RE: chosing assetSteps and timeSteps for FdAmeri canOption

[Vadim Ogranovich-3]

Thank you Marco.

I believe this is correct in the general case of the heat equation in a
block domain (where you have the boundary condition u(s_0) = u(-s_0) = 0 or
similar) and where one wants to find the solution on the entire segment
[-s_0, s_0] for t=T.

However BS American has a number of traits that might provide for less than
O(T) number of asset steps.
Firstly, the discontinuity at expiration is taken care of by the use of the
control European option.
Secondly, we only need the solution at one point S=S_0, which in principal
may allow to shrink the asset grid as we come close to t=0.
Thirdly, the asset grid at expiration need not to cover the entire support
set of the asset distribution (the size of the support set indeed grows as
O(t)) since outside some fixed range the American option is approximated
really well by the control European option.

Don't know if this makes any sense at all, but these were the motives behind
my original question.

Thanks, Vadim

-----Original Message-----
From: Marco Marchioro [mailto:[hidden email]]
Sent: Tuesday, May 07, 2002 1:51 AM
To: [hidden email]; [hidden email]
Subject: Re: [Quantlib-users] chosing assetSteps and timeSteps for
FdAmericanOption


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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