Previous Topic

Next Topic

At Expiry Double Barrier Options

At-expiry double barrier options are analysed in a similar manner to their single barrier counterparts. The instruments are priced by forming a replicating portfolio which is structured to yield exactly the same cashflow as the barrier option in all states of nature. As the prices for the components (vanilla caps/floors and European digital options) of the replicating portfolio are known, then the value of the barrier option can be simply determined by summing the values of the component options.

The composition of the various replicating portfolios is set out in Table 1.

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

Barrier Option Type

 

Replicating Portfolio

 

Restrictions

 

 

Position

Option Type

Strike

Payoff

 

 

 

 

 

 

 

 

 

 

Down & In / Up & In Cap

 

Long

Put Digital

LB

 

 

 

Short

Vanilla Floor

LB

)

 

 

 

 

Long

Vanilla Floor

X

 

 

 

 

Long

Vanilla Cap

UB

 

 

 

 

Long

Call Digital

UB

 

 

 

 

 

 

 

 

 

 

Down & In / Up & Out Cap

 

Same as Down & In Single Barrier Cap

 

 

 

 

 

 

 

 

Down & Out / Up & In Cap

 

Same as Up & In Single Barrier Cap

 

 

 

 

 

 

 

 

Down & Out / Up & Out Cap

 

Long

Vanilla Cap

LB

 

 

 

Long

Call Digital

LB

 

 

 

 

Short

Vanilla Cap

UB

 

 

 

 

Short

Call Digital

UB

 

 

 

 

 

 

 

 

 

 

Down & In / Up & In Floor

 

Long

Put digital

LB

 

 

 

Long

Vanilla Floor

LB

)

 

 

 

 

Long

Call Digital

UB

 

 

 

 

Short

Vanilla Cap

UB

 

 

 

 

Long

Vanilla Cap

X

 

 

 

 

 

 

 

 

 

 

Down & In / Up & Out Floor

 

Same as Down & In Single Barrier Floor

 

 

 

 

 

 

 

 

Down & Out / Up & In Floor

 

Same as Up & In Single Barrier Floor

 

 

 

 

 

 

 

 

Down & Out / Up & Out Floor

 

Long

Put digital

UB

 

 

 

Long

Vanilla Floor

UB

)

 

 

 

 

Short

Put digital

LB

 

 

 

 

Short

Vanilla Floor

LB

 

 

 

 

 

 

 

 

 

 

For the double barrier options, define:

the upper barrier level

the lower barrier level

Note that for at-expiry options, some of the listed double barrier caps and floors are effectively the same as a single barrier option. For example, consider the following payoff diagram for a Down & In / Up & Out Cap.

Figure 1: Payoff Diagram for a Down & In / Up & Out Cap

Given the nature of the barriers, this option will only have a positive payoff if the reference interest rate falls between the lower boundary () and the exercise price (). That means that the upper boundary () is effectively irrelevant and that this option is equivalent to a Down & In single barrier cap. A similar argument can be made for 3 of the other listed at-expiry double barrier options. Given the redundancy, these double barrier options are not supported in this development.

For completeness, it may be useful to again demonstrate that the replicating portfolios listed in Table 1 are an appropriate basis for valuing the double barriers. Consider first the Down & In / Up & In Cap, which has a payoff function as presented in Figure 2.

Figure 2: Payoff Diagram for a Down & In / Up & In Cap

The reference interest rate at the maturity of each caplet can fall into 1 of the 4 depicted zones. We demonstrate in Table 2 that the payoff from the replicating portfolio and the Down & In / Up & In Cap are the same in each of those zones.

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

Zone

Level of Reference Rate at Maturity

 

Payoff to Barrier Cap

 

Payoff to Replicating Portfolio

 

 

 

 

 

Option Type

Payoff

 

 

 

 

 

 

 

1

 

 

Put Digital (Strike = LB)

 

 

 

 

 

Vanilla Floor (Strike = LB)

 

 

 

 

 

Vanilla Floor (Strike = X)

 

 

 

 

 

Vanilla Cap (Strike = UB)

 

 

 

 

 

Call Digital (Strike = UB)

 

 

 

 

 

Net Payoff

 

 

 

 

 

 

 

2

 

 

Put Digital (Strike = LB)

 

 

 

 

 

Vanilla Floor (Strike = LB)

 

 

 

 

 

Vanilla Floor (Strike = X)

 

 

 

 

 

Vanilla Cap (Strike = UB)

 

 

 

 

 

Call Digital (Strike = UB)

 

 

 

 

 

Net Payoff

 

 

 

 

 

 

 

3

 

 

Put Digital (Strike = LB)

 

 

 

 

 

Vanilla Floor (Strike = LB)

 

 

 

 

 

Vanilla Floor (Strike = X)

 

 

 

 

 

Vanilla Cap (Strike = UB)

 

 

 

 

 

Call Digital (Strike = UB)

 

 

 

 

 

Net Payoff

 

 

 

 

 

 

 

4

 

 

Put Digital (Strike = LB)

 

 

 

 

 

Vanilla Floor (Strike = LB)

 

 

 

 

 

Vanilla Floor (Strike = X)

 

 

 

 

 

Vanilla Cap (Strike = UB)

 

 

 

 

 

Call Digital (Strike = UB)

 

 

 

 

 

Net Payoff

 

 

 

 

 

 

 

As a final example consider the Down & Out / Up & Out Floor shown in Figure 3. Unlike the previous example, note that the replicating portfolio is not just a combination of the constituent components that are used to replicate a Down & Out single barrier floor together with those for an Up & Out single barrier floor. Rather, the portfolio consists of a long and short position in an Up & Out floor, one struck at the upper boundary () and the other struck at the lower boundary level ().

Figure 3: Payoff Diagram for a Down & Out / Up & Out Floor

All outcomes at maturity can again be broken down into 4 zones based on the position of the reference interest rate. Table 3 presents the payoff structures in each case, and these verify the validity of the chosen portfolio.

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

Zone

Level of Reference Rate at Maturity

 

Payoff to Barrier Cap

 

Payoff to Replicating Portfolio

 

 

 

 

 

Option Type

Payoff

 

 

 

 

 

 

 

1

 

 

Put digital (Strike = UB)

 

 

 

 

 

Vanilla Floor (Strike = UB)

 

 

 

 

 

Put digital (Strike = LB)

 

 

 

 

 

Vanilla Floor (Strike = LB)

 

 

 

 

 

Net Payoff

 

 

 

 

 

 

 

2

 

 

Put digital (Strike = UB)

 

 

 

 

 

Vanilla Floor (Strike = UB)

 

 

 

 

 

Put digital (Strike = LB)

 

 

 

 

 

Vanilla Floor (Strike = LB)

 

 

 

 

 

Net Payoff

 

 

 

 

 

 

 

3

 

 

Put digital (Strike = UB)

 

 

 

 

 

Vanilla Floor (Strike = UB)

 

 

 

 

 

Put digital (Strike = LB)

 

 

 

 

 

Vanilla Floor (Strike = LB)

 

 

 

 

 

Net Payoff

 

 

 

 

 

 

 

4

 

 

Put digital (Strike = UB)

 

 

 

 

 

Vanilla Floor (Strike = UB)

 

 

 

 

 

Put digital (Strike = LB)

 

 

 

 

 

Vanilla Floor (Strike = LB)

 

 

 

 

 

Net Payoff

 

 

 

 

 

 

 

Return to www.derivativepricing.com website

Copyright 2013 Hedgebook Ltd.