Re: FdDividendAmericanOption

Posted by Luigi Ballabio on Jan 03, 2005; 1:28am
URL: http://quantlib.414.s1.nabble.com/FdDividendAmericanOption-tp3453p3455.html

On 12/31/04 08:16:31, Joseph Wang wrote:
> I think the basic problem is that when you turn x = log(S) into an  
> operator, dx = dS / S and you start getting extra terms in the difference  
> equation  that aren't being put into BSMOperator.  One could go and work  
> out what the operator should be with a logarithmic grid, but I suspect  
> that the second derivative is going to add a lot of nasty terms.

Joseph,
        no, the BSM operator is already the correct discretization for the  
logarithm of S. It discretizes in space the equation

df/dt = - sigma^2/2 * d^2f/dx^2 - nu * df/dx + r*f

that one gets from changing the variable from S to x = log S in the BS  
equation. At the point of discretization, it is only incidental that x  
comes from a logarithm; it is simply the variable being discretized, and  
building a "logarithmic" grid is just a way (albeit a roundabout one) to  
build an equally spaced grid for x. In fact, the name "grid_" in the code  
is a misnomer; it is not the grid { x_1, ..., x_N } used for the operator,
but rather the set { S_1, ..., S_N } of prices corresponding to the grid  
points. The actual grid is { log(S) for S in grid_ }, and if you print it  
out, you'll see that it is correctly spaced.  In short, it is beyond me how  
your patch provides results that look sensible :)

I had naively hoped to give the problem some time during the weekend, but I  
found that having a wife and a couple of kids seriously cuts into one's  
debugging time :)
All I was able to put together was the small patch below, which does not  
completely remove the instability, but at least triggers it only above 250  
asset steps for the example code given.

diff -r1.28 fddividendoption.cpp
74c74
<         setGridLimits(center_ + dividends_[step], dates_[step]);
---
>         setGridLimits(center_ + dividends_[step], residualTime_);

Basically, the code tried to be smart (probably too much for its own good)  
and resize the grid based on the time to the dividend payment. Now it  
resizes it based on the time to maturity, so that it doesn't end up with a  
grid too different from the one it started with.

But the sad truth is, the whole thing should be rewritten. I had a look at  
the code in the past few days (I had'n written it myself) and it's a tangle  
of methods, each one trying to outsmart the user and correct the number of  
grid points, the grid limits and whatever else based on what was thought to  
be a "safe" choice of parameters. Not that it can't be useful, but the user  
should at least have the choice to say "heck no, just do as I said."

Needless to say, at this time I wouldn't use the FD dividend options where  
actual money is involved---at least, not without cross-checking the results  
with some other pricer or some other source.

Later,
        Luigi