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Here is an interview question:
For a double knock out barrier option (ITM), which model gives higher price? local vol or stochastic vol model?
The answer is that local vol gives the higher price. However I only know that stochastic vol is the correct model to price barrier option to control the forward vol (vol of vol) as the product depends on the conditional distribution.
Empirically, knock-out barrier options are more expensive in stochastic volatility models and less expensive in local volatility models. Since knock-out + knock-in = vanilla, and vanillas are priced the same in local vs stochastic volatility, knock-in barrier options are priced higher in local volatility.
There is no known mathematical proof for this, and I have seen rare claimed counterexamples. But it is certainly true almost always.
