Volatility Smile and Volatility Models: A Short Note

Local Volatility (LV) Model

1. LV preserves market completeness (is it a good/must-have thing?)
2. LV gives forward smiles that are too flat compared to market
3. In fact, LV doesn't really qualify as a model: it fits implied vol perfectly but trivially, with no capability/intention to consider the underlying dynamics
4. LV also produces poor smile dynamics prediction

 More General Notes

1. The most sought-after quality for a model should be accurate hedging, not mathematical tractability etc.
2. In fact, there is no good reason to limit ourselves to vanillas only when calibrating a vol model; calibrating only to vanillas has been common practice, which is unfortunate
3. The information carried by vanillas is the unconditional (risk-neutral) probabilities that can be recovered by taking second derivatives w.r.t. strikes (i.e. the stock price is here at t=0, what is the probability that it will end up there at t=T?)
4. Conditional probabilities, on the other hand, is not contained in vanillas (i.e. conditional on the stock price being here at t=s
5. Another way of saying this is that vanillas are not sensitive to unconditional distributions
6. Therefore, one way to proceed is to include some exotics in the calibration: forward starting options are sensitive to conditional probabilities; while American digitals (or barriers) are sensitive to jumps
7. As a rule of thumb, jumps are "responsible" for short-dated smile, and stochastic volatility is "responsible" for longer-dated smile

Reference:
Ayache 2004, Can anyone solve the smile problem?
Ayache 2005, Can anyone solve the smile problem? A Post-Scriptum

Also, Rebonato's Volatility and Correlation: The Perfect Hedger and the Fox is relevant, and it's a great read, but that would be another episode.