Barrier Options and Why They Are Hard to Hedge

A barrier option's payoff depends on whether the stock ever touches a set level, which makes the position gap risk hard to hedge away.

A barrier option's payoff depends both on where the underlying finishes at expiry and on whether it touches a specified level, the barrier, at any point before then, which means two paths that end at the identical price can produce completely different outcomes depending on what happened in between. That path dependency is what makes barrier options cheaper than an equivalent vanilla option and considerably harder for a market maker to hedge.

Take a call struck at $100 with a knockout barrier at $90, on a stock at $102 with 30 days to expiry. As long as the stock never trades at or below $90 during that period, the option behaves exactly like a vanilla $100 call. The moment the stock touches $90, the option is knocked out and becomes worthless immediately, regardless of what the stock does afterward. Because of that added risk of extinction, the down and out call is quoted cheaper than the equivalent vanilla call, illustratively $4.35 against a vanilla price of $5.20 for the same strike and expiry.

If the stock dips to a low of $91 during the period, never touching the $90 barrier, and finishes at $108, the option survives intact and pays $8, a profit of $3.65 after the $4.35 paid. If instead the stock dips to $88 on day 10, breaching the barrier, and then rallies hard to finish at $115 by expiry, the option was knocked out the moment it touched $90 and is worth nothing at expiry regardless of the strong finish, a total loss of the $4.35 premium. The second scenario is the entire point of a barrier option: a deep in the money finish offers no protection once the barrier has already been breached.

Away from the barrier, a down and out call behaves much like a vanilla call, with a positive delta that rises as the stock rises. Close to the barrier the behavior inverts: as the stock approaches $90 from above, the option's entire remaining value, both intrinsic and time value, is at risk of vanishing at once, so a further drop toward the barrier accelerates the loss faster than a vanilla option ever would, and the delta can turn sharply negative right before the barrier is touched. A position that looks like being long a call can behave, in delta terms, like being short the stock in the moments before a knockout.

The option's value also jumps at the barrier itself, dropping from some positive amount to exactly zero the instant the barrier trades, an actual discontinuity rather than a smooth transition, a much sharper version of the pin risk problem that shows up in binary options.