^*Pseudonym

^*This chapter contains some references to options. Readers completely unfamiliar with options may find it helpful (although not essential) to first read the four-page primer in the appendix.

^*The facts related to Fletcher’s employment at Kidder Peabody were obtained from court-case summaries and articles appearing in Business Week (October 24, 1994), Fortune (July 5, 1999).

^*Readers unfamiliar with options may find it useful to consult the short primer on options in the appendix.

^*The strike price is the price at which the option buyers could buy the stock by exercising their options. Of course, they would exercise their options only if the market price was above the strike price at the time of the option expiration.

^*As of March 2000, Lescarbeau’s average annual compounded return had risen to 70 percent.

^*Although bonds pay a steady return, many less sophisticated investors do not sufficiently appreciate the fact that price declines in bonds due to higher interest rates can outweigh interest income, resulting in negative total returns. With interest rates at long-term lows, the danger of negative total returns in bonds is significant, particularly if Masters is correct in his expectations for increased inflation over the long term.

^*It is recommended that readers unfamiliar with options first review the brief primer on options in the appendix before reading this chapter.

^*A probability distribution is simply a curve that shows the probabilities of some event occurring—in this case, the probabilities of a given stock being at any price on the option expiration date. The x-axis (horizontal line) shows the price of the stock. The y-axis (vertical line) shows the relative probability of the stock being at different prices. The higher the curve at any price interval, the greater the probability that the stock price will be in that range when the option expires. The area under the curve in any price interval corresponds to the probability of the stock being in that range on the option expiration date. For example, if 20 percent of the area under the curve lies between 50 and 60, it implies that there is a 20 percent chance of the stock being between 50 and 60 on the option expiration date. As another example, if 80 percent of the area under the curve corresponds to prices under 60, the 60 call option, which gives the holder the right to buy the stock at 60, would have an 80 percent chance of expiring worthless.
The shape of the probability distribution curve, which is a snapshot of the probabilities of prices being at different levels on the option expiration date, will determine the option’s value. The true shape of this curve is unknown, of course, and can only be estimated. The assumptions made regarding the shape of this curve will be critical in determining the value of an option. Two traders making different assumptions about the shape of the probability distribution will come to two different conclusions regarding an option’s true value. A trader who is able to come up with a more accurate estimate of the probability distribution would have a strong edge over other traders. The standard approach, which is based on the Black-Scholes formula, assumes that the probability distribution will conform to a normal curve [the familiar bell-shaped curve frequently used to depict probabilities, such as the probability distribution of IQ scores among the population]. The critical statement is that it “assumes a normal probability distribution.” Who ran out and told these guys that was the correct probability distribution? Where did they get this idea?

^*See note starting on page 237.

^*To be precise, the representation is a lognormal curve, which is a normal curve of the log values of stock prices. In a lognormal curve, an increase by a factor x is considered as likely as a decrease by a factor 1/x. For example, if x = 1.25, a price increase by a factor of 1.25 (25 percent) is considered as likely as a price decrease by a factor of 1/1.25, or 0.80 (20 percent). The lognormal curve is a better fit than the normal curve because prices can rise by any amount, but can decline only by 100 percent. If applied to prices instead of the log of prices, the symmetry of a normal curve could only be achieved by allowing for negative prices (an impossible event), which in fact is what some early option theoreticians did.

^*Evan G. Gatev, William N. Goetzmann, and K. Geert Rouwenhort. Pairs Trading: Performance of a Relative Value Arbitrage Rule. National Bureau of Economic Research Working Paper No. 7032; March 1999.

^*There are three variations of this theory: (1) weak form—past prices cannot be used to predict future prices; (2) semistrong form—the current price reflects all publicly known information; (3) strong form the current price reflects all information, whether publicly known or not.

^*To be precise, this statement would be true even for small net losses in the short component of the portfolio, but an adequate explanation is beyond the scope of this book.