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Baltazar


Total Posts: 1771
Joined: Jul 2004
 
Posted: 2009-04-03 10:45
I am working with a student on a local vol pricer for american option.

As an input it takes a given local vol and should give a price.
We did a binomial tree and a MC to check the tree (Using LS for american feature).
The goal is to get the tree working.
The thing is the tree and the MC do not agree all the time.
For flat vol they do agree on american and euro prices
For non flat vol, they agree on european prices.
For non flat american they sometimes agree so it seem to be a problem in the american feature with non flat vol.

It seem the MC is wrong (they might both be wrong off course) as it give sometimes prices for an american call with non dividend below the european call price.

I would really appreciate if someone could check some results with me.
Our set up so far is that we use a SABR to generate an implied volatility surface that is arbitrage free, from this we get a local vol surface and plug it in the pricer.
Of course we cannot check our prices against a sabr pricer for american options because even if we start from the same implied vol surface, dynamics are different hence american option prices.

I could also dig more in my MC and see if the LS we use is screwed up because of the non flat vol but since the weapon of choice is tree and the MC is just there to check, i'd rather not spend too much time on it.



Qui fait le malin tombe dans le ravin

Lapin


Total Posts: 277
Joined: Feb 2006
 
Posted: 2009-04-03 11:33

Baltazar,

"For non flat american they sometimes agree so it seem to be a problem in the american feature with non flat vol."

What do you mean by sometimes? If they don't agree all the time with flat vol there might be a problem already there.

 


mib


Total Posts: 354
Joined: Aug 2004
 
Posted: 2009-04-03 12:15
optimal exercise is quite sensitive to non-constant vol assumptions. are you somehow adjusting LS?

local vol exacerbates the problem because it makes volatility changes with time and spot much stronger and predictable than they are in real life or stochastic vol worlds and thus I would not be surprised to see optimal exercise boundary to shift further under local vol than under the generating SABR process.

Head of Mortality Management, Capital Structure Demolition LLC

mj


Total Posts: 1049
Joined: Jun 2004
 
Posted: 2009-04-03 13:15
LS is not very reliable. Are you running an upper bounder to check how good it is?

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Baltazar


Total Posts: 1771
Joined: Jul 2004
 
Posted: 2009-04-03 14:19
Lapin: they fully agree for flat vol.
For non flat they don't sometimes. The MC sometimes gives american call cheaper then the european one.


Mib&MJ
I'll check with the student exactly which LS he uses and if he does something special

Thanks for the feedback

Qui fait le malin tombe dans le ravin

deeds


Total Posts: 436
Joined: Dec 2008
 
Posted: 2009-04-03 14:35

Baltazar, please forgive me for the threadjack...

MJ, is there a good published critique of LS?  I can see how there is substantial lattitude in implementation, but I was under the impression that, even with its limitations, so far, it is the best monte carlo approach for options with early exercise opportunities.

if the observations about the reliability of LS are your own, if you'll signal that you don't mind summarising, I'd be happy to start another thread and ask you about them...

thanks for any help...

d


Lapin


Total Posts: 277
Joined: Feb 2006
 
Posted: 2009-04-03 14:48

B,

Do you have discrete or continuous divs?

In case of discrete, do you have a simulation point on the date of payment?


Baltazar


Total Posts: 1771
Joined: Jul 2004
 
Posted: 2009-04-03 15:04
Lapin:

No dividend yet, AFAIK non flat vol should not make a call early exercise with out dividend, therefore the pricer should give the same prices for euro or american calls.
For the moment the tree does this but not the MC.

Deeds: it is not a threadjack as my problem can very well be due to a short coming of the LS (at least the way it's coded in my MC)

Qui fait le malin tombe dans le ravin

Roel


Total Posts: 3
Joined: Apr 2009
 
Posted: 2009-04-03 15:08
Dear all,

I am the student in question, so maybe I can clarify things a bit.

My standard method for pricing with a given local volatility surface is the trinomial tree. It gives accurate results for European prices. But for American Puts I don't have values to compare it with. The optimal exercise boundary from the tree is smooth and has the typical shape. Furthermore the American Call price is equal to the European price (which is equal to the Black Scholes price), suggesting it works OK.

To compare the American Puts I used LS (which calculates prices for American Calls at the same time to check), with weighted Laguerre polynomials as basis functions for the regression. It is the typical model as described in the LS paper, no adjustments (wouldn't know what sensible adjustments would be). For each regression N sample paths are used (1/2N normal 1/2N antithetic). The procedure is repeated M times, giving N*M sample paths in total. For flat vol it agrees with the tree.

The weird thing is that keeping N*M constant, the result changes when N and M are changed. Eg for N=1000, M=1500 it results in values that are significantly higher than those given by the tree, while N=150000, M=10 results in values lower than the tree (the American call is now cheaper than the BS price, which is nonsensical). I would imagine the choice of N, M matters somewhat because N paths are used for the regression, but I don't think it should matter this much.

MJ: what exactly do you mean by an 'upper bounder'? Couldn't find anything on it.
How can I fix my problem? Does some other method exist to price American options with local volatility?

All the help is greatly appreciated.

deeds


Total Posts: 436
Joined: Dec 2008
 
Posted: 2009-04-03 15:22

Roel -

...convergence curves all look good?

d


Lapin


Total Posts: 277
Joined: Feb 2006
 
Posted: 2009-04-03 16:00

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


Total Posts: 3
Joined: Apr 2009
 
Posted: 2009-04-03 16:09
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


Total Posts: 436
Joined: Dec 2008
 
Posted: 2009-04-03 17:43

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


Total Posts: 436
Joined: Dec 2008
 
Posted: 2009-04-06 11:21

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


Total Posts: 3
Joined: Apr 2009
 
Posted: 2009-04-08 12:56
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


Total Posts: 1049
Joined: Jun 2004
 
Posted: 2009-04-09 06:33
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.

Quant Job Interview Questions and Answers now available on lulu and createspace: www.markjoshi.com

deeds


Total Posts: 436
Joined: Dec 2008
 
Posted: 2009-04-09 08:31
thanks, Mark!

quallenjaeger


Total Posts: 1
Joined: Jun 2019
 
Posted: 2019-06-03 16:06
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?
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