Local vol pricer for american options

Baltazar · Post by **Baltazar** » Fri Apr 03, 2009 10:45 am

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.

Lapin · Post by **Lapin** » Fri Apr 03, 2009 11:33 am

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 · Post by **mib** » Fri Apr 03, 2009 12:15 pm

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.

mj · Post by mj » Fri Apr 03, 2009 1:15 pm

LS is not very reliable. Are you running an upper bounder to check how good it is?

Baltazar · Post by **Baltazar** » Fri Apr 03, 2009 2:19 pm

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

deeds · Post by **deeds** » Fri Apr 03, 2009 2:35 pm

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 · Post by **Lapin** » Fri Apr 03, 2009 2:48 pm

B,

Do you have discrete or continuous divs?

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

Baltazar · Post by **Baltazar** » Fri Apr 03, 2009 3:04 pm

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)

Roel · Post by **Roel** » Fri Apr 03, 2009 3:08 pm

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 · Post by **deeds** » Fri Apr 03, 2009 3:22 pm

Roel -

...convergence curves all look good?

d