I know that a local volatility model does not allow to control the correlation between Spot and Vol. I know also that the correlation Spot Vol is important for products like autocalls. Why is correlation spot vol important for autocalls? In which cases the correlation spot vol is important? Is it important for quanto options?
I saw this interesting answer: What are the advantages/disadvantages of these approaches to deal with volatility surface? What is the forward volatility? How can we see mathematially that the forwrd volatility of a local volatility model flattens out?