# Pricing Barrier Option **Barrier Option Valuation** Barrier options are set up conditionally, so that within the life of the option, a barrier may or may not be reached. The barrier option is dependent on an “extreme” price, either high or low, attained by the underlying. If a barrier is reached, or touched, the constraints are triggered. The two types of contracts, knock-in barrier options and knock-out barrier options, each respond differently when the barrier is "triggered". If a *knock-out* option's underlying touches the barrier, the option is eliminated and the holder receives a rebate. Conversely, a *knock-in* option touches the barrier to activate the option. If the *knock-in* option never reaches the barrier, the holder will receive a rebate. For single barrier option, there are four possible variations to the barrier: * Up and In * Up and Out * Down and In * Down and Out The option is considered knocked-in (knocked-out) if, on a barrier observation date, the underlying asset’s closing price exceeds (for up-barriers) or falls below (for down-barriers) the barrier level corresponding to that observation date. Once knocked-in (knocked-out), the option’s status is no longer subject to further modification by subsequent barrier observation events. Payoff at maturity may take the form of a Call, Put or Note. The payoff is contingent on the option having been knocked in (in the case of knock-in option), or having *not* been knocked out (in the case of a knock-out option), on some barrier observation date prior to maturity. For European barrier options, a vanilla option is equal to the sum of the premiums for a knock-in and its corresponding knockout (assuming no rebates and equivalent input parameters). Since at expiration, only one option can have a value. The Double Barrier is a discretely-observed barrier option on a single asset. An initial stock price can be fixed at some future date. The option is said to be *triggered* when the stock price, relative to its initial price, exceeds the high barrier or falls below the low barrier on any barrier observation date. The option may be Knock-In (where triggering enables the payoff) or Knock-Out (where triggering disables the payoff). For Knock-Out options, a rebate amount may be specified to be paid if the option becomes triggered. The payoff at maturity, if active, may be a relative-return call or put payoff on the deal Notional. An average Barrier is a call or put option on the average of the asset level over a discrete set of observation dates, with a knock-in or knock-out barrier applied to the running average on a subset of those dates. More precisely, the option is said to be *triggered* when the running average of the asset price exceeds the high barrier or falls below the low barrier on any barrier observation date. In the case of a *knock-in* barrier, the buyer of the option receives the call/put payoff at maturity if the option has been triggered, and receives zero otherwise. In the case of *knock-out* barrier, the buyer receives the call/put payoff at maturity if the option was not triggered during the lifetime of the deal, while the buyer receives a knock-out rebate (at the barrier date or at maturity) if the option is triggered. The option is said to be *triggered* when the asset level exceeds the high barrier or falls below the low barrier at any time within a barrier observation interval (or on a barrier observation date). The option may be Knock-In (where triggering enables the payoff) or Knock-Out (where triggering disables the payoff). For Knock-Out options, a rebate amount may be specified to be paid if the option becomes triggered. The payoff at maturity, if active, may be a call, a put, a digital call, or a digital put payoff on the deal Notional. **In-out parity** **There are some relationships among knock-in, knock-out, and vanilla options.** ![](https://2046872026-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2Fbarrier-option-pricing%2F-MXcmcFFH3GE4f2lmwwM%2F-MXcmetT44B_27Eo_WUL%2F0.png?generation=1617739558850158\&alt=media) ![](https://2046872026-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2Fbarrier-option-pricing%2F-MXcmcFFH3GE4f2lmwwM%2F-MXcmetUinQt203SbSRr%2F1.png?generation=1617739558851313\&alt=media) ![](https://2046872026-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2Fbarrier-option-pricing%2F-MXcmcFFH3GE4f2lmwwM%2F-MXcmetV_9J_4IzhMk23%2F2.png?generation=1617739558850228\&alt=media) ![](https://2046872026-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2Fbarrier-option-pricing%2F-MXcmcFFH3GE4f2lmwwM%2F-MXcmetWMEEJMSlH9tpu%2F3.png?generation=1617739558850308\&alt=media) Reference: