Implied Distribution from Skew

[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.

thanks in advance.

 

[Martinghoul]

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

 

[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)/d²

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

Masteratwork

 

[Bben1006]

And the absurdity is that someone did manage to even register a patent on these type of approximations of the RND —
Of course—this can’t be enforceable
see:http://www.google.com/patents?id=4G...rce=gbs_overview_r&cad=0#v=onepage&q=&f=false

Quote from Martinghoul:

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

 

[erol]

Quote from 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)/d²

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

Masteratwork

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?

 

[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.

 

[dennisb]

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

 

[Kevin Schmit]

Quote from MasterAtWork:

Implied distribution at K2 is simply ImpDist (K2)=exp(-rT)(C1-2C2+C3)/d²

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

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.

 

[MasterAtWork]

Quote from erol:


so Ci are the prices of the ith call correct? correct

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

If not, then ImpDist(K2) is essentially the PV of (C1-2C2+C3)/d^2 correct? 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
You won't get a price change, you 'd get approximate implied risk neutral probabilities

I think I may be asking the wrong question... If you want to get logarithmic price changes, you just need to translate implied annualized volatility into implied daily deviation ( almost current implied volty divided by sqrt(256)). That way you'd get an expected logarithmic price change

 

[henderson]

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