Implicit Finite Differences Method For Pricing Barrier Option

While trying to price a simple knock down and out barrier option, I encountered several difficulties I did not expect with the implicit finite differences method. The explicit method has less issues with barrier options pricing. I will show here what the tricky parts are and why explicit seems simpler in this case.

The full article is here (pdf) or here (html) (the later is not very well formatted).

Algorithms used in this article can be found at http://code.google.com/p/javamc/