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