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Barrier options are path-dependent options, that are either initiated (knocked in) or eliminated (knocked out) upon reaching a certain barrier level.

Barriers are classed as "up" or "down", depending upon their position relative to the spot prices at the start of the Option. A standard barrier option has only one barrier which can be "up" or "down". A double barrier option has both an "up" and "down" barrier.

Barriers are either "in" or "out". A barrier is considered touched when the spot price of the underlying asset equals or crosses the barrier value. Once an "in" barrier is touched the option is "knocked in", and will pay out in the same way as a vanilla option. If an "out" barrier is touched then the option is "knocked out" and has no value.

In cases where the option expires and an "in" barrier has not been touched, or an "out" barrier has been touched then the Barrier Option may pay a rebate to the holder.

### Example Knockin option In this example we have a knock-in option, where the option must first breach the barrier to become active. Once the barrier is breached, it is priced as a regular call option.

### Example Knockout option In this example we have a knockout option, where if the barrier is breached the option disappears.

Barrier option pricing formula

 Mertons (1973) and Reiner and Rubinstein (1991) formulae for pricing Standard barrier options: where ,  ,  ,  "In" Barriers Down and in call S >H Cdown&in(X>H) = C+E =1, =1 Cdown&in(XH) = A+E =-1, =1 Cup&in(XH Pdown&in(X>H) = B-C+D+E =1, =-1 Pdown&in(X>H) = A+E =1, =-1 Up and in put S>H Pup&in(X>H) = A-B+D+E =-1, =-1 Pup&in(X>H) = C+E =-1, =-1 "Out" Barriers Down and out call S >H Cdown&out(X>H) = A-C+F =1, =1 Cdown&out(XH) = F =-1, =1 Cup&out(XH Pdown&out(X>H) = A-B+C-D+F =1, =-1 Pdown&out(X>H) = F =1, =-1 Up and out put S>H Pup&out(X>H) = B-D+F =-1, =-1 Pup&out(X>H) = A-C+F =-1, =-1 where X = Strike price T = Time to maturity r = risk free rate N = The cumulative normal distribution function S = Sport price H = Barrier K = Predetermined cash payoff

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