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We consider a financial market in which two securities are traded: a stockand an index. Their prices are assumed to satisfy the Black-Scholes model.Besides assuming that the index is a tradable security, we also assume that itis efficient, in the following sense: we do not expect a prespecifiedself-financing trading strategy whose wealth is almost surely nonnegative atall times to outperform the index greatly. We show that, for a long investmenthorizon, the appreciation rate of the stock has to be close to the interestrate (assumed constant) plus the covariance between the volatility vectors ofthe stock and the index. This contains both a version of the Capital AssetPricing Model and our earlier result that the equity premium is close to thesquared volatility of the index.

The Capital Asset Pricing Model as a corollary of the Black-Scholes model