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.

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.

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.

Under conditions consistent with the Black‐Scholes formula, a simple formula is developed for the expected rate of return of an option over a finite holding period possibly less than the time to expiration of the option. Under these conditions, surprisingly, the expected future value of a European option, even prior to expiration , is shown equal to the current Black‐Scholes value of the option, except that the expected future value of the stock at the end of the holding period replaces the current stock price in the Black‐Scholes formula and the future value of a riskless invesment of the striking price replaces the striking price. An extension of this result is used to approximate moments of the distribution of returns from an option portfolio.

Under conditions consistent with the Black-Scholes formula, a simple formula is developed for the expected rate of return of an option over a finite holding period possibly less than the time to expiration of the option. Under these conditions, surprisingly, the expected future value of a European option, even prior to expiration, is shown equal to the current Black-Scholes value of the option, except that the expected future value of the stock at the end of the holding period replaces the current stock price in the Black-Scholes formula and the future value of a riskless invesment of the striking price replaces the striking price. An extension of this result is used to approximate moments of the distribution of returns from an option portfolio.

This paper develops a new option pricing formula that pushes the underlying source of risk back to the risk of individual assets of the firm. The formula simultaneously encompasses differential riskiness of the assets of the firm, their relative weights in determining the value of the firm, the effects of firm debt, and the effects of a dividend policy with both constant and random components. Although this setting considerably generalizes the Black-Scholes [1] analysis, it nonetheless produces a formula via riskless arbitrage arguments that, given estimated inputs, is as easy to use as the Black-Scholes formula.

This paper develops a new option pricing formula that pushes the underlying source of risk back to the risk of individual assets of the firm. The formula simultaneously encompasses differential riskiness of the assets of the firm, their relative weights in determining the value of the firm, the effects of firm debt, and the effects of a dividend policy with both constant and random components. Although this setting considerably generalizes the Black‐Scholes [1] analysis, it nonetheless produces a formula via riskless arbitrage arguments that, given estimated inputs, is as easy to use as the Black‐Scholes formula.