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reaverprog
Topic Author
Posts: 194
Joined: October 28th, 2009, 8:53 am

### Barrier Options : problem with boundary condition

Hi guys,I have a basic question concerning barrier options.Denote V the value of the Barrier Option. Say we have an Up and Out call with barrier level = Su we get :1/ V verifies the BS PDE2/ V(Su,t) = 0, for all t < T (T the expiry date)I understood the first condition, by can't see why the second boundary condition is enough ? Because it doesn't take into account the fact that if the underlying asset exceeds Su and later on falls below the barrier Su the contract is no longer worthless.THanks in advance.

daveangel
Posts: 17031
Joined: October 20th, 2003, 4:05 pm

### Barrier Options : problem with boundary condition

if the asset reaches Su then it knocks out - thats why its Up and Out.
knowledge comes, wisdom lingers

reaverprog
Topic Author
Posts: 194
Joined: October 28th, 2009, 8:53 am

### Barrier Options : problem with boundary condition

But I can't see why it's enough to say V(Su,t) = 0 for all t< TSuppose at time t1 the barrier Su is reached.And time t2>t1 the asset price falls below Su: how is this situation taken into account by the conditions 1/ and 2/ ??

reaverprog
Topic Author
Posts: 194
Joined: October 28th, 2009, 8:53 am

### Barrier Options : problem with boundary condition

Denote Vc the BS price of a call.The solution is, for a DOwn and out call :V = Vc(S,t) - (S/Sd)^(1-2r/sigma^2).Vc(Sd²/S,t)if S<Sd, the value is not equal to zero ! and if the asset reaches the barrier level and then goes beyond it, this's not taken into account. Why ?

ACD
Posts: 107
Joined: April 19th, 2004, 8:09 am

### Barrier Options : problem with boundary condition

Think of it as a piecewise solution, the BS pde and boundary conditions give the value prior to the barrier being hit. After it's hit the formula for the price is V=0. That way one it's hit the value will always be zero.

reaverprog
Topic Author
Posts: 194
Joined: October 28th, 2009, 8:53 am

### Barrier Options : problem with boundary condition

OK ! you make things clear for me now.

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