some code to help me understand how to do it? For each option i have information like the delta, strike price, volume, spot price.

Thank you!

EDIT.1: I am not sure in terms of data; typically more than 1 option is present on a certain date. I.e. on the 03-02-2010 we can have 30 options on expiration date A, 50 on expiration date B and so forth, which each having their respective values introduced prior. When finding the kurtosis/skewness the procedure includes at the same time all options; taking into account the different expiration dates (seems more logical), or not?

EDIT.2: One good way to measure those metrics, is through the Corrado and Su (1996) model, or, even better, with the corrected Brown and Robinson (2002) equations of the Corrado and Su model. Link is in the comment below.

• What's the formula for the option implied distribution? – will May 13 '17 at 11:07
• I think this will help you understand it better than me explaining it: quant.stackexchange.com/questions/1621/… – Hercules Apergis May 13 '17 at 12:21
• You didn't understand the point of my question. If you know the formula to go from option prices to implied vol, then you can go the other way. R has a lot to offer in terms of flexible distributions, ie try a johnson distribution that allows you to set the first 4 moments. Then imply option prices from that, and then wrap it all into a function that returns the residues. Then chuck that into one of the general minimisers (i think there is a levmar wrapper). – will May 13 '17 at 12:24
• What is wrong with Brian B's answer in the link you provided? He tells you how to get a pdf for the implied distribution (conditional on a fit of your data) - so you need to calculate $E(X^3)$ and $E(X^4)$ with that pdf which you can do easily numerically. – FinanceGuyThatCantCode May 13 '17 at 17:39
• @HerculesApergis fyi, the distributions given in the paper are not theirs, i didn't read the whole paper to see if they site a source, but the particular example they give in the first few equations are known as the Gram Charlier A series. – will May 15 '17 at 11:32