Local vol pricer for american options

Lapin · Post by **Lapin** » Fri Apr 03, 2009 4:00 pm

Local Vol could be a pain on the very short term with Vols exploding. Do you have some kind of a cap in order to avoid this nasty effect?

Roel · Post by **Roel** » Fri Apr 03, 2009 4:09 pm

Exploding volatility is no problem in the MC method. The local vol surface derived from implied volatilities in the SABR model, explodes for very high values of the asset price. These values are not attained in MC. In the tree an adjustment was made for this, by capping the asset dependency of the local vol surface.

deeds: what convergence curves?

deeds · Post by **deeds** » Fri Apr 03, 2009 5:43 pm

Because Monte Carlo is a statistical approach, as the number of runs increases, I usually look at the standard error of the mean in the estimates of any quantities being simulated, and, in particular of the value.

I often output the intermediate numbers and "confidence interval" to ensure convergence statistically. Of course it's quite a separate matter to make sure that it is converging to something correct, and this where MJ is likely coming from with his mention of a bounder.

Peter Jaeckel, in his book, "Monte Carlo Methods in Finance" (which takes a lot of stick on the fora, I should comment), calls this a convergence diagram.

I don't have any reason to think that you have a particular problem with convergence, but showing the intermediate values could also be helpful in showing any implementation problems, or unusual behavior requiring further diagnosis.

I will say that varying volatility should require more simulations than the flat case, and with LS, you've got a statistical estimation within an estimation going on.

deeds · Post by **deeds** » Mon Apr 06, 2009 11:21 am

It may not be the approach that MJ has/had in mind, but, in one sense,"upper bounder":

http://www.statslab.cam.ac.uk/~chris/papers/mcamer.pdf

Roel · Post by **Roel** » Wed Apr 08, 2009 12:56 pm

I finally seem to have solved my problem. I was working with a relatively steep implied volatility surface. Since I got my local volatility surface from making numerical derivation of this impvol surface, the way I defined my grid did matter. By making my grid finer (smaller intervals) my previous problem went away. It is still necessary to use a lot of sample paths and basis functions, but at least now it seems to work.

Thank you all for the help.

mj · Post by mj » Thu Apr 09, 2009 6:33 am

See

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1331904

for discussion of limitations of LS and how to get upper bounds.

It's for the LMM but the same techniques will apply but more simply for equities.

deeds · Post by **deeds** » Thu Apr 09, 2009 8:31 am

thanks, Mark!

quallenjaeger · Post by **quallenjaeger** » Mon Jun 03, 2019 4:06 pm

Hi,

How did you get an arbitrage free local vol surface from implied vol surface for american options?

Have you assumed that Dupire formula holds for american options? Would I introducing some arbitrage by doing so?