Boundary conditions for pricing barrier swaptions

youser · December 12th, 2007, 8:29 am

Greetings to all, happy holidaysI am trying to price a barrier swaption using the finite difference method (Hull white model).I need some help determining the boundary conditions. Please shed some light on this topic for me.Thanks guys I'd really appreciate any responce.

Cuchulainn · December 12th, 2007, 7:50 pm

http://www.wilmott.com/messageview.cfm? ... adid=49363 Write up the PDE, would help ==> BC

youser · December 13th, 2007, 8:25 am

The PDE is:Vr + (Theta*(t) - ar)Vr + 0.5*Sigma^2Vrr - rV = 0Theta*(t) = Theta(t) - Lambda(t)Sigma(I don't know latex, sorry)Using the FDM I set up a grid with short rate on the y-axis and time on the x-axis.I got all the formulas for solving the pde this way, all I need is the boundary conditions.If the short rate is close to zero or close to infinity, what does that say about the value of the swaption at any point in time?Also at time of maturity of the option, what will the value be there?

pekola · December 13th, 2007, 10:08 am

Do not think to much of the behaviour of the swaption at your boundary points,rhather think about WHERE you choose your boundary points..Ususally assuming Gamma to be zero at the boundaries is sufficient, i.e. This results in Linear Extrapolation at the boundaries: As i said it is critical where you truncate your space domain...this should depend on your distribution, which is mainly influenced by the overall variance..(for Hull-White you can compute that analytically)Then simply choose +- 5 StandardDeviations around the Forward for your DomainBoundaries - at these points the behaviour of your swaption value w.r.t the short rate is negligible.Hope this helps.Peter

youser · December 13th, 2007, 10:59 am

I think we also require the relationship between Libor/jibar(as it is in SA) and the short rate.I am using the payoff to the barrier swaption as one of the boundary values. I do not understand precisely what you meant earlier. please could you elaborate on that. greatly appreciated

Cuchulainn · December 13th, 2007, 5:47 pm

QuoteI am using the payoff to the barrier swaption as one of the boundary valuesPayoff is a terminal condition (or initial condition), yes? I think saying payoff is not a BC but an IC.BCs are at. r = rmin (0?). r = rmaxQuoteVr + (Theta*(t) - ar)Vr + 0.5*Sigma^2Vrr - rV = 0Should be??V_t + (Theta*(t) - ar)Vr + 0.5*Sigma^2Vrr - rV = 0 pekolahow do you motivate BC at rmin?

youser · December 14th, 2007, 5:44 am

Yes, it is Vt, sorry about that.rmin and rmax is exactly where I need the boudary conditions, and exactly where I don't how to find the boudary conditions.also what would be min and max values for r? zero and ....

Cuchulainn · December 14th, 2007, 7:19 am

QuoteOriginally posted by: youserYes, it is Vt, sorry about that.rmin and rmax is exactly where I need the boudary conditions, and exactly where I don't how to find the boudary conditions.also what would be min and max values for r? zero and ....At rmax, pekola's analysis should do the job. BTW, he has written a book about PDE for IR and it is there that he motivates the far field rmax conditions.At the near field rmin, the Fichera analysis could be useful. I'll check it myself. Here is a discussion of this topic in PP. rmin = 0 should ne OK?? not sure.

Cuchulainn · December 15th, 2007, 4:40 pm

The Fichera function is not strictly applicable because the characteristic form is positive definite. So no degenerate boundaries and the problem is not so complicated.In this case we can use Dirichlet (absorption), when r = 0 we need to specify a value for V? If r = 0 would be any point in entering a swaption?