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 AtExpiry 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 noarbitrage 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 inthemoney at expiration if the reference rate fixes at a level lower than the prespecified 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 prespecified 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 
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