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I think this must be a really stupid question but I cannot see what I am missing.

Let's assume we're pricing some barrier option under Black -- Scholes model. Under risk neutral measure, the drift $\mu$ becomes the risk free rate $r$ and we compute discounted expectation of the payoff of the barrier option under this measure which is our fair value price.

Now my confusion is as follows: let's assume drift $\mu >> r$ is very large (for sake of argument), then under risk neutral measure we could be in a situation where we never go anywhere near a barrier that, under real-world measure, we would hit very quickly (with high probability). This doesn't make sense intuitively to me at all, if, say, barrier was a knock out.

What am I missing? Thanks!

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  • $\begingroup$ No matter what drift you assume the risk-neutral measure does not care. $\endgroup$
    – Kurt G.
    Nov 17, 2023 at 10:59
  • $\begingroup$ The barrier option must be priced relative to the forward price of the stock which depends only on r. $\endgroup$
    – dm63
    Nov 17, 2023 at 11:42
  • $\begingroup$ If you price the barrier option based on $\mu$, it would be expensive and you could sell it versus buying the stock as delta hedge and this portfolio would make a profit $\endgroup$
    – dm63
    Nov 17, 2023 at 11:47
  • $\begingroup$ So the answer is that, however unintuitive it might seem, there is nothing special about barrier options and we still evaluate whether barrier has been hit using Brownian paths with drift $r$ (e.g. if doing MC)? $\endgroup$
    – Jack
    Nov 17, 2023 at 11:51
  • $\begingroup$ Yes indeed correct $\endgroup$
    – dm63
    Nov 17, 2023 at 12:28

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