VOLUME 1, NUMBER 22/August 7, 1995
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DERIVATIVES 'R US is a weekly non-profit publication on the
Internet user group misc.invest.futures that provides a simple
non-technical treatment of various topics in derivatives. DRU
is written by Don M. Chance, Professor of Finance at the Center
for the Study of Futures and Options Markets at Virginia Tech.
He can be contacted at dmc @ vt.edu or by phone at 540-231-5061
or fax at 540-231-4487. DRU is for educational purposes only
and does not provide trading advice.
Back issues of DRU are available by anonymous ftp from
fbox.vt.edu/filebox/business/finance/dmc/DRU or can be accessed
using a Web browser at
http://fbox.vt.edu:10021/business/finance/dmc/DRU. The file
contents.txt can be viewed to see a list of old filenames and
topics available for reading or downloading.
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Before I start this week's essay, I'm announcing a new feature of DRU. For quite a few years, I've been collecting quotes about derivatives. Some are quite insightful, some are funny and most hold up extremely well when taken out of context, as all quotes are. Some of the speakers, particularly famous people, were not talking about derivatives but their words have applicability to the subject. I'm looking some day to publish these quotes in a short book (anybody know a good publisher for such an oddball topic?), but in the meantime I'll share one a week with you. Right now I have about 150 of them so that's about a three year supply and the list is growing. I will also occasionally throw in my own editorial response to whatever was said. This feature will appear at the end of the essay under the name DERIVATIVES QUOTE FOR THE WEEK.
COMPOUND & INSTALLMENT OPTIONS
This week't topic is compound and installment options. A compound option is quite simply an option on an option. You can buy a call that gives you the right to buy another call or a call that gives you the right to buy a put. You could also buy a put that gives you the right to sell a call or a put that gives you the right to sell a put. Thus you can have a call on a call, a call on a put, a put on a call or a put on a put. On the expiration date of the compound option, the holder simply compares its exercise price to the price of the underlying option and makes an exercise decision.
The compound option was created and analyzed in 1977. At that time it was recognized to be a brand new type of option. Whereas a standard option is modeled by assuming that the underlying asset follows a particular stochastic process and then setting up a no-arbitrage strategy, the compound option is slightly more complex. However, it can be shown that you can also construct a no-arbitrage strategy and solve for the compound option price. The formula bears some resemblance to the Black-Scholes formula because it is basically an option on a Black-Scholes-type option. While the Black-Scholes formula incorporates the univariate normal probability distribution, the compound option formula requires both the univariate and bivariate normal probability distributions. The bivariate probability reflects the joint probability that the underlying option will end up in-the-money and that the compound option would have been exercised, granting the holder a position in the underlying option.
One application of the compound option is in modeling a stock as an option. Yes, a common stock is indeed an option. It's simply an option on the assets of the firm. Consider an all-equity firm with assets worth $A. The firm then decides to issue a one-year zero coupon bond with a face value of $F. Now, at the end of the year, the bond matures and let $A* be the value of the assets at that time. If $A > $F, the firm is able to pay off its debt. The bondholders receive $F and the stockholders get the rest, $A - $F. If $A <= $F, the firm is in default. The bondholders get $A and the stockholders get nothing. The payoff to the stockholders is exactly like a call option on the assets with an exercise price of $F. The option is written by the bondholders.
Now suppose someone creates an ordinary call option on the stock. Then that call option is a compound option. The Black-Scholes formula is not correct for valuing that option. The assumption of constant volatility of the stock is not correct because the stock's volatility will be primarily determined by the firm's leverage, which will be constantly changing as the market values of the equity and debt change.
Thus, a share of stock is indeed an option. (Take that, Mr. Buffett, Mr. Lynch and all you equity types who think options ought to be outlawed) and an ordinary option on a stock whose firm has debt is really a compound option.
The compound option formula is also useful for pricing an American call option. It can be shown that an American call is a combination of two standard European options and one compound option. I'll cover this in more detail in a later issue.
In recent years compound options have begun to surface in the real markets, not just as the theoretical creations of academics. More commonly referred to as "installment options," these instruments have proven to be somewhat popular. An installment option is an option in which the premium is paid in installments over the life of the option. For example, consider a two-year European call on the S&P 500. Suppose instead that we structure the call as an installment option, specifying that a premium is paid up front and another premium of the same amount is paid one year later. In other words, at the end of the first year, the holder has the right to decide if he would like to pay the second premium and continue with the option. This is the same as a compound option. Of course, the option could be designed to have multiple premium dates, would make it a multiple compound option. Calculating prices is normally done using numerical methods. The premiums on the compound option will collectively add up to more than if the option had the same maturity but was a straight European option. In case, you are wondering if there is a distinction between a compound option and an installment option, there indeed is but it is slight and need not be the case. An installment option usually has the installment premiums all set equal to each other though there is no reason why that has to be the case. If a compound option fit this requirement, then the premium would be forced to equal the exercise price so that upon exercise of the compound option, the holder would end up paying the same at that point as he had up front. Also an installment option would be only a compound call on a call or a call on a put.
The advantage of a compound or installment option is the greater flexibility if offers the holder. Part of the way through the life of the option, the holder can decide if the position is worth continuing. If he decides to abandon the position by simply not paying the premium, he holds his loss to the premiums already paid. For OTC options, which typically have little or no secondary market, this can be particularly useful. Also the compound option, being cheaper up front, could be more attractive to some investors who are interested in conserving as much capital as possible.
Installment options have been particularly popular in the Canadian retail market.
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DERIVATIVES QUOTE FOR THE WEEK
"No recent non-political topic that I know of has inspired more hyperbole."
David Mullins, Long Term Capital Management and former Fed governor, quoted in Risk, March, 1994, p. 30.
(Obviously, spoken before l'affaire Simpson)