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Kamil90
Topic Author
Posts: 178
Joined: February 15th, 2012, 2:02 pm

### Compatibility condition

For the barrier option we often have 0 on the boundary and likely some positive payoff on the right of the boundary. So the data is not compatible. Why is this a problem? the solution for any t<T is infinitely smooth as it can be shown mathematically deriving the exact solution, so why do I care about what happens at S=barrier, t=T?

bearish
Posts: 5615
Joined: February 3rd, 2011, 2:19 pm

### Re: Compatibility condition

To the extent that you are dealing with an analytical solution (which usually means you are in a Black-Scholes world), your intuition is pretty much right in that there is no pricing problem, per se. But, even there you will have a lurking hedging problem in the relatively unlikely case that you end up in that awkward corner of your state space where your delta goes to infinity (not to mention the higher order Greeks). Once you move out of analytical solutions and into numerical ones, I suspect that our local PDE experts (of which we have several) will tell you that the "local" discontinuity can induce problems of convergence more widely. I am deliberately being imprecise here, since I do not count myself among said experts.

Kamil90
Topic Author
Posts: 178
Joined: February 15th, 2012, 2:02 pm

### Re: Compatibility condition

ok, so what you are saying the function is smooth, meaning I have all kinds of partial derivatives, however, it is magnitude is very large(think 1/x as s approaches 0). And numerically it is hard to capture this type of functions?

Cuchulainn
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Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
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### Re: Compatibility condition

ok, so what you are saying the function is smooth, meaning I have all kinds of partial derivatives, however, it is magnitude is very large(think 1/x as s approaches 0). And numerically it is hard to capture this type of functions?
Yes, it's similar to boundary layer behaviour and singular perturbation theory, for which classical FD tends to break down.
https://www.nag.co.uk/market/Peter_Duck ... tation.pdf
The exponentially fitted schemes for convection-diffusion PDE (e.g. BS) I developed are based on singular perturbation theory.