Barrier options are pathdependent 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 knockin 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(X<H) = AB+D+E =1, =1

Up and in Call S<H
Cup&in(X>H) = A+E =1, =1
Cup&in(X<H) = BC+D+E =1, =1

Down and in put S>H
Pdown&in(X>H) = BC+D+E =1, =1
Pdown&in(X>H) = A+E =1, =1

Up and in put S>H
Pup&in(X>H) = AB+D+E =1, =1
Pup&in(X>H) = C+E =1, =1

"Out" Barriers

Down and out call S >H
Cdown&out(X>H) = AC+F =1, =1
Cdown&out(X<H) = BD+F =1, =1

Up and out Call S<H
Cup&out(X>H) = F =1, =1
Cup&out(X<H) = AB+CD+F =1, =1

Down and out put S>H
Pdown&out(X>H) = AB+CD+F =1, =1
Pdown&out(X>H) = F =1, =1

Up and out put S>H
Pup&out(X>H) = BD+F =1, =1
Pup&out(X>H) = AC+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

For more information see our pricing plans.



Free Trial 

ResolutionPro is a derivative pricing library which supports the valuation, risk management and hedge accounting of derivatives & other financial instruments.
You can try ResolutionPro right now on a free trial basis.
 


Most Popular Posts 
 Interest Rate Swap Tutorial, Part 5 of 5, building your swap curve
 Interest Rate Swap Tutorial, Part 4 of 5, swap curve construction
 Interest Rate Swap Tutorial, Part 3 of 5, Floating Legs
 Interest Rate Swap Tutorial, Part 1 of 5, terminology
 Interest Rate Swap Tutorial, Part 2 of 5, Fixed Legs
 Business day conventions used for interest rate swaps & other derivatives
 Building a swap curve
 Currency options pricing explained
 Finding swap rates
 Black Scholes equity example

