538 Part V: Index Options and Futures 
pared to an equal dollar amount of stock. By selling the futures on an index - say, the 
S&P 500- he removes the "market risk" from his portfolio (assuming the S&P 500 
represents the "market"). What is left over after selling the futures is the "tracking 
error." The discrepancy between the movement of the general stock market and any 
individual portfolio is called "tracking error." This investor will still make money if his 
portfolio outperforms the S&P 500, but he will find that he did not completely elim­
inate his losses if his portfolio underperforms the index. Note that if the market goes 
up, the investor will not make any money except for possible tracking error in his 
favor. 
REMOVING THE MARKET RISK FROM A PORTFOLIO 
Stock portfolios are diverse in nature, not :p.ecessarily reflecting the composition of 
the index underlying the futures contracts. The characteristics of the individual 
stocks must be taken into account, for they may move more quickly or more slowly 
than "the market." Let us spend a moment to define this characteristic of stocks that 
is so important. 
VOLATILITY VERSUS BETA 
Recall that when we originally defined volatility for use in the Black-Scholes model, 
we stated that Beta was not acceptable because it was strictly a measure of the cor­
relation of a stock's performance to that of the stock market and was not a measure 
of how fast the stock changed in price. Now we are concerned with how the stock's 
movement relates to the market's as a whole. This is the Beta. 
Unfortunately, Beta is not as readily available to the option strategist as is 
volatility. Many option traders merely have to punch a button on their quote 
machines and they can receive estimates of volatility. However, Beta estimates are 
more difficult to obtain, and the ones that are available are often for very long time 
periods, such as several years. These long-term Betas cannot be used for the pur­
poses of the index hedging discussed in this chapter. Therefore, if one does not have 
access to shorter-term Beta calculations, then he can approximate Beta by compar­
ing an individual stock's volatility with the market's volatility. 
Example: XYZ is a relatively volatile stock, having both an implied and historical 
volatility of 36%. The overall stock market has a volatility of 15%. Therefore, one 
could approximate the Beta of XYZ as 
Beta approximation = 36/15 = 2.40