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.

Yutaka Imai, Mark Rubinstein, Equilibrium in the Pricing of Capital Assets, Risk-Bearing Debt Instruments, and the Question of Optimal Capital Structure: A Comment, The Journal of Financial and Quantitative Analysis, Vol. 7, No. 4 (Sep., 1972), pp. 2001-2003