Single barrier caps and floors are modified versions of vanilla interest rate caps and floors. A barrier cap or floor only provides a payoff when the reference interest rate is either above or below the defined barrier level when each individual caplet or floorlet expires. For this reason, the cost of barrier caps and floors is lower than for the equivalent vanilla instrument. Common usage classifies these instruments based on the required relativity between the reference rate and barrier rate, and the 'event' that occurs if this relativity is observed on the fixing date. In general, the four classes of barrier options are as follows:
Down & In
Down & Out
Up & In
Up & Out
For at-expiry options, the use of 'Up' and 'Down' to classify the option type is a bit misleading because it suggests that the reference rate must travel in a particular direction between inception of the deal and it's expiration in order for the 'In' or 'Out' event to be triggered. In reality, all that is checked at the expiration date of each caplet and floorlet for a 'Down' barrier option is that the reference rate at fixing is below the barrier. Similarly for 'Up' options, the 'In' or 'Out' event is only triggered if the reference rate at expiration is above the barrier level.
At-expiry barrier options written on an interest rate are valued by viewing them as a combination of vanilla caps or floors and digital options. Because all of these constituent options are European style, each can be valued using a simple variant of the Black-Scholes model.
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