Another example of an exotic option that trades in the OTC market is called the pay-later option. A pay-later option is essentially a vanilla option in which the option premium is paid at the time of exercise only when the option finishes in-the-money (regardless of the amount of in-the-moneyness). Thus, if the option is never exercised (i.e., finishes out-of-the-money), nothing needs to be paid by the purchaser of the option. Based on the description of the profit associated with the purchase of the option, it is straightforward to see that the profit profile^{[2]} associated with the purchase of a pay-later call option is given by

To obtain the formula to value this type of a call option, one would need to evaluate the expression

and solve for. Doing this yields

(5.2a)

The analogous formula for a pay-later put option is given by the expression

(5.2b)

Table 5.2 shows the implementation of equations (5.2a) and (5.2b).

TABLE 5.2 Valuation of Pay-Later Options

From the results in Table 5.2, it can be seen that the pay-later option values are higher than their vanilla counterparts.^{[3]} The reason for this can be easily explained if the reader observes from equations (5.2a) and (5.2b) that the conditional premiums paid for these options are basically the future value of the vanilla options that are conditioned on the probability of exercise. Hence the lower the probability of exercise, the higher this conditional premium.

Nonlinear Payoff Options

^{[4]}

Another example of an exotic option that is sometimes quite popular is the nonlinear payoff option. See Ravindran (1997) for details. The purchaser of the nonlinear payoff call option gets on maturity date a payoff of

where

To derive the formula associated with the nonlinear payoff, one would need to first observe that the formula for the option price, (NLP), can be obtained by evaluating the expression . Doing this, allows one to get

As before, observing that where and , it readily follows that

(5.3a)

Where

It is easy to show that for the nonlinear payoff put option, the expression simplifies to

(5.3b)

The pricing formulae in equations (5.3a) and (5.3b) have been implemented in Table 5.3.

As can be seen from Table 5.3, when a is 1, the answers agree with those in Table 3.4. Furthermore, when a is 2, these types of options are also called power options. Although the discussion so far has been restricted to the valuation of European-style, path-independent options, valuing such options with early-exercise features is just an easy extension of what I discussed in Chapter 3 using equations (3.14a), (3.14b), and (3.14c). Succinctly put, to value these options, one only needs to modify the maturity payoffs associated with the options that are being valued – while keeping all the other things similar.

[1] To see an application of this type of option, consider the purchase of an interest rate cap. If the cap premium turns out to be expensive and the hedger is of the opinion that the cap purchase is not necessary (i.e., the LIBOR on the each of the caplet maturity dates never will exceed the strike rate), as an insurance, the hedger can instead purchase a pay-later cap in which the conditional premium for each caplet is only paid if the caplet goes in-the-money on each caplet expiry date.

[3]As the astute reader will realize, the pay-later option can easily be replicated by buying a vanilla call option and selling a binary call option (where the size of the bet payoff is the pay-later premium). The reader is referred to Ravindran (1997) for further details.

[4] An example of this type of option was seen in the European and Asian retail markets when currency warrants were issued by banks to give investors a leveraged upside with a limited downside. Another example includes the use of these options to hedge nonlinear exposure to movements in the underlying asset price.

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