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Topic Title: Kurtosis and Skewness in Option Pricing
Created On Thu Jan 05, 06 10:05 PM
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NeedSomeHelp
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Thu Jan 05, 06 10:05 PM
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Hello:
I was taking a look at CSCO's most recent filing. In their assumptions for Employee Stock Options, they provide a skewness and kurtosis assumption. They use a binomial tree model to value the options.

Now I understand that skewness and kurtosis describe the shape of the curve, but how are these practically used in pricing options? It seems that in Black Scholes, you could use a distribution that be normal but adjusted for the skewness and kurtosis. Is there anyway to do this in Excel? In building the binomial tree, how would this be implemented?

Much appreciated!
 
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kanukatchit
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Fri Jan 06, 06 12:17 AM
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Originally posted by: NeedSomeHelp
Hello:
I was taking a look at CSCO's most recent filing. In their assumptions for Employee Stock Options, they provide a skewness and kurtosis assumption. They use a binomial tree model to value the options.

Now I understand that skewness and kurtosis describe the shape of the curve, but how are these practically used in pricing options? It seems that in Black Scholes, you could use a distribution that be normal but adjusted for the skewness and kurtosis. Is there anyway to do this in Excel? In building the binomial tree, how would this be implemented?

Much appreciated!



Ok this is a guess , but when a contstructing a binomial tree we usually determine u and d i.e the up and down stock moves using moment matching. This moment matching is done using the expected stock price and and the volatility or the variance of the stock. skewness and kurtosis are just the third and 4th moments of a rv.

The expected value of the stock is just given as p(S)u + (1-p)Sd = S exp(mu*dt)
and similarly for variance you can write an expression using var(X) = E(X^2) - E(X)^2 and the compute u and d from these two equations.
And then there are the Cox RR and equal probability approximations.

I guess similarly you can write the equations for the 3rd and 4th moment and solve for u and d, though it would be an overdetermined system. You could impose some more conditions to solve it.

somebody please correct me if I am wrong.

K.
 
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Athletico
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Fri Jan 06, 06 03:00 AM
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> similarly you can write the equations for the 3rd and 4th moment and solve for u and d

Unfortunately this doesn't work; the Central Limit Theorem gets in the way. The more time steps you add to the tree, the more the 3rd central moment -> 0 and 4th central moment -> 3 sigma^4, no matter what you use for u, d, p (assuming you keep them constant throughout the tree).
 
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kanukatchit
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Fri Jan 06, 06 06:42 AM
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Originally posted by: Athletico
> similarly you can write the equations for the 3rd and 4th moment and solve for u and d

Unfortunately this doesn't work; the Central Limit Theorem gets in the way. The more time steps you add to the tree, the more the 3rd central moment -> 0 and 4th central moment -> 3 sigma^4, no matter what you use for u, d, p (assuming you keep them constant throughout the tree).


I am not sure how you say in the limit the 3rd CM -> 0 and 4th CM -> 3sigma^4
I think this would be true in the case that we are dealing with a N(mu,sigma) using Stein's lemma for higher moments

But here we are matching moments for the stock price which is modeled as lognormal.

Am I missing something ?



Edited: Sat Nov 21, 09 at 07:05 PM by kanukatchit
 
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mutley
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Fri Jan 06, 06 08:06 AM
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You could get skewness in using displaced diffusion

where a >= 0 (a = 0 ==> Lognormal, a --> Inf ==> Normal)

But I'm not sure how you'd get kurtosis in - other whether there's data to justify kurtosis in employee stock options. But I am no expert on such things!

J
 
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spursfan
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Fri Jan 06, 06 10:37 AM
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look for mark rubinstein's paper on edgeworth binomial trees
 
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Surfer
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Fri Jan 06, 06 03:33 PM
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Edgeworth or Gram-Charlier Expansion will get u there.

-------------------------
"You'll See it, When You Believe it"
 
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halopsy
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Sun Jan 08, 06 10:02 PM
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get the paper "edgeworth binomial trees", mark rubinstein.

you cannot use kurtosis higher than 5.4.

how can i price an option with an underlying return with kurtosis > 6?

tnx
 
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