Stock Market Wizards
Page 37
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.