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# VALUING PATH-DEPENDENT, EUROPEAN-STYLE OPTIONS ON MULTIPLE VARIABLES

In this section, I will continue my discussion on valuing path-dependent, European-style options on two variables and then extend my discussion to situations that comprise multiple variables. From the earlier sections, it is not difficult to see and understand how quickly the complications associated with the pricing of path-dependent options become. It is also easy to hypothesize how exponentially difficult the problem can get when the number of variables underlying these options starts increasing. Given this backdrop, it should not come as a surprise to the reader that it is more practical to use Monte Carlo methods to tackle such problems.

## Averaging Spread Options

[1]

In this example, I will discuss one variation of the spread option and averaging theme that is quite popular in the commodity markets. Called averaging spread options, this option gives the owner a payoff that is comprised of the difference between two correlated average prices. The payoff for such an option can be more succinctly written as follows:

where

In particular when n = m = 3. Table 5.13 shows the implementation for pricing an averaging spread call option. Although the illustration in Table 5.14 shows the results associated with one path, running this over 5,000 paths yields an option premium of \$4.83.

The reader may be interested in noting that it is possible for one to use the average-option-pricing algorithm and combine it with the spread- option algorithm. Clearly, doing this would only yield a proxy solution as compared to the results obtained using Monte Carlo simulation – as the effectiveness very much depends on the at-the-moneyness, correlation, and volatilities.

• [1] This option is quite widely used in the commodity markets when the hedger wants protection against the relative movements in the averages of two commodity prices (e.g., crude oil versus heating oil – a variation of a crack spread).

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