# Option implied distributions

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!

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 )