VALUING PATH-INDEPENDENT, EUROPEAN-STYLE OPTIONS ON A SINGLE VARIABLE

In this section, I will discuss three examples of widely used exotic options on a single index that are path independent,^{[1]} so as to give the reader a better appreciation for some of the mathematical techniques used by practitioners to value these options.

Binary Options

^{[2]}

One example of a path-independent exotic option is the S&P 500 linked binary option that trades under the symbol BSZ on the Chicago Board Options Exchange (CBOE). The binary call option pays off a bet value of $ 1 to the option holder if the option finishes in-the-money. Thus, unlike the vanilla call option, this option's payoff can be written as

where refers to the S&P 500 index value on option maturity.

To obtain the formula to value this type of a European-style call option, , one needs to evaluate the expression

Doing this yields

(5.1a)

The analogous formula for a binary put is given by the expression

(5.1b)

TABLE 5.1 Valuation of Binary Options

Table 5.1 shows the implementation of equations (5.1a) and (5.1b).

As can be seen in Table 5.1, the values of the call and put options are given respectively by the values of 0.4933 and 0.3936. In the context of BSZ options, since each contract has a lot size of 100, the value of the BSZ options can be obtained by multiplying the values in cells B10 and B11 by 100. Thus, to execute an order on BSZ options, the trader has to stipulate the number of contracts associated with that trade. This is quite different when compared to the binary option transactions in an OTC market since when trading in OTC markets, the trader has to only stipulate the bet amount (which acts as a total notional amount associated with the trade). Thus if the bet size was USD$1 million, then the quantities in cells B10 and B11 are simply multiplied by USD$1 million to arrive at the respective option premiums.

[1] Path-independent options refer to options that pay the option owner at the time of exercise a value that is only dependent on the value of the index at the moment of exercise.

[2] In addition to being traded on the exchange, binary options are also used by investors to take a view on market directions. Binary options do exist as principal- protected note structures. An example of such a structure is a one-year note that is linked to S&P 500 index value. The purchaser of this note would be guaranteed to receive his/her principal and coupon of (5% * R) at the end of one year (note maturity) – where R refers to the fraction of times the closing S&P index value exceeds 1700 during each trading day in the life of the note. See for example Ravindran (1993b).

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