chosing assetSteps and timeSteps for FdAmericanOption

[Vadim Ogranovich-3]

Hi,

This is more of a numerical question. Is there any rule of thumb for
choosing assetSteps and timeSteps values for FdAmericanOption? For example
do they need to grow as time to maturity grows and if yes how (linearly,
etc.)?

Thanks, Vadim

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[Kris .]

Hull does mention that it is numerically most efficient to set
dS = sigma * sqrt(3*dT)
for explicit FD page(422) chap 16
I have seen a simillar such condition in another paper too.



> Hi,
>
> This is more of a numerical question. Is there any rule of thumb for
> choosing assetSteps and timeSteps values for FdAmericanOption? For example
> do they need to grow as time to maturity grows and if yes how (linearly,
> etc.)?
>
> Thanks, Vadim
>

[Marco Marchioro-2]

Hi,
I believe that the answer depends on the type of equation and the
type of time-scheme used for the resolution. In our case,
since Black-Scholes (partial-differential) equation is parabolic, the
condition
dS = sqrt(K dt),
where K is a constant, is strictly required for stability and convergence
only for an *explict* finite-difference scheme. If, however, an implicit scheme
is used than convergence and stability are automatically obtained
and it is possible to use something less restrictive like
dS = K2 dt.

These considerations, however, do not tell us anything about the number
of time steps to be used as time to maturity grows. Again, the answer
depends on the shape time grid.
For parabolic (partial-differential) equations it is always better to have
as many time steps as possible close to the places where discontinuities occur.
The smoother the initial condition the better an FD method will work, that's
why a binary option is the nightmare of finite-difference.
In the case of a plain vanilla option the discontinuity is only in the
first derivative,
and at maturity. Therefore, the best thing to do is to have a much finer
time grid
close to maturity. When far from maturity the option price has already been
smoothed
out by the dynamic and there is no need for a very fine grid.
If, on the other hand, an uniform grid is to be used, to allow the same
time spacing
close to maturity it is necessary to increase the number of total time-steps
linearly with the total time to maturity.

I hope this has been helpful,
                         Marco Marchioro

At 07:42 PM 5/6/02 -0700, you wrote:

>Hull does mention that it is numerically most efficient to set
>dS = sigma * sqrt(3*dT)
>for explicit FD page(422) chap 16
>I have seen a simillar such condition in another paper too.
>
>
>
> > Hi,
> >
> > This is more of a numerical question. Is there any rule of thumb for
> > choosing assetSteps and timeSteps values for FdAmericanOption? For example
> > do they need to grow as time to maturity grows and if yes how (linearly,
> > etc.)?
> >
> > Thanks, Vadim
> >
>
>
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[Johann Hibschman-2]

Okay, my specific knowledge isn't that high, but I can comment on some
of this.  (Heh.  Yet another physicist jumping ship here.)

On Tuesday, May 7, 2002, at 01:50  AM, Marco Marchioro wrote:

(This isn't really aimed at you specifically, Marco, but is more of a
general rant.)

> I believe that the answer depends on the type of equation and the
> type of time-scheme used for the resolution. In our case,
> since Black-Scholes (partial-differential) equation is parabolic, the
> condition
> dS = sqrt(K dt),
> where K is a constant, is strictly required for stability and
> convergence
> only for an *explict* finite-difference scheme. If, however, an
> implicit scheme
> is used than convergence and stability are automatically obtained
> and it is possible to use something less restrictive like
> dS = K2 dt.

Certainly, the exact rule you need depends on the equation you're
using.  I'm more used to hyperbolic PDEs, but there the basic von
Neumann stability analysis is to just substitute in an exponential local
solution and solve for it's growth.  You want to make sure that spurious
modes are always damped.  In fluids, this gives you the famous Courant
condition, that dx >= v dt, so nothing jumps over a cell in your grid.

I'm not sure about explicit/implicit, but really the only way to do
decent numerics is to carefully watch the stability condition.  I've
done stuff like this in Mathematica, and am just looking into QuantLib
library, but it seems like there should be a way to automate this by
adding some methods to the differencing operators.  It might get a bit
hairy, though.

This isn't quite so crucial for parabolic equations, but it's still a
good thing to keep in mind.  Sorry for the lecture, I just had this
stuff repeatedly drummed into my head by one of my numerical methods
courses.  I'm sure most readers of this list know this already.

> These considerations, however, do not tell us anything about the number
> of time steps to be used as time to maturity grows. Again, the answer
> depends on the shape time grid.

Actually, if you can get the stability analysis right, it should tell
you how many steps you can get away with, and you almost always want to
make as big a step as is numerically safe.  Otherwise, you're just
wasting time.  I've not played with it much, but I'm guessing the
"right" rule would naturally give a finer required grid near
discontinuities, so a variable-grid method work well.  Adaptive-mesh and
all.

I suppose I'm ranting.  I just thought I should post something other
than "help, I can't get it to compile!", just to save face.

--Johann