The BARRA Brainteaser for Spring 1997

Potpourri of Exotic Options on CBOE
(Conundrums, Brainteasers, and Other Enigmas) Exchange

by Eugene Reznik

A new exchange—the Conundrums, Brainteasers, and Other Enigmas Exchange, or CBOE—recently opened in Berkeley, California, to accommodate investors interested in derivatives with odd payoffs and stocks with bizarre behavior. It was commonly agreed that BARRA Brainteaser T-shirts were to be used as the principal currency for settling all transactions on CBOE.

Soon after the new exchange opened, you set up a shop to provide quantitative pricing for securities traded on CBOE. Your first client wants you to accurately value each of the options below, and promises you one unit of CBOE currency (a Brainteaser T-shirt, of course) if you can do it. You can assume that the interest rates are zero and the T-shirts are infinitely divisible.

The CBOE digital option

This is a European-style option for which the payoff at maturity is contingent upon the prices of a pair of underlying assets. The option will pay a fixed number N of Brainteaser T-shirts only if the price of Asset A at maturity exceeds that of Asset B one week prior to maturity. The assets for which such options are available must have the same price at the time the option is written, and are assumed to have a fifty-fifty chance of going up or down during any one-week period, by the same fixed amount.

The Banach1 option

Many of the companies whose stocks are traded on CBOE are such that only a pair of them make up a market sector. For these sectors, it is true that during any period, the stock price of either of the rival companies is equally likely to gain value. It is also certain that the price of only one stock goes up, while that of the other remains unchanged.

Banach options on CBOE are written for pairs of stocks described above. These options expire as soon as one of the stock prices increases R times after the option is written. At that point, the options pay out S T-shirts, where S is the number of stock jumps by which one of the rival companies has "beaten" the other.

Knockout options on P&S 500 stocks

The quick rise in popularity of the new exchange has led to the creation of several stock indexes. Puzzle & Solvers' 500 (P&S 500) Index was set up to reflect the performance of CBOE stocks with the following peculiar characteristic: The monthly returns on each of the P&S 500 stocks take one of M possible values with equal probability.

The knockout options have been created in two flavors. Like the Banach option, neither type has a preset exercise date. The first expires when the underlying asset has exhibited all M possible values of the returns since the time the option was written. The second type expires as soon as the price of the underlying asset undergoes K identical changes consecutively.

The payoff to each of the CBOE’s knockout options is equal to the sum of all the dividends paid by the underlying asset during the life of the option. It is well-known that the P&S 500 stocks pay a constant dividend of d T-shirts per month.

Footnotes

¹ This problem is based on one introduced by the Austro-Hungarian mathematician Stefan Banach. (back to text)