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he was convinced of his work s breakthroughness, for one crucial reason: The stock s expected return was nowhere to be seen in the equation That is, he seemed to have found a formula for warrant (option) pricing that did not require one to have to forecast the future stock path, a decisively subjective and inaccuracy-prone task Previous efforts suffered precisely from that problem, as we will see in a second It was thus with more than a passing interest that Black reacted to Scholes disclosure of his own work on warrant pricing a few months later, apparently inspired after supervising a thesis from one of his students They both tackled the problem from the same philosophical angle: CAPM-compliance.

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Scholes was familiar with arbitrage-enforced pricing and tried to apply the notion to the warrant affair by building a portfolio made up of the warrant and stock; by going long the warrant (which goes up in value if the stock price goes up) and short the stock, he reasoned that you could build a market-neutral position, or, in CAPM-speak, a zero-beta position, uncorrelated with the overall market and thus, according to CAPM, one that should earn a risk-free rate of return But at that point, he didn t know how to construct such portfolio Like Black, his solo efforts were roadblocked Black was familiar with the modeling of the relationship of variables that depend on each other, thanks to work that he had done with, certainly, Jack Treynor He had used such tricks (including the famous Taylor-series expansion) to get to his differential equation.

15

Black and Scholes understood that the partial (mathematical) derivative of the warrant price with regards to the stock price was the key to building the zero-beta portfolio: that derivative (or delta ) tells you how much you have to trade of the stock in order to keep the portfolio s value insulated from market movements The end result of this reasoning led again to Black s equation Scholes arbitrage arguments thus provided con rmation that Black had been on to something They were now certain that they were in a zero-beta scenario and, thus, according to CAPM, a riskfree one The portfolio s expected return should be equal to the riskless rate But they still could not solve the equation.

70 E

Now that we have determined the appropriate style index and can con rm its validity based on a variety of other analytical techniques, we can take this style analysis to the next level. We know from 5 ( Risk Analysis ) that the coef cient of determination (R2) is the statistical measure of the predictive power of the regression results. R2 is calculated by dividing the sum of the squared error numbers by the sum of the squared total numbers. Our style analysis worksheet automatically calculates each of these terms. Referring to Exhibit 10.1, the sum of the squared error numbers is calculated in cell I70 and the sum of the squared total numbers is calculated in cell J70. The R2 statistic represents the percentage of the underlying portfolio s return that is due to the style index or, stated another way, the percentage of the underlying portfolio s return that is explained by asset allocation decisions.

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