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Cuchulainn
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### Re: The Nelson-Ramaswamy Method

A related question is the applicability of NR to Monte Carlo simulation, i.e. transform the SDE to one with constant coefficients instead of nonlinear coefficient like S^b.

NR does it for binomial method, but does it improve performance in MC?

And it might be numerically easier?

Alan
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Joined: December 19th, 2001, 4:01 am
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### Re: The Nelson-Ramaswamy Method

I assume by 'constant coefficients', you actually mean just the diffusion term.

With that, I'd say 'it depends' on whether you want to do a Monte Carlo or a lattice method.

You have to analyze the model to see if S=0 is reachable. If not (and I am assuming S=infinity is not reachable) then the transformation is useful. But if S=0 is reachable, then I'd

- do the transformation anyway for a lattice method, but
- stick with the original coordinates for Monte Carlo.

A simple illustrative case:  dS = sqrt(S) dW, where S=0 is reachable. So I will guess you will have Monte Carlo problems near (the Y-image of) small S if you transform it to
dY = b(Y) dt + dW.

Try it!

On the other hand,  if you are doing a lattice method, that coord. transform can be very useful even when the origin reachable.

Also, haven't looked at the paper in a while, but my recollection is they use both the original and transformed coordinates and also allow the binomial process to jump several lattice spacings when needed. So that's a different wrinkle. In other words, you transform to unitized variance, derive some binomial transition rules, and then transform back. So that's for a lattice method. When you do all that, issues with reaching S=0 may be resolved because you are actually making (final) transitions in the original (finite) coordinates.

Cuchulainn
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Joined: July 16th, 2004, 7:38 am
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### Re: The Nelson-Ramaswamy Method

I assume by 'constant coefficients', you actually mean just the diffusion term.

Yes.
Using NR transform allows us to turn it into a Bessel process which helps.

When the SDE Lispschitz condition break down (e.g. large $\beta$) the use of drift-adjusted predictor corrector FDM is worse than Euler. In fact 10^3 draws give better accuracy than 10^6 draws for PC. Just because we have a 'betfer fdm' does not mean it works better than Euler on rough cases.

I need to leave this on the back burner for a while.