Also, while we might accept a certain smoothness hypothesis (fuzzy logic as applies to human perceptions) in financial valuations, these functions certainly aren't (if I may be pedantic) analytic at all.

If a function is analytic, then the derivatives at one point tell the whole story on the complex plane--the information is all encoded at (an arbitrarily small neighborhood around) a single point. But most smooth functions aren't analytic; indeed, a smooth function that is the trace of a process taking any stochastic external input will typically (probability 1) not be.