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

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 16 ROBERT A. JARROW AND DILIP B. MADAN (1994). This paper uses a result of Dana (1994) to show that the CAPM implies the existence of arbitrage opportunities for economies trading options on the market portfolio with an unbounded sequence of strikes, when the distribution for the market portfolio’s return is unbounded. The problem arises because mean variance preferences imply that the pricing kernel is linear in the return on the market portfolio, with a negative slope, thereby forcing call options with high strikes to have negative prices. This inconsistency between the capital asset pricing model and option trading has been previously observed by Dybvig and Ingersoll (1982) (Theorem l) in the context of a complete market. The fact that their result requires market completeness, gives CAPM enthusiasts an ‘out’. The ‘out’ is that with returns having a continuum support (the positive half line), markets are incomplete. The contribution of this paper is to show that market completeness is not required for this inconsistency. It is the mean variance preferences themselves that are inconsistent with the absence of arbitrage opportunities. The traditional sources for mean-variance preferences – either quadratic utility or return distributions that are multivariate normal or elliptical – do not concern us here. The consequence of this result, then, is that CAPM enthusiasts do not have an ‘out’. From a theoretical perspective, with traded deep out of the money options, mean-variance analysis is vacuous. The practical implication of this inconsistency critically depends on whether options on the market portfolio trade for the strikes admitting arbitrage opportunities. To investigate this possibility we develop an option pricing formula based on the CAPM for an arbitrary and discrete planning horizon h > 0. This is in contrast to the Black–Scholes equilibrium which is consistent with no arbitrage and a CAPM with an instantaneous planning horizon, Merton (1973).1 This distinction between a finite and instantaneous planning horizon is crucial to the existence of the arbitrage opportunities proven below. We find that the general shape of observed Black–Scholes implied volatilities, as for example in Derman and Kani (1994), are consistent with a CAPM model with call option strikes about two standard deviations out-of-the money generating arbitrage opportunities. Such options do trade, although most of the actively traded options are nearer to the money. We estimate our CAPM option pricing model on S&P 500 futures options data for the period January 1992–September 1994. We find the planning horizon h to be fairly consistently estimated at just under a year, with an average value of 0.95 years and a standard deviation of 0.04. The null hypothesis of the planning horizon being a year is not rejected. Mean rates of return, though positive are found to be poorly identified, while the implied CAPM volatility estimate is fairly stable. The average pricing error is 4.5% which is slightly inferior to the Black–Scholes model with an average percentage error of 3.74%. This outperformance of the Black–Scholes model is taken as evidence consistent with the mean-variance analysis being vacuous in a practical sense as well. IS MEAN-VARIANCE ANALYSIS VACUOUS? 17 An outline for this paper is as follows. The economy is described in Section 1. Section 2 demonstrates the existence of the arbitrage opportunities. Section 3 derives an explicit option pricing formula consistent with a mean-variance preference CAPM. Section 4 studies the behavior of Black–Scholes implied volatilities for options trading in a CAPM type economy. Results of estimating the CAPM option pricing model on S&P 500 futures option data are presented in Section 5. Section 6 comments on the implications for other asset pricing theories invoking mean variance considerations. Finally, Section 7 concludes the paper. 1. The Economic Model This section presents the economic model. Consider a one period economy with dates t = 0, 1. Let ( , F , P ) be a probability space with the set of events . All uncertainty is resolved at time 1. The economy has n investors, indexed by i = 1; : : :; n, who know the probability measure P on F . Traded at time 0 in the economy is a closed subspace,Z , of P -square integrable, F-measurable real-valued functions on . The investors’ endowments are given by non-negative elements "i 2 Z , for i = 1, : : : , n. The aggregate endowment is " = Pn i=1 "i. We let " have a distribution under P with unbounded support on the positive half line. It follows under the above structure that all endowments have finite mean and variance under P . We let Z include the constant payoff 1 (a riskless asset), and a sequence of call options on the aggregate endowment " with time one payoffs (" kj) for an unbounded increasing sequence of strikes kj . The closed subspaceZ therefore has infinite dimension. The investors have mean-variance preferences given by a utility function of the form ui(z) = Ui(E[z]; var(z)); (1) for z 2 Z and i = 1; : : :; n, where E is the expectation operator under the measure P and var(z) is the variance of z under P which equals E[z2] (E[z])2. We suppose, as in Dana (1994), that the partials Ui1 and Ui2 are respectively positive and negative, reflecting a preference for mean and an aversion to variance.2 A competitive equilibrium for this economy is given by a continuous linear pricing functional on the space of traded claimsZ and a vector c = ( ci; : : :; cn) 2 Z denoting the consumption of the n investors, such that (a) each investor’s utility is maximized subject to their budget constraint, i.e. [ ci] ["i] and if [ci] ["i] for ci 2 Z then ui(ci) ui( ci), and (b) markets are cleared, i.e. Pn i=1 ci = ". The existence of such an equilibrium for this economy is proven in Dana (1994). 2. The Arbitrage Opportunity This section shows that the above equilibrium contains an arbitrage opportunity. As a result, the mean-variance analysis is vacuous. 18 ROBERT A. JARROW AND DILIP B. MADAN First note that the space Z is a closed subspace of the space of square integrable random variables. It follows from the Riesz representation theorem, that the continuous linear functional has a representation in terms of product moments, specifically, for some 2 Z , [z] = E[ z]; for all z 2 Z: (2) Dana (1994) characterizes as linear in the aggregate endowment ". We reproduce this proof here in somewhat greater detail. Theorem 1. The pricing functional has the form