Thread by @muskk

(Let’s do some math!) Thread on how to retrieve probability density function (PDF) of any underlying from its option prices. We will use this result later on in another thread I’ll post in the future to derive an equation for VIX.

Let's go...

Let’s first look at equation to price a call option at any time t, maturing at time T and with Strike K:

(refer equation1 pic below)

here F is forward, E[] is expectation & B(t,T) is discount factor. I’m excluding a few math details like measures & numeraires to keep it simple.

Let φ be the probability density function (PDF) of the underlying we are trying to recover. Let’s try to solve the expectation above (ignoring discount factor and other parameters C depends on to make equations look simpler)

(refer equation 2 pic below)

Taking first derivative with respect to K we get:
(refer equation 3 pic below)

Taking second derivative wrt K gives the PDF as:
(refer equation 4 pic below)

Let’s write the discretized version of the equation above:
(refer equation 5 pic below)

Guess the numerator on the RHS of equation 5? Yes, it’s a butterfly spread! The PDF at any strike can be obtained from a butterfly around that strike(with a very small spread, δ) divided by squared-strike (one should get similar result for PDF starting with a put instead of call)

That’s our result. Please feel free to correct me if I missed something in the derivation. Note that the above derivation is model independent i.e. no assumptions made on how the underlying behaves (unlike say black’s model). We just retrieve the distribution from option prices.

The next step is to plot the PDF vs strikes. I’ll try to cover that on another thread. That’s a little easier and more interesting part!

(Writing such math posts on twitter is a pain btw. Please let me know if there is a way to write equations rather than posting pics!)

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