Gram-Charlier expands the locus of acceptable edgeworth expansions of skew kurtosis pairs. Is there a method that extends it further beyond the usual skewness(-.80, .80) and kurtosis(3.0, 5.5) ? It seems to me that in order to do so you would have to do some sort of transform of skewness and kurtosis, and then maybe apply these methods ... Any help would be appreciated.
What do you want to know about edgeworth or Gram Charlier series expansion ? What kind of problem you got with other skewness ( <-0.8 or >0.8) and other kurtosis (<3,>5.5) ? Models based on edgeworth series expansion work very well behind those levels . Take a look at what kind of smile you got if kurtosis<3 .
Huh? My problem is that if the skew or kurtosis falls outside the locus of points outside where the GC expansions falls, I get negative probabilities. I don't think I understand your question, or maybe you don't understand mine.
Huh ? I'm sorry, I didn't experience the same problem with Edgeworth series expansion for option pricing. The only thing I wanted to stress is that with a kurtosis<3, you get an inverse smile. You're certainly right, I don't get your question.
See Longstaff or Brown and Robinson for modified Corrado Su model to avoid negative probabilities and arbitrage. Or try this: http://eprints.lse.ac.uk/24938/1/dp419.pdf
Thanks for the suggestion. I think you may be missing a small part of what I am asking for. Gram-Charlier replaces Edgeworth because it expands on the values that skew and kurtosis can take in Edgeworth before giving negative probabilities. But eventually GC also gives negative probabilities, and in fact the expansion of GC is minor over Edgeworth. It is not a fault of a particular model that it gives negative probabilities, the "fault" lies in these expansions to handle skew and kurtosis too constrained (although I seem to remember seeing some adjustments to certain models where you can guarantee non-negative probabilites by calibration of other parameters). Hence, suggesting model X over model Y, using GC, AFAIK, will _not_ resolve negative probatilities when the skew or kurtosis falls outside their domain of sensible values for the given expansion. Hence the title of this thread. Also, AFAIK, Corrado-Su only work on European style options, although I think Jarrow Rudd has a trinomial tree implementation. What I need is an implied binomial or trinomial tree model with an expansion that works for much more expansive skewness and kurtosis values. Haug in his Option Pricing book shows how to do some of this, but AFAIK, he doesn't give code for a model that can handle both american options and skewness and kurtosis. This paper gives some insight into the problem with these expansions (look at exhibit 2): http://www.haas.berkeley.edu/groups/finance/WP/rpf275.pdf
I understand. I know what a gram charlier or an edgeworth series expansion is and I didn't mean that you'd better use one model instead of another. But take a look at what Brown and Robinson wrote on how to avoid such a problem ( I will try to post the pdf here). Longstaff wrote on problems with gram charlier and edgeworth expansions limits, and Brown wrote on arbitrage opportunities made by using it in option models. For american pricing you just need to use Haug's code with skewness and kurtosis if you want with the american option feature, the boundary condition : an america call value is max((S-K); same european node value)) everywhere on the tree ( S=spot K= strike). The same for american put pricing. I hope it helps.