Another path-dependent exotic option that gets used extensively by hedgers is the installment option which was discussed by Geske in 1979. In the instance of a vanilla or binary or averaging option, the option purchaser paid the premium at the inception of the option contract. In the instance of a pay-later option, the option purchaser paid the premium on the maturity of the option date only if the option finished in-the-money. Unlike these options, an installment option purchaser pays the option premiums in two installments (a compulsory one which is paid at the inception of the contract and an optional one which is paid at some prespecified time in the future). More precisely, suppose that one is interested in purchasing a vanilla call option expiring at time T using two installments, where и is the time at which the second optional installment premium X is paid (where t < и <T) and t is the time at which the first (initial) compulsory installment premium CCt is paid, the payoff to the payor of the premium at time и is given by

(5.9)

where (van) represents the market value of vanilla call option of life T – и at time и when the stock price is. Clearly if the value of (van) is lesser than X, the holder of this option will choose not to exercise the option (i.e., pay X).^{[2]}

Taking the expectations of equation (5.9), it readily follows that

Taking the expectations, with respect to Su yields

(5.10)

where , for

S_{*} is the solution of the equation

To evaluate equation (5.10), first observe that the third integral and the terms accompanying it are

Letting, the above simplifies to

(5.11a)

Now observe that the second integral (and the terms accompanying it) in equation (5.10) can be rewritten as

Letting, one gets

where

Rewriting the lower limit yields

(5.11b)

whererepresents the cumulative probability of a standard bivariate normal variate.^{[3]}

Finally observe that the first integral (and the terms accompanying it) in equation (5.10) can be rewritten as

Letting , one gets

Making the substitution w = z – β, one gets

(5.11c)

Putting equations (5.11a) to (5.11c) together, it can be seen that for the call-on-call option the initial premium can be computed using the formula,

(5.12a)

where

(5.12b)

and, is the solution to

(5.12c)

Going through similar analysis, it can be shown that the initial premiums associated with a put-on-call option, call-on-put option, and put-on-put option can be obtained using the following formulae.

(5.13)

(5.14)

(5.15)

The implementation of equations (5.12a) to (5.12c) and (5.13) is shown in Table 5.7.

As can be seen from Table 5.7, implementing this algorithm involved the calculation of cumulative bivariate probabilities^{[4]} (a function that Microsoft Excel does not come with). To do this, I used the algorithm provided in Hull (2012). Furthermore, to obtain the value of S* (in cell BIO), I manually guessed at this value (using Excel's Goal Seek feature) to

TABLE 5.7 Valuation of Call-on-Call and Put-on-Call Options

ensure the value of cell B13 is 0. In practice, this is done using Newton's method.^{[5]}

It is not uncommon to see installment options whereby the option premium is paid in 12 installments (one of which is the compulsory premium that is paid at the inception of the contract). As the astute reader will realize, in this instance, the problem quickly gets exponentially unmanageable due to the fact that one would have to evaluate a 12-dimensional integral – further iterating the importance of use of efficient numerical methods and techniques in solving such problems.

[1] These options are used when a hedger wants to lock in the potential movements in volatility and market prices today but is not sure if these options are actually required (e.g., the hedger may have submitted a bid and is waiting for the results of the tender so as to acquire the raw materials needed to fulfill the contract). Because the hedger is not sure if the option is required, the hedger is interested in paying for the option in installments. Another application of these options is discussed in Chapter 9.

[2] This type of option is also called a call-on-call option. Other combinations include the put-on-call option, the call-on-put option, and the put-on-put option where the payoffs at time и to the payor of the option premiums are given by the expressions , , respectively. In writing these payoffs, I have assumed that X represents the second installment premium that is paid (received) at time и to receive (sell) the underlying vanilla call or put option with life T – и when the stock price is Su (where the values of the call and put option at time и are given by(van) and (van) respectively).}],{{I assumed throughout this illustration thatandare all constant throughout the time interval (t,T). Thusfor

[3] is the cumulative bivariate standard normal pdf that is given by the expression, which can be alternatively written as

[4] The online content contains a spreadsheet that allows the reader to compute these probabilities.

[5] By defining , one can use the recursive relation forwhereis the initial guess at the solution. It can also be easily shown that given this form of,

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