# At-Expiry Single Barrier Options

The following table lists the 8 potential single barrier options covered in this development. For each option, we list the combination of digital options and vanilla caps and floors that can be used to replicate the payoff structure for the barrier option, and which can therefore be used to value the barrier option at any time prior to maturity. We also impose a number of restrictions on the relativity between the option's barrier and strike price to either ensure that a payoff is feasible or to ensure that the barrier option is distinct from an equivalent vanilla cap or floor.

Table 1: Replicating Portfolios for At-Expiry Barrier Caps and Floors

 Barrier Option Type Replicating Portfolio Restrictions Position Option Type Strike Payoff Down & In Cap Long Put digital B Short Vanilla Floor B Long Vanilla Floor X Up & In Cap Long Vanilla Cap B Long Call Digital B Down & Out Cap Same as Up & In Cap Up & Out Cap Same as Down & In Cap Down & In Floor long Put digital B long Vanilla Floor B Up & In Floor Long Call Digital B Short Vanilla Cap B Long Vanilla Cap X Down & Out Floor Same as Up & In Floor Up & Out Floor Same as Down & In Floor

Notation used in this and the other tables in this section is as follows:

maturity date of the optlet reset period

barrier level for a single barrier option

the strike level of the reference interest rate

value of the reference rate at the maturity date of the optlet

To help explain the information contained in the table, consider a Down & In Cap. For each caplet associated with this deal, the holder will receive the standard cap payoff at maturity of if the reference rate is below the barrier rate. With reference to Figure 1 below, the payoff is positive for those values of the reference rate that fall between the barrier level () and the exercise price (). At all other ending values, the payoff to this caplet is zero.

Figure 1: Payoff Diagram for a Down & In Cap

We now need to demonstrate that the payoff to the replicating portfolio exactly matches the payoff to the Down & In Cap for all potential values of the reference rate at maturity. These rates fall into 3 zones, as set out in the following Table:

Table 2: Equivalence Between Payoffs to a Down & In Cap and the Chosen Replicating Portfolio

 Level of Reference Rate at Maturity Payoff to Barrier Cap Payoff to Replicating Portfolio Option Type Payoff Put Digital (Strike = B) Vanilla Floor (Strike = B) Vanilla Floor (Strike = X) Net Payoff Put Digital (Strike = B) Vanilla Floor (Strike = B) Vanilla Floor (Strike = X) Net Payoff Put Digital (Strike = B) Vanilla Floor (Strike = B) Vanilla Floor (Strike = X) Net Payoff

These results demonstrate that the payoff to the replicating portfolio is equivalent to the payoff from the Down & In Cap in all possible states of nature on the maturity date of the caplet. A simple no-arbitrage argument can therefore be invoked to ensure that the current values of the Down & In Cap and the replicating portfolio must also be equal.

Figure 2: Payoff Diagram for a Down & In Floor

The constituents of the replicating portfolio can be similarly derived for all of the other considered options. As one more example, consider the Down & In Floorlet depicted in Figure 2.

Note that the floorlet is only in-the-money at expiration if the reference rate fixes at a level lower than the pre-specified barrier level, in which case the payoff structure is the same as a vanilla floorlet (). Also note that a logical restriction on this type of instrument requires that the pre-specified barrier rate must be lower than the strike price. Barrier floors with the barrier set above the strike rate are effectively the same as vanilla floors.

As before, the payoff from the replicating portfolio can be assessed by considering the level of the reference rate in three stages. As shown in Table 2, the payoff to the replicating portfolio is the same as the payoff to the Down & In Floor, and we can therefore value the floor as the sum of the values of the replicating portfolio.

Table 3: Equivalence Between Payoffs to a Down & In Floor and the Chosen Replicating Portfolio

 Level of Reference Rate at Maturity Payoff to Barrier Floor Payoff to Replicating Portfolio Option Type Payoff Put Digital Vanilla Floor (Strike = B) Net Payoff Put Digital Vanilla Floor (Strike = B) Net Payoff Put Digital Vanilla Floor (Strike = B) Net Payoff