mind you Scholar, you can always use Puts and verify a posteriori that your measure is norm preserving. no?

In the MinEnt method whether the measure is norm-preserving or not depends on our choice. If we impose a constraint ensuring a right normalization, it will be norm-preserving, otherwise it won't. Imagine we use puts and enforce the right normalization. This will give us one Minimum Entropy measure. But according to what you're saying we could use calls and don't impose the right normalization. In this way, we would end up with another Minimum Entropy measure (or even a continuum of measures). Does it at all make sense ? Or maybe I did not understand you ? I wish I had Lewis' book but the winter here was really cold

actually whatever the measure we always have
put = exp{-rT} E{[K-S(T)]+}
regardless of measure and uniqueness
so I guess using puts is always safe?
(again because they are bounded)

Exactly ! It was my impression from what you said that puts are better because they're bounded.

Reza, Scholar: Models that blow up or go to zero with non-zero probability never bother me; I think the defects are mathematical in nature.

For example, take the case of a model where some variable X may run away to infinity. In practice, as soon as the variable gets unreasonably large, the model is guaranteed wrong. (Ie, no model of the US interest rates can be expected to hold true when interest rates get over, say, 500% a year). For this reason, one can at least theoretically draw a boundary (at say 500% a year) and change the model so that:

If variable As soon as variable>= maximum bound, model goes to incredibly simple model
(Like if rates get to 500%, then volatility goes to zero).

Cutoffs can be introduced at will provided they are distant enough so they have no significant influence on the pricing (if they do, the original model is no good). Any mathematical difficulties eliminated by using cutoffs (which have no significant effect on the prices) should be considered technicalities, and should not get in the way of the theory.

The types of incompleteness that I worry most about are jumps.

Pat,
what you say is true but sometimes models blow up without you knowing it !
again to come back to the same example under a GARCH diffusion limit you can calculate
Call = E{(S-K)+}
but it is not the right solution
the right solution is
Call = E{(S-K)+} + S p
where p is the vol explosion probability under another measure
so if we just use this model there is no way for us to know that, we will not observe an explosion directly, only wrong prices !

Cutoffs can be introduced at will provided they are distant enough so they have no significant influence on the pricing (if they do, the original model is no good). Any mathematical difficulties eliminated by using cutoffs (which have no significant effect on the prices) should be considered technicalities, and should not get in the way of the theory. >>
I have some experience with cutoffs in field theory, but not in financial modeling. In the former, very roughly a strong (power-like) dependence on a cutoff hints on a new physics involved, while a mild
(logarithmic) cutoff dependence can be tolerated/cured by a renormalization within the same theory. I don't have any feeling re a type of a cutoff-dependence you mentioned. Are there any examples at hand ?

Unfortunately, nothing cool or deep seems to be happening here. In finance you can't renormalize, so even a mild "log" blow up will invalidate your model. If you set a boundary, and find that the option prices are still a function of the boundary's postion, even for an unrealistically distant boundary, then clearly more modeling needs to be done.

As examples, apart from the trivial (any numerical scheme requires some kind of handling at infinity; hopefully the convergence is very rapid, and infinity can be set to 3 or thereabouts) there is always CEV.

For interest rates, the CEV model is "bad" for exponents between 0.5 and 1, because there is a finite probability of R reaching R=0, and once there, the rate gets stuck forever. Consequently one builds up a delta function at R=0. For rates below 0.5, one has a choice of whether one wants the rates to cross into negative territory or whether one wants the rates to remain positive. One way to eliminate these problems is to use
dR = a f(R)dW
where
f(R) = R^beta for R>eps
f(R) = eps^beta * R/eps for RFor options which have a significant epsilon dependence, one surely must model what's actually happening around R=0. Fortunately, apart from Morgan Stanley's zero strike options, patching a linear piece in for small R doesn't appear to affect USD prices. Probably this is NOT true for JPY; for these products one should develop a model for what actually happens as the rates cross through zero.

Similarly, some of Leif Anderson's work involves CEV models with beta>1 (which clearly blows up). He also regularizes these models by changing the model to linear-in-R for large R

I am looking to estimate state price density under the market measure. An interest model that I'm using does not permit an analytic solution. Any references/ideas on possible functional form for the density? To put it differently, if I know dynamics of a process under both risk neutral(Q) and market(P) measures how to numerically estimate Radon-Nikodym derivative dQ/dP? Due to complexity of the process very little analytic derivation is possible, everything is within simulation context.Thank you.

there are non-parametric implied distributions for both risk-neutral (from options) and real (from time series)see for example Ait-Sahalia et al.http://www.princeton.edu/~yacine/comp.pdf

reza, thanks a lot! it is very helpful.