VALUING PATH-INDEPENDENT, EUROPEAN-STYLE OPTIONS ON TWO VARIADLES

In the earlier sections, I discussed examples of exotic options where the payoffs on the option maturity were a function of the prices of a single variable. In this section, I will discuss the extension of these concepts to entertain the valuation of options on two variables to give the reader an awareness of the ad-hoc assumptions and methods practitioners use to price such options.

Exchange Options

^{[1]}

One example of a widely used exotic option on two variables is the exchange option, which was discussed by Magrabe in 1978. This option also happens to be a special case of spread option (something I will discuss in the section following this).

An exchange-option purchaser gets a maturity payoff that is of the form max() where the owner of the option has the right to exchange anasset, Rj, at time T for another asset ST – where RT and ST are the prices of the two underlying assets on which the exchange option is written.

To develop the pricing formulae for the exchange option, (exc), I will as before assume thatand each has normal pdf with means of

and

respectively and variances of and respectively. Since there are two variables at play here, it is reasonable to expect the correlation between these variables to be a contributing factor when pricing such options.

To incorporate the correlation as an assumption, I will assume that the natural logarithm of the continuously compounded stock price returns have a correlation ofwhere (i.e., – an assumption that is consistent with the way the standard deviation of the natural logarithm of the continuously compounded stock price returns defined.^{[2]}

As before, to compute the expression for one has to take the present value of the expectations of the payoff with respect to and . Doing this yields

Letting ^{[3]} represent the joint probability density function of , and, it readily follows that

(5.16)

Since the joint probability density function,, can be rewritten as a product of a conditional and marginal pdf, namely ^{[4]}, equation (5.16) becomes

(5.17)

To simplify the first integral (and the terms surrounding it) in equation (5.17), first observe that

and

As a consequence, it readily follows

(5.18a)

^{[5]}

where

Equation (5.18a) can further be simplified to obtain

where.

Hence one can conclude that the first integral (and the terms accompanying it) in equation (5.17) can be simplified to

(5.18b)

Similarly, it can be concluded that the second integral (and the terms accompanying it) in equation (5.17) simplifies to

(5.18c)

Putting equations (5.18b) and (5.18c) together, one can arrive at the expression

(5.19)

where

Equation (5.19) has been implemented in Table 5.8.

As the reader will note from the payoff of the exchange option, the notion of a call or put is immaterial since the value of this option is driven by the fact that upon exercise, one is exchanging one asset for the other (and hence the underlyings are interchangeable).

TABLE 5.8 Valuation of Exchange Options

[1] This option is used when the investor is interested in taking a view on the relative movements in the two underlying assets. One application that is quite popular in the fixed income markets is to bet on the yield curve steepening or flattening. See Das (1996) and Ravindran (1997) for a more detailed discussion on this.

[2] It is important for the reader to note that a correlation ofbetween the stock returns does not imply that the stock prices have the same correlation. As pointed out by Ravindran (1997), if, then and

[3] Sinceandhave a bivariate lognormal pdf, the joint pdf of these variables takes the form of, where for,.

[4] In this representation, n (RT) represents the pdf of RT which is given by the expressionfor.represents the conditional pdf ofwhich is given by the expressionfor , whereand.

[5] This can be obtained by using the result in Ravindran (1997).

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