By Elie Ayache: The smile problem isn’t something that happens to the Black–Scholes–Merton (BSM) model from outside. It is not a falsification of the BSM model. The smile problem isn’t that BSM assumes the underlying price process to be lognormal and that it suddenly happens in reality, externally to BSM, that the process is different, i.e., admitting of stochastic volatility and jumps. The smile problem is produced from inside.
If it were limited to the outside descriptive view, the smile problem would have been solved a long time ago. In that view, it would simply appear that the underlying probability distribution admits of a third and a fourth moment; derivatives would be evaluated in some martingale measure that is equivalent to the real one. As a result, implied volatility smiles would manifest themselves relative to BSM. That’s a statistician’s, econometrician’s, smile problem; and I don’t think it is the one that interests us.
Continue Reading: The Volatility Smile Problem: From Within – Wilmott