Correlation Spot Vol - when is it important?

Question

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?

EDIT

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?

Answer 1

When pricing autocalls you are basically dealing with digitals and barriers. The probability of crossing barrier/digit up or down is governed by how much volatility is expected in next spot moves. This expected volatility in BS model is constant regardless of your spot level now. On the other hand, the vol is negatively correlated with spot level in a stochastic vol model. e.g if spot is below the barrier, the expected vol is more important (you are more likely to cross the barrier up and become ITM) than if your are above the barrier (less likely to cross the barrier down and become OTM).