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# Implied Distribution from Skew

Discussion in 'Options' started by erol, Mar 28, 2010.

1. ### erol

Hello,

I'm wondering if there's a practical way to find out the implied distribution of daily returns from a given option skew for a specific month.

I'm trying to figure out the skew and kurtosis the implied Vols for a given month are... well implying.

2. ### Martinghoul

You can get the risk-neutral pdf by differentiating the smoothed Price(strike) function twice, if memory serves...

3. ### MasterAtWork

True .

As a practical way to do it, you can estimate implied distribution with at least 3 calls with closed strikes.
Assume 3 calls C1 C2 C3 with strikes K1=K2-d and K3=K2+d, with interest rate r, and T the maturity.

C1(K2-d)
C2(K2)
C3(K2+d)

Implied distribution at K2 is simply ImpDist (K2)=exp(-rT)(C1-2C2+C3)/dÂ²

One can do the same for each call. That way you 'd have an Implied distribution including the skew.

Masteratwork

5. ### erol

thank you all!

so Ci are the prices of the ith call correct?

So, the formula (C1-2C2+C3)/d^2 is not part of the exponent, right?

If not, then ImpDist(K2) is essentially the PV of (C1-2C2+C3)/d^2 correct?

I really appreciate your help, and pardon my ignorance... but I'm not seeing how I go from this formula to get a set of logarithmic price changes, since my only variables are r and T

I think I may be asking the wrong question...

My understanding of skew was to correct for statistical-skew and kurtosis, to fatten up the tails of the normal distribution to account for observed statistical-skew and kurtosis.

Is there a way to figure out what statistical-skew and kurtosis the market is implying from the option IV skew?

6. ### Martinghoul

I prefer to go all the way and do the dirty differentiation... I have a Matlab snippet that does the trick, if you like.

7. ### dennisb

In what sense differs the implied distribution from the implied deviation wich is the inverse log of IV/&radic;days?

8. ### Kevin Schmit

I could swear that we used to call this technique "butterfly quadrature," but to my surprise I can't find a single Google hit on the term.

10. ### henderson

I would be interested in seeing your matlab code for the dirty implementation

#10     Apr 1, 2010
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