Alternative sets of sufficient conditions are developed under which equilibrium security rates of return are determined as if there exist only identical individuals whose resources, beliefs, and tastes are a composite of the actual individuals in the economy. These conditions include as special cases all those previously examined in the literature (including conditions sufficient to produce the two-parameter mean-variance model), as well as others. Whenever such a composite individual exists it is shown that (1) valuation equations take a specific form and contain only exogenous parameters of the economy; (2) market exchange arrangements are Pareto-optimal; and (3) competitive value-maximizing firms make completely specified Pareto-optimal production decisions both over dates and states. These results rely on the observation that under popular homogeneity assumptions regarding beliefs and tastes, even though the securities market may be incomplete, equilibrium rates of return are determined as if there were an otherwise similar Arrow-Debreu economy.

This paper shows how to decompose the dollar profit earned from an option into two basic components: i) mispricing of the option relative to the asset at the time of purchase; and ii) profit from subsequent fortuitous changes or mispricing of the underlying asset. This separation hinges on measuring the true relative of the option from its realized payoff. The payoff from any one option has a huge standard error about this value that can be reduced by averaging the payoff from several independent option positions. Simulations indicate that 95% reductions in standard errors can be further achieved by using the payoff of a dynamic replicating portfolio as a Monte Carlo control variate. In addition, the paper shows that these low standard errors are robust to discrete rather than continuous dynamic replication and to the likely degree of misspecification of the benchmark formula used to implement the replication. Option mispricing profit can be further decomposed into profit due to superior esti? mation of the volatility (volatility profit) and profit from using a superior option valuation formula (formula profit). To make this decomposition reliably, the benchmark formula used for the attribution needs to be similar to the formula implicitly used by the market to price options. If so, then simulation indicates that this further decomposition can be achieved with low standard errors. Basic component ii) can be further decomposed into profit from a forward contract on the underlying asset (asset profit) and what I term pure option profit. The asset profit indicates whether the investor was skillful by buying or selling options on mispriced underlying assets. However, asset profit could also simply be just compensation for bearing risk?a distinction beyond the scope of this paper. Al? though simulation indicates that the attribution procedure gives an unbiased allocation of the option profit to this source, its standard error is large?a feature common with others' attempts to measure performance of assets.

The attempt to incorporate securities market imperfections other than proportional taxes within a mean-variance security valuation context has met with modest success. Lintner [5], however, has recently considered imperfections by the device of segmented markets. His paper has motivated the following taxonomy. Securities markets are defined as weakly segmented if some of the securities in at least one market are available to some investors but not to others, partially segmented if the sets containing both investors and available securities in each market are disjoint, and completely segmented if additionally the sets of firms in each market are disjoint. Segmented markets effectively relax the separation property of mean-variance equilibrium models (i.e., all investors, irrespective of differences in present wealth or preferences, divide their wealth between the same two mutual funds; one is risk-free and the other is the market portfolio of risky securities). This property unfortunately implies that each investor must hold a portion of every available risky security. This is empirically unrealistic, primarily due to restrictions on borrowing and shorting and scale economies in security analysis and brokerage. Moreover, even in the absence of these complications, ownership of nonmarketable assets, nonhomogeneous beliefs, or breakdown of the separation property due to tastes or nonnormality will motivate individuals to hold different risky portfolios. The device of segmented markets embodies in extreme form these obstacles to diversification and portfolio similarity.

Under the assumption that individuals are single-period maximizers of the expected utility of their future wealth, this essay extends the mean-variance security valuation model developed by Sharpe [10], Lintner [4, 5, and 6], and Mossin [7 and 8] to a general parameter-preference model, with and without the simplifications of homogeneous subjective probabilities and the existence of a risk-free security. Results with quadratic and cubic utility are developed as special cases.

to prepare this retrospective, and for bringing to the task his unique erudition and perspective. THIS YEAR MARKS the fiftieth anniversary of the publication of Harry Markowitz’s landmark paper, “Portfolio Selection, ” which appeared in the March 1952 issue of the Journal of Finance. With the hindsight of many years, we can see that this was the moment of the birth of modern financial economics. Although the baby had a healthy delivery, it had to grow into its teenage years before a hint of its full promise became apparent. What has always impressed me most about Markowitz’s 1952 paper is that it seemed to come out of nowhere. Compared to the work of his 1990 co-Nobel Prize winners ~Sharpe primarily for his paper on the capital asset pricing model and Miller for his paper on capital structure!, Markowitz’s paper seems to have more of this flavor. In 1676, Sir Isaac Newton wrote his friend Robert Hooke, “If I have seen further it is by standing on the shoulders of giants ” ~Newton ~1959!! and that is true of Markowitz as well, but,

This article develops a new method for inferring risk‐neutral probabilities (or state‐contingent prices) from the simultaneously observed prices of European options. These probabilities are then used to infer a unique fully specified recombining binomial tree that is consistent with these probabilities (and, hence, consistent with all the observed option prices). A simple backwards recursive procedure solves for the entire tree. From the standpoint of the standard binomial option pricing model, which implies a limiting risk‐neutral lognormal distribution for the underlying asset, the approach here provides the natural (and probably the simplest) way to generalize to arbitrary ending risk‐neutral probability distributions.

This article develops a new method for inferring risk-neutral probabilities (or state-contingent prices) from the simultaneously observed prices of European options. These probabilities are then used to infer a unique fully specified recombining binomial tree that is consistent with these probabilities (and, hence, consistent with all the observed option prices). A simple backwards recursive procedure solves for the entire tree. From the standpoint of the standard binomial option pricing model, which implies a limiting risk-neutral lognormal distribution for the underlying asset, the approach here provides the natural (and probably the simplest) way to generalize to arbitrary ending risk-neutral probability distributions.

This article develops a new method for inferring risk-neutral probabilities (or state-contingent prices) from the simultaneously observed prices of European options. These probabilities are then used to infer a unique fully specified recombining binomial tree that is consistent with these probabilities (and, hence, consistent with all the observed option prices). A simple backwards recursive procedure solves for the entire tree. From the standpoint of the standard binomial option pricing model, which implies a limiting risk-neutral lognormal distribution for the underlying asset, the approach here provides the natural (and probably the simplest) way to generalize to arbitrary ending risk-neutral probability distributions.

The tests reported here differ in several ways from those of most other papers testing option pricing models: an extremely large sample of observations of both trades and bid-ask quotes is examined, careful consideration is given to discarding misleading records, nonparametric rather than parametric statistical tests are used, reported results are not sensitive to measurement of stock volatility, special care is taken to incorporate the effects of dividends and early exercise, a simple method is developed to test several option pricing formulas simultaneously, and the statistical significance and consistency across subsamples of the most important reported results are unusually high. The three key results are: (1) short-maturity out-of-the-money calls are priced significantly higher relative to other calls than the Black-Scholes model would predict, (2) striking price biases relative to the Black-Scholes model are also statistically significant but have reversed themselves after long periods of time, and (3) no single option pricing model currently developed seems likely to explain this reversal.