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

TL;DR

This paper derives a version of the Capital Asset Pricing Model from the Black-Scholes framework, linking stock appreciation rates to market efficiency and covariance with the index over long horizons.

Contribution

It demonstrates that under Black-Scholes assumptions, the CAPM naturally emerges, connecting stock returns to market covariance and efficiency principles.

Findings

Stock appreciation rate approximates interest rate plus covariance term

Equity premium is close to squared volatility of the index

Results hold for long investment horizons

Abstract

We consider a financial market in which two securities are traded: a stock and 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 it is efficient, in the following sense: we do not expect a prespecified self-financing trading strategy whose wealth is almost surely nonnegative at all times to outperform the index greatly. We show that, for a long investment horizon, the appreciation rate of the stock has to be close to the interest rate (assumed constant) plus the covariance between the volatility vectors of the stock and the index. This contains both a version of the Capital Asset Pricing Model and our earlier result that the equity premium is close to the squared volatility of the index.