Re: Jobs status and brain dump

Posted by Joseph Wang on
URL: http://quantlib.414.s1.nabble.com/Interpolator-semantics-tp4212p4218.html

Daniel J. Duffy wrote:

> 7) You should be able to go straight from a stochastic differential
> equation to a FDM scheme *without* going through a PDE.  I've never seen
> someone do this or work out step by step the math for doing this, but it
> can be done.  Calculating a PDE is totally unnecessary in doing FDM.
> wow! Examples???

The basic idea is that people go directly from a SDE to a binomial tree
model by matching moments and without explicitly writing down a PDE.  
Since a tree is just a badly implemented explicit differencing scheme,
you can take the operator that you would have used on the binomial tree
and use this directly in your finite differencing scheme.

So you can take a process such as a short rate bond model like Vasicek
and/or CIR and then write down the differential operator, and instead of
dumping that into a binomial tree, you run it into a finite difference
engine.  (You can also go the other way, take a PDE, extract the
differential operator and then run that into Monte Carlo.  The time
you'd want to do that is when your volatility or drift is a function of
the history of the process, and the system is no longer Markovian.)

I suspect that this technique hasn't been published in the quantitative
finance literature, since you have a lot of literature in which people
are trying to do weird things where a tree hits a boundary condition or
weird complex things involving subtrees and American options where you
want more time steps near the exercise date.  All of this is
unnecessary.  Once you have written down the differential operator from
the SDE, then you can control your time steps however you want.  You can
also fix your boundary conditions exactly, shift your grid to deal with
dividend payments.  Whatever.....

Also, I suspect that this formalism also handles jump-diffusion problems
nicely.  Jump is represented in the PDE world as just another term.  So
you take your SDE operator for the diffusion part, add the PDE operator
for the jump part, and what you end up is nicely numerically behaved.

>  
> What about straight from a PDE to FDM without SDE "preprocessor"?

>  
> That's the way it is in CFD?

I think in a lot of CFD, they treat the differencing engine as a black
box.  In CFD, your differential equations don't change, but your
boundary conditions do.