Serving the Quantitative Finance Community

 
User avatar
gannarumma
Topic Author
Posts: 0
Joined: July 14th, 2002, 3:00 am

skewness/kurtosis time scaling

February 9th, 2010, 2:39 pm

I know this question has already been posted but I can't find it somehowE(R) f(T^1)Vol f(T^(1/2))Skewness f(T^?)Kurtosis f(T^?)Many thanks for your help.
 
User avatar
fomisha
Posts: 29
Joined: December 30th, 2003, 4:28 pm

skewness/kurtosis time scaling

February 9th, 2010, 5:40 pm

look at Gatheral's book/notes
 
User avatar
dpm25
Posts: 0
Joined: January 10th, 2008, 11:21 pm

skewness/kurtosis time scaling

February 10th, 2010, 6:11 am

i have pondered that question a lot but not come up with any satisfactory answers.If you assume a stoch vol model, you can derive the term structure of skewness and kurtosis as a function of the sv parameters (see Daas-Sandaram paper), but it is not a nice easy function.i think the problem comes as the familiar sqrt-time law for vol derives from assuming returns in each time sub-period are iid. we know in reality this is not quite true but it still works for vol. however the higher moments such as sk n kurt are more sensitive to this assumption.Carr-Wu wrote a paper where they advocate the iid scaling laws which just boil down to 1/sqrt(N) for skew and 1/N for kurt to scale to time period of length N. I think this is the assumption that drives the realized sk n kurt measures under VOLC in bbg...but these just look plain wrong compared to the derived implied series.I have read the Gatheral book but do not recall seeing an exact answer to this -- would love to know if there is
 
User avatar
silenoz
Posts: 0
Joined: November 22nd, 2004, 4:42 pm

skewness/kurtosis time scaling

February 10th, 2010, 9:16 pm

Quote...Carr-Wu wrote a paper where they advocate the iid scaling laws which just boil down to 1/sqrt(N) for skew and 1/N for kurt to scale to time period of length N. ...Could you please give a more detailed reference to the paper? Assuming for instance N(mu,sigma) distributed sub-period logreturns, period kurtosis will stay 3, no matter how many subperiods the period comprises ...
 
User avatar
dpm25
Posts: 0
Joined: January 10th, 2008, 11:21 pm

skewness/kurtosis time scaling

February 10th, 2010, 10:35 pm

sure, but I dont think that is quite what i am saying.just taking back to vol as an example:you can measure vol using daily returns over the last month or daily returns over the last year and assuming the process has constant vol there will be no systematic bias between the two results (yet there are a different number of subperiods in the overall period, as you say).However the question is how to compare a vol measured using daily observations with a vol mesured with say monthly or yearly observations. this is where the sqrt time law comes in for vol and where we are trying to find a law for sk n kurt.the full name of the paper I am referring to is " the Information Content of Straddles, Risk Reversals and Butterlfy Spreads" by Carr and Wu dated March 18 2005. However I do not know where it was published and a quick google search didnt turn it up. Maybe the author's homepage might have it.
 
User avatar
silenoz
Posts: 0
Joined: November 22nd, 2004, 4:42 pm

skewness/kurtosis time scaling

February 11th, 2010, 7:43 am

Quote...However the question is how to compare a vol measured using daily observations with a vol mesured with say monthly or yearly observations. this is where the sqrt time law comes in for vol and where we are trying to find a law for sk n kurt. ...I was referencing to that case, maybe not very clear. Assume monthly logreturns to be independent N(mu,sigma). Then you have yearly logreturns as the sum of 12 independent N(mu, sigma) random variables, so yearly logreturns will be distributed as N(12*mu, sqrt(12)*sigma). As they are still normal, Kurtosis of yearly logreturns will be 3, just as the monthly logreturns, so no scaling effect on kurtosis in that case. I could not find the paper on the author's homepage.
Last edited by silenoz on February 10th, 2010, 11:00 pm, edited 1 time in total.
 
User avatar
taneururer
Posts: 0
Joined: October 4th, 2005, 4:07 pm

skewness/kurtosis time scaling

February 11th, 2010, 1:10 pm

If it helps, I did try this will Monte-Carlo simulations of T-distributed returns (say, with 5 or 10 degrees of freedom).The square root rule held for the daily to weekly volatility.For kurtosis, the 1/N rule was pretty close.For five dof:Daily Kurtosis: 5.25Weekly Kurtosis: 1.116Of course, the results are not great given that with 5 dof the correct kurtosis for daily returns is 6 I believe.Anyone else want to confirm this or try a different distribution?
 
User avatar
dpm25
Posts: 0
Joined: January 10th, 2008, 11:21 pm

skewness/kurtosis time scaling

February 11th, 2010, 9:39 pm

thanks, that is interesting.presumably your daily simulated T returns were iid ....i think that is where the problem lies when moving to actual market returns.silenoz, i now get you i think, and agree for N dist sub-period returns the aggregate over any scaling period should have zero kurtosis. but this is a special case isn't it ? And doesn't mean that there is no scaling law generally....
 
User avatar
silenoz
Posts: 0
Joined: November 22nd, 2004, 4:42 pm

skewness/kurtosis time scaling

February 11th, 2010, 10:16 pm

No, it doesn't mean that there is no scaling law in general. However, I wouldn't see the normal dist as a special case, but the limit, as in my opinion the scaling is a consequence of the central limit theorem. As the sum converges to a normal random variable, excess kurtosis reaches zero.
 
User avatar
dpm25
Posts: 0
Joined: January 10th, 2008, 11:21 pm

skewness/kurtosis time scaling

February 11th, 2010, 10:52 pm

well if anything the 1/sqrt(N) and 1/(N) rules agree entirely with your argument dont they -- as saying that both kurt n sk go to zero in the limit of large N (ie the central limit theorem) ... the secondary question is how fast do they go to zero and are these laws correct.the further question is then the validity of the iid assumption...
 
User avatar
willsmith
Posts: 2
Joined: January 14th, 2008, 11:59 pm

skewness/kurtosis time scaling

February 11th, 2010, 11:29 pm

"An introduction to high frequency finance" by Olsen et al ( covers the scaling of volatility as you change timeframes, in great detail, section 5.5 and elsewhere.
 
User avatar
silenoz
Posts: 0
Joined: November 22nd, 2004, 4:42 pm

skewness/kurtosis time scaling

February 12th, 2010, 10:24 pm

I think if the third moment exists and rvs are iid, the rate of convergence is (at best) 1/sqrt(n) (Berry-Essen theorem).
Last edited by silenoz on February 11th, 2010, 11:00 pm, edited 1 time in total.