FDM vs MC (Part II)

Previously, we compared for what finite differencing (or Monte Carlo) is good/bad.

After some practices, here are some more thoughts:

1. MC is more intuitive and easier to visualize
2. FDM has a higher sunk cost, i.e. the basic architecture has to be there to price even the simplest stuffs
3. Debugging FDM can be really hard since the different features of your instrument mingles together
4. FDM is rewarding in the sense that you get the instrument price at any spot value (and hence also some of the greeks) for free
5. FDM is much more problem specific - the code for one derivative needs much work to be used to price something else

FDM vs MC

FDM is more tricky, and less intuitive, than MC.

Handling early exercise feature: FDM > MC
Handling path dependence: FDM < MC

The most difficult part of doing FDM is determining the correct boundary conditions. It can get really complicated, especially because one has to think backward in time.

Also, although it is better than MC in terms of handling early exercise, tools such as PSOR are not trivial - especially in the presence of other things such as barriers.

Affine models

The Piazzesi paper (Affine term structure models) is an excellent overview of affine interest rate models, both on the theoretical foundations and a handful of specific models. It is also revised recently, to be included as a chapter in a book. It's amazing that single-factor models still remain important in the fixed income toolbox.