November 5th, 2008, 8:25 pm
QuoteOriginally posted by: AlanThis seems like an excellent special case to start with.The general case stand-alone volatility process pdeis dp/dt = (1/2) sig^2 v d^2 p/dv^2 + (a - b v) dp/dv,But, now you are taking a = b = 0. This special case model describes a volatility process trapped when it hits v=0,unlike the (usual full) Heston model. So the v=0 BC is different -- you canset option values to their (Black-Scholes) v=0 limit, which is generally wrong for Heston but right here.This limit, since r=0, is simply the intrinsic value.But, the large v behavior of p(t,v) is similar to the full model.Since the large v behavior is the problem at issue, I think your special case should indeed tell us if you can map v=infinity successfully to v=1 in a FDM scheme.I arrive at the same conclusions. For the original problem (no transformation, yet), I looked at it and 1. The boundaries B1 {x=0} and B2 {y= 0} are degenerate (quadratic form == 0) so here we must calculate the Fichera Function2. B1: FF = 0B2: FF = - 1/2 sig^2 (!! Heston would have kappa*theta and Feller ==> FF > 0 ==> no BC)Thus we must specify a BC on B2. How?3. On B1 the PDE becomes du/dt = 0 ==> u is constant right up to (0,0). So4. based on a 'analytic continuation'?? (heuristic?) from x = 0 outward we get the BC on x = 0 to be the intrinsic value, indeed.//////Q. It is WLOG (without loss of generality) to take a barrier option so that I don't have to transform x as well, for the moment? It would alow me to focus on v only, for the moment?It's trivial to put in -ru term, but no big deal? 5. The quadratic form is not degenerate on Xmax, Ymax (assuming they were part of the problem) then BC would need to be described, but that's another discussion.
Last edited by
Cuchulainn on November 4th, 2008, 11:00 pm, edited 1 time in total.