## Other Option Contracts

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The options described so far are standard, plain-vanilla options. Since the 1970s, however, markets have developed more complex option types.

Binary options, also called digital options pay a fixed amount, say Q, if the asset price ends up above the strike price cT = Q x I(ST - K) (6.21)

where I(x) is an indicator variable that takes the value of 1 if x > 0 and 0 otherwise. Because the probability of ending in the money in a risk-neutral world is N(d2), the initial value of this option is simply

These options involve a sharp discontinuity around the strike price. As a result, they are quite difficult to hedge since the value of the option cannot be smoothly replicated by a changing position in the underlying asset.

Another important class of options are barrier options. Barrier options are options where the payoff depends on the value of the asset hitting a barrier during a certain period of time. A knock-out option disappears if the price hits a certain barrier. A knock-in option comes into existence when the price hits a certain barrier.

An example of a knock-out option is the down-and-out call. This disappears if S hits a specified level H during its life. In this case, the knock-out price H must be lower than the initial price S0. The option that appears at H is the down-and-in call. With identical parameters, the two options are perfectly complementary. When one disappears, the other appears. As a result, these two options must add up to a regular call option. Similarly, an up-and-out call ceases to exist when S reaches H > S0. The complementary option is the up-and-in call.

Figure 6-9 compares price paths for the four possible combinations of calls. The left panels involve the same underlying sample path. For the down-and-out call, the only relevant part is the one starting from S(0) until it hits the barrier. In all figures, the dark line describes the relevant price path, during which the option is alive; the grey line describes the remaining path.

The call is not exercised even though the final price ST is greater than the strike price. Conversely, the down-and-in call comes into existence precisely when the other one dies. Thus at initiation, the value of these two options must add up to a regular European call

Because all these values are positive (or at worst zero), the value of cDO and cDI each must be no greater than that of c .A similar reasoning applies to the two options in the right panels.

Similar combinations exist for put options. An up-and-out put ceases to exist when S reaches H > S0. A down-and-out put ceases to exist when S reaches H < S0.

Barrier options are attractive because they are "cheaper" than the equivalent ordinary option. This, of course, reflects the fact that they are less likely to be exercised than other options. These options are also difficult to hedge due to the fact that a discontinuity arises as the spot price get closer to the barrier. Just above the barrier, the option has positive value. For a very small movement in the asset price, going below the barrier, this value disappears.

FIGURE 6-9 Paths for Knock-out and Knock-in Call Options

Down and out call Up and out call

FIGURE 6-9 Paths for Knock-out and Knock-in Call Options

Time

Down and in call

Time

Down and in call

Time

Up and in call

Time

Time

Up and in call

 J\ Barrier i \ / V/V Time Time Finally, another widely used class of options are Asian options. Asian options, or average rate options, generate payoffs that depend on the average value of the underlying spot price during the life of the option, instead of the ending value. The final payoff for a call is cj = Max(SAVE(f, T) - K, 0) (6.24) Because an average is less variable than an instantaneous value, such options are "cheaper" than regular options due to lower volatility. In fact, the price of the option can be treated like that of an ordinary option with the volatility set equal to a/ J3 and an adjustment to the dividend yield.3 As a result of the averaging process, such 3 This is only strictly true when the averaging is a geometric average. In practice, average options involve an arithmetic average, for which there is no analytic solution; the lower volatility adjustment is just an approximation. options are easier to hedge than ordinary options.