Forum Navigation: Select WHO'S ON: 02:54 PM Wilmott / BookMap Competition General Forum Technical Forum Economics Forum Numerical Methods Forum Trading Forum The Quantitative Finance FAQs Proje... Student Forum Book And Research Paper Forum Programming and Software Forum The Quantitative Finance Code Libra... Careers Forum Events Board Brainteaser Forum Off Topic Forum and Website Bugs and Suggesti...

 new topic
 search
 jobs board
 magazine
 news
 help
 file share
 audio visual
 articles
 blogs
 wiki
 forums
 home

 FORUMS > Student Forum < refresh >
 Topic Title: Kurtosis and Skewness in Option Pricing Created On Thu Jan 05, 06 10:05 PM Topic View: Branch View Threaded (All Messages) Threaded (Single Messages) Linear

NeedSomeHelp
Junior Member

Posts: 6
Joined: Dec 2004

Thu Jan 05, 06 10:05 PM

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!

kanukatchit
Member

Posts: 143
Joined: Dec 2003

Fri Jan 06, 06 12:17 AM

Quote

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.

Athletico
Senior Member

Posts: 867
Joined: Jan 2002

Fri Jan 06, 06 03:00 AM

> 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).

kanukatchit
Member

Posts: 143
Joined: Dec 2003

Fri Jan 06, 06 06:42 AM

Quote

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

mutley
Senior Member

Posts: 662
Joined: Feb 2005

Fri Jan 06, 06 08:06 AM

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

spursfan
Senior Member

Posts: 842
Joined: Oct 2001

Fri Jan 06, 06 10:37 AM

look for mark rubinstein's paper on edgeworth binomial trees

Surfer
Member

Posts: 154
Joined: Apr 2002

Fri Jan 06, 06 03:33 PM

Edgeworth or Gram-Charlier Expansion will get u there.

-------------------------
"You'll See it, When You Believe it"

halopsy
Member

Posts: 44
Joined: Apr 2005

Sun Jan 08, 06 10:02 PM

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