Black-Scholes PDE with boundary conditions

[quotes]

A typical European option pricing via Black-Scholes approach is
to solve a Black-Scholes PDE with terminal option payoff f(T,x)

Simply put, a Black Sholes PDE involving the first derivative of option value on t and on x (the value of underlying security) and the second derivative on x, with a domain [0,T]*[-inf,+inf] with a boundary value condition on the right vertex can be solved.

My question is, is it possible to solve the Black-Scholes PDE with a different boundary value condition?

e.g. domain [0,T]*[U,L] with the upper boudary the lower boundary and the right vertex boudary.

And so on.

 

[Daniel Duffy]

Much has be done in this area, much of which is based on heuristics and somewhat ad hoc arguments.

One approach is to transform y = S/(S+a) and use the Fichera theory.
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1552926

 

[Darou]

This really depends on your option pricing problem: E.g. for a Barrier Option a Dirichlet Condition should be applied. And for pricing an American option, second derivative V_SS =! 0 is useful.
See e.g. the papers from Forsyth: http://www.cs.uwaterloo.ca/~paforsyt/
especially http://www.cs.uwaterloo.ca/~paforsyt/con7.pdf.

 

[quotes]

My problem is a double barrier option between two slope lines (instead of known interval X in (a,b) ）

Thank you.

 

[Daniel Duffy]

My problem is a double barrier option between two slope lines (instead of known interval X in (a,b) ）

Thank you.

You you have non-flat and/or tme-dependent boundaries?

 

[quotes]

Well could you tell me if this is a time dependent boundary or not?

An option pays 1 dollar immediately when the underlying hit a predetermined level before expiry.

 

[quotes]

Another option pays the same but lose effect when the underlying hits another level.