I have two models of some spot. One is under local vol and the other is under stoch vol. Both are calibrated to the prevailing vanilla prices. I then consider the option that pays $1$ if the spot price exceeds a certain level before expiry, $T$. If there is any payment to be made, this happens at the time the barrier is reached.

$S$ denoting the spot price process, we can define

$$\tau = \inf\{t \geq 0: S_t = A\}$$

with $A$ being the barrier and $S_0 < A$. The price of the option is then given by $$E[e^{-r\tau} \mathbb{1}_{\tau \leq T}]$$

with $r$ being the interest rate. Under which model (local or stoch) is this price higher?

Does the answer change if the payout occurs at $T$? Would it change if $S_0 > A$?

I am looking for an answer that is not entirely intuitive but one that justifies itself at least with some hand-wavy calculations.


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