Is Mean-Variance Analysis Vacuous: Or was Beta Still Born?

Abstract

We show in any economy trading options, with investors having mean-variance preferences, that there are arbitrage opportunities resulting from negative prices for out of the money call options. The theoretical implication of this inconsistency is that mean-variance analysis is vacuous. The practical implications of this inconsistency are investigated by developing an option pricing model for a CAPM type economy. It is observed that negative call prices begin to appear at strikes that are two standard deviations out of the money. Such out-of-the money options often trade. For near money options, the CAPM option pricing model is shown to permit estimation of the mean return on the underlying asset, its volatility and the length of the planning horizon. The model is estimated on S&P 500 futures options data covering the period January 1992- September 1994. It is found that the mean rate of return though positive, is poorly identified. The estimates for the volatility are stable and average 11%, while those for the planning horizon average 0.95. The hypothesis that the planning horizon is a year can not be rejected. The one parameter Black-Scholes model also marginally outperforms the three parameter CAPM model with average percentage errors being respectively, 3.74% and 4.5%. This out performance of the Black-Scholes model is taken as evidence consistent with the mean-variance analysis being vacuous in a practical sense as well. The capital asset pricing model (CAPM) of Sharpe (1964), Lintner (1965) and Mossin (1966) was derived as a general equilibrium consequence of the portfolio decisions made by investors, with mean variance preferences, investing in a single risk free asset and a finite number of risky assets whose joint probability distribution is known to all investors. The CAPM model has had a long history of use and statistical evaluation in the finance literature, with the notable recent contributions of Fama and French (1992, 1995), Jaganathan and Wang (1995), Kothari, Shanken and Sloan (1995) and Roll and Ross (1994). In particular, the recent paper by Fama and French (1995) argues that 'Beta is dead' as it has no explanatory value in explaining stock price returns. This paper argues that the CAPM is vacuous as it implies the existence of arbitrage opportunities for deep out-of-the money options. Hence the title 'was beta still born?'. The existence of a CAPM equilibrium was first established in the finite asset context by Hart (1974), Nielsen (1990) and for an infinite asset economy by Dana