Dear all,
I wonder what are the boundary conditions used by QuantLib FDM pricing engines (e.g., FdBlackScholesBarrierEngine). I understand this may be scheme-dependent. If that's the case, you can use ImplicitEuler as an example.
Assume we have N price grids at each time step, representing stock prices ranging from Smin to Smax (both inclusive). When we discretize the PDE, we have N-2 equations (1 for each inner grid). To make it complete, we need boundary conditions at Smin & Smax for each time step.
We could do this in a product-specific way. For example, we can specify simple Dirichlet boundary conditions for vanilla European options as below:
- Vanilla Call
- V(Smin, t) = 0
- V(Smax, t) = S - K exp (-r * (T - t))
- Vanilla Put
- V(Smin, t) = K * exp (-r * (T - t))
- V(Smax, t) = 0
Alternatively we may impose some "generic" boundary conditions, as suggested by Wilmott. See Wilmott on QF (2nd edition) Vol 3 Chapter 77.10.
Or maybe there are other techniques to handle it, which I are not aware of.
I tried to trace the code myself, but got lost in the method ImplicitEulerScheme::step().
Could someone help on this?
Best wishes,
Henry
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