Hi,I am solving the BS using Crank-Nicolson on the S-t plane with the drift term, r-d, being time dependent. Barriers are put at S=0 and S=1. Rebate of value 1 is paid when S=0 is struck and no rebate is paid at the other boundary. When I plot P(S,t) vs S for a given t, I find that P(S,t) exhibits obvious fluctuations in values near S=0. For example, from S=0 to S=10^-4, P(S,t)=1.2, 0.8, 0.9, 1.01, 0.99... and it rapidly levells off and then decays towards S=1. Is that normal? Can I view such oscillations as a sort of reflection when the boundary is hit? Thanks.Shaleen

If your contract is a call the solution is not continuous at S=0.Indeed your boundary condition forces V=1 at S=0.But close to S=0 you have V close to 0 (the spot cannot touch the boundary 0 and is too small to finish in the money)...

I don't really know this stuff, but I think that a barrier at S=0 requires a jump-model (jumps to S=0) to make sense for the reason DiceMan gives, i.e. that in the the BS-world (log-normal random walk) the risk-neutral probabality of hitting S=0 is zero.

If you'r using a lognormal random walk for S then you've got problems assigning a value at S=0. Just think about transforming to new variable X = ln(S), this takes the S=0 boundary to X = -infinity! What we are all saying here is that the asset can't ever hit 0 with the lognormal model so the b.c. V=1 at S=0 is meaningless.P