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I am having a bit of trouble understanding how to obtain the option implied distributions.

I have strike levels, deltas and implied vols for a call option that expires in 6 months. Roughly 40 data points for each of the three parameters.

I would like to interpolate this data to a cubic spline, obtain a p.d.f. and then obtain the 'standard deviation' of the density function. What does a p.d.f. graph contain? Implied vols on the y-axis and deltas on the x-axis? Is that something possible? I'm seeking to perform it on Python. Any ideas would be highly appreciated!

Thanks so much!

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The theorem you want to use is Breeden Litzenberg which says that the density $\phi_{T}$ of your underlying is given by $\phi_{T}(K) = \frac{1}{B(0,T)}\frac{\partial^{2} C}{\partial K^{2}}$ where C is your call price with maturity T and strike K ( you can obtain rthe price with BS formula as implied volatility is given )

From this theorem, deriving your implied density would involve a continum of calls but you have only some strikes which are quoted on the market. What you do in this case is that you fit the implied volatility through some model like the Gatheral SVI and then interpolate though the Andreasen Huge equation to get all the call prices for all the strikes at a specific maturity T.

If you dont want to perform this two steps , you can try to directly interpolate but that's not likely to give you a good result.

Having the density for all strikes at a fixed maturity you can calculate any moments you want.

( I dont see how the delta would be useful here tho :p )

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