One of my commenters wrote in response to my piece Book Reviews: The Complete Guide To Option Pricing Formulas, and Derivatives, Models on Models:
Kurt, I’ve met Mandelbrot, and have discussed these issues with him. The two books that I recommended are also up on those issues. Implied volatility estimates as applied to option pricing formulas are a fall-out. No one thinks they are true, but they are a paramater used to keep relationships stable across options of similar expirations.
Intelligent hedgers hedge options with options; they don’t try to apply the theoretical equivalence that lies behind the traditional Black-Scholes formula and do dynamic hedging with the common stock itself. That is the philosophy behind the books that I reviewed.
I’m on your page, Kurt. Variance is infinite, and B-S blows up. But within the options world, there has to be a way of calculating relative value, and these books aid us in that calculation.
If you think I am wrong here, go to your local library, and get these books via Interlibrary loan. Read them, and you will see that we are all in agreement.
Kurt Osis 
December 13th, 20085:05 pm at EditDavid:
How can advocate people using these models which clearly don’t work? Estimating volatility is a suckers bet. Even if you could estimate the underlying “actual” volatility with 100% accuracy there would be sample error in your realized volatility. And of course the volatility isn’t just changing, the fundamentals of the underlying are changing.
I once heard of a man named Mandelbrot who said volatility was infinite, in which case these sigmas and lemmas are a bit beside the point, no?