The data sample is not draws from a pdf, but option prices. However you approach the problem, you have to tackle the issue of extrapolation (of the risk-neutral pdf) to non-marketed strikes. Good point. Does this mean that kernel functions are not 'suitable' for data with no discernible underlying ...

Fair point. The OP should indicate if he/she is talking about a negative interest rate environment. (I indeed was not). If so, there is the possibility that the "non-optimal" early exercises being seen are actually optimal. Perhaps bearish will elaborate ...

The data sample is not draws from a pdf, but option prices. However you approach the problem, you have to tackle the issue of extrapolation (of the risk-neutral pdf) to non-marketed strikes. Anyway, I am close to finishing a write-up for a larger problem (which contains this risk-neutral density est...

Consider covered call writing: buy the stock, sell a call. There will be a maximum return if the call is optimally exercised. If the call is exercised early (non-optimally), that maximum return will be obtained *sooner*, which is a good thing.  To take a trivial case, suppose there are no dividends,...

The problem is finding a risk neutral density from option quotes. The spline approach which I abandoned was: fit a smoothing spline to option smile data, with given derivative endpoints.  The purpose of the derivatives, as opposed to so-called 'natural' boundary conditions, was to be improved extrap...

Daniel, Thanks for the book link. Sorry, don't have a good numerical example because I abandoned this approach for another. But, may get back to it.

Just thinking out loud, if you *don't* want the vol to explode maybe there are some modifications that don't completely wreck the nice properties. Either truncating the lattice or keeping the full lattice but somehow truncating just the volatility. Or, a rectangular grid version of your model with e...

I'll bite. Why is volatility exploding "a more realistic feature"?

So they way it works is say you have 2 different oil market locations.  Spot1 is $55 and Spot2 is $50.  Right now we have a -$5 basis differential and I bought a put or call option with a  (negative)$5 strike on the basis diff.  As long as both prices stay negative the model is fine but if your bas...

Since I give  a (quasi)-analytic "absorbing" solution, all the various cases are checkable. BTW, when differentiating the latter w.r.t. r (which you asked about before), the identity [$] M'(a,b,z) = \frac{a}{b} M(a+1,b+1,z)[$]  might be helpful (from Abramowitz & Stegun).   Using that will keep the ...

Alan, Regarding eq. 10.4 of your book, how can it be differentiated WRT r and the other parameters? >> Standard differentiation: "r" appears in 3 places. BTW for [$]B_r[$] the Feller condition is always satisfied at [$]r = 0[$]. Correct? >> The Feller condition, applied to the [$]B_r[$] pde, and us...

BTW is it not so that CIR has a unique analytical solution for all values of the parameter regime? No. As explained on pg. 456 and also on pg 349 of my Vol. II, the CIR pde does *not* have a unique solution when (to use your notation) [$]0 < a < \sigma^2/2[$]. This is where the boundary is classifi...

The PDE for this greek does indeed contain terms in the bond price. The far-field BC sounds reasonable. At the near field the Feller condition for the 'greek PDE' is always satisfied which means we have a drift PDE at that boundary. It must be solved numerically (upwinding) in the worst case. I get...

The nice thing about NDSolve (in Mathematica) is that it automatically adopts a high order inward pointing derivative at boundaries, which then couples the system smoothly to the behavior of the solution at the interior points.

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Consider the well-known  PDE for the bind price [$]B[$] in the CIR model.Then differentiate the PDE with respect to [$]r[$] to get a PDE in [$]\frac{\partial B}{\partial r}[$] (this is a bespoke CSE equation). Some points: 1. The Fichera/Feller conditions implies that no BC is given at [$]r= 0[$] a...