Fair enough.This heuristic has been made directly on the SDE, not on the corresponding PDE. It's a classical "drift freezing" method. Obviously, some properties are broken with this approximation, otherwise we wouldn't misprice a swap (this approximation breaks the fitting of the initial yield curve, because by construction on the HJM framework, y must be equal to is expression under the qG model).
The justification is that empirically, it provides good enough results to calibrate on swaptions (even in the linear local vol + stoch vol case).
In both x and y or just x?You'll need boundary info coming in from the edges of the domain.
Of course we are loosing information, but the variance of y is negligible vs the variance of x, so the error on prices won't be that big for not too high maturities / negatives mean reversions. Furthermore, if you price vanilla derivatives (in the sense hedgeable by swaptions, such as cms caps/floors/swaps, cancelled swaps, ...), a small errors on swaptions => a small error on your price as it is a integral over swaption prices.Fair enough.This heuristic has been made directly on the SDE, not on the corresponding PDE. It's a classical "drift freezing" method. Obviously, some properties are broken with this approximation, otherwise we wouldn't misprice a swap (this approximation breaks the fitting of the initial yield curve, because by construction on the HJM framework, y must be equal to is expression under the qG model).
The justification is that empirically, it provides good enough results to calibrate on swaptions (even in the linear local vol + stoch vol case).
I understand why one would adopt such a tactic as it makes life easier. But numerical success with swaptions does not mean the trick will work on the current problems, or will it? In general, it feels like we are losing information by removing the y dimensions and replacing it by an approximation without boundaries.
The BGM book mention freezing in 5 lines and the authors seem to equate it with the Euler method..Is that a fair assessment? Or am I missing something crucial? Even authors sat it's 'very simple'. At fist glance, even the predictor-corrector method would be worth a try as well?
For Anchor PDE, 2-point upwinding is better but 1st order accurate, Can be rectified by taking Towler-Yang or Roberts-Weiss for the convection term.He does say it's ADI Craig-Sneyd.On time discretisation you don't mention if implicit, CN etc.
Ferdelo, I'm not against 5-point stencils in general (or even ghost points for that matter) but this 5-point stencil is still a central discretization with 2 points downwind for a convective term, so I agree with Cuchulainn, that's obviously asking for trouble. The real question is why doesn't simple 2-point upwind work for you. Have you by any chance messed up the downwind/upwind sign? I always do a stupid check like that just to make sure. Your pdf does seem to indicate that but I may be wrong.
If that's not it then I'd plot the results, have a visual. Are there oscillations, does the solution really have enough space to reach zero gamma at the boundaries as your boundary conditions assume, etc. And by the way, don't any of the references (books/papers) you have on this mention the boundary conditions they used?
In general. quants tend to be fixated on centred differencing even for convection, even at the downstream boundary.You'll need boundary info coming in from the edges of the domain.