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If you put some numbers into down-in/out barrier call option formulae that can be found in many books, you will see that the down-in curve is not symmetric. It just looks like it in that plot. Below the barrier, the prices are obviously just Black-Scholes values, as the spot price goes higher the chance of it going below the barrier is obvious becoming ...


Intuitively, underlying call keeps losing value as the spot goes down, but the barrier option value (which starts at almost nothing for high spot) keeps growing as the spot approaches the barrier level (the chance to get something, even if it's an out-of-money call, is growing). When the spot hits the barrier level, the value of the call is still ok (unless ...


Definelty barrier shift is necessary when hedging PDI and CUI, thats because you always want to hedge your barrier as it was no hitted because you would experience enormous gamma in that region. So you shift the barrier and in this way you will never hit the barrier in your hedge

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