Merton Miller's status as a father of finance reflects the academic depth, breadth, and rigor of his writings and two important facets of his character. Merton was a man of great warmth and humor. He communicated his often challenging views via memorable phrases and anecdotes that have become part of the everyday language of the profession. Merton was also a man of great dedication. The whole profession has benefited from his devotion to his doctoral students, to his colleagues, and to his coauthors. This essay demonstrates how Merton's admiration for markets provided the foundation for all his research.

This work examines the relation between option prices and the true, as opposed to risk-neutral, distribution of the underlying asset. If the underlying asset follows a diffusion with an instantaneous expected return at least as large as the instantaneous risk-free rate, observed option prices can be used to place bounds on the moments of the true distribution. An illustration of the paper's results is provided by the analysis of the information concerning the mean and standard deviation of market returns contained in the prices of S&P 100 Index Options. ALTHOUGH IT SEEMS NATURAL that the prices of claims to various parts of the underlying asset's distribution should contain information about the shape of that distribution, Cox and Ross (1976) established that an option's price equals its expected payoff discounted at the risk-free rate where the expectation is taken over the 'risk-neutral', rather than the true, distribution of the underlying asset. Linking the risk-neutral distribution implicit in option prices to the true distribution remains a comparative mystery. A necessary condition for the risk-neutral pricing methodology to be applicable is that the true and the risk-neutral distribution share a common support. The only information about the true distribution that can be obtained from observed option prices alone is information about that support. This paper demonstrates though that observed option prices, when used in conjunction with simple assume,d restrictions on the true distribution, do contain information about the noncentral moments of the true distribution not directly implied by those assumed restrictions alone.' The intuition for the result that option prices can be useful in placing bounds on the moments of the true distribution is straightforward. First, we generalize the results in Lo (1987) to show how the expected payoff to a call can be bounded above in terms of any chosen set of the noncentral moments of the return on the underlying asset. Second, we establish restrictions on the

This work examines the relation between option prices and the true, as opposed to risk‐neutral, distribution of the underlying asset. If the underlying asset follows a diffusion with an instantaneous expected return at least as large as the instantaneous risk‐free rate, observed option prices can be used to place bounds on the moments of the true distribution. An illustration of the paper's results is provided by the analysis of the information concerning the mean and standard deviation of market returns contained in the prices of S&P 100 Index Options.

When management has private information it has an incentive to finance investment by issuing a security that is overpriced in the market. The market's valuation of the issued security may lead management either to forego profitable investments or to invest suboptimally. With investment fixed, there exist fully revealing signaling equilibria in which the covenants of the issued claim serve as signals. A straight bond issue cannot provide the signals but a convertible bond issue can. With investment endogenous, fully revealing equilibria exist in which the par value of a straight bond issue and the announced level of investment jointly serve as signals and investment is optimal. The article also investigates the role of a stock repurchase in these equilibria.

The informational role of prices contributes positively to their variability. In a noisy rational expectations equilibrium, traders rationally respond to price changes by revising their estimates of other traders' private signals and hence their own expectations of future dividends. The resultant shifts in traders' demands amplify any supply shock-induced price changes. We develop an infinite horizon noisy rational expectations model and calibrate, simulate, and test it using U.S. stock market data. The price variability in a heterogeneous information economy is shown to be 20% to 46% higher than in an otherwise equivalent economy in which all signals are publicly announced.

This article analyzes the volume of trade in a multiperiod noisy rational expectations model. When traders receive private signals at the first trading date and are allowed a second round of trade, two type of equilibria exist. In the first, traders do not learn about the average private signal from the second round of trade, and all trade takes place at the first date. In the second, traders do learn from the second round, and trade thus takes places at both the first and second dates. The article characterizes volume when a public signal is disclosed at the second date.

We develop a model that predicts corporate investment level increases with investors’ optimism and that the relationship between investment level and executive compensation depends on investor sentiment and other parameters. The empirical test shows that optimism is significantly and positively related to the level of investment and that executive compensation is insignificantly related to the level of investment. The managerial share ownership is positively related to the level of investment, conditional on the degree of optimism. The empirical results suggest that executives make investment decisions that not only cater to investor sentiment but also reflect their own interest in the company.

Buying recent winners and shorting recent losers guarantees time-varying factor exposures in accordance with the performance of common risk factors during the ranking period. Adjusted for this dynamic risk exposure, momentum profits are remarkably stable across subperiods of the entire post-1926 era. Factor models can explain 95% of winner or loser return variability, but cannot explain their mean returns. Momentum strategies which base winner or loser status on stock-specific return components are more profitable than those based on total returns. Neither industry effects nor cross-sectional differences in expected returns are the primary cause of the momentum phenomenon.

When the underlying price process is a one-dimensional diffusion, as well as in certain restricted stochastic volatility settings, a contingent claim's delta is bounded by the infimum and supremum of its delta at maturity. Further, if the claim's payoff is convex (concave), the claim's price is a convex (concave) function of the underlying asset's value. However, when volatility is less specialized, or when the underlying process is discontinuous or non-Markovian, a call's price can be a decreasing, concave function of the underlying price over some range, increasing with the passage of time, and decreasing in the level of interest rates.

When the underlying price process is a one‐dimensional diffusion, as well as in certain restricted stochastic volatility settings, a contingent claim's delta is bounded by the infimum and supremum of its delta at maturity. Further, if the claim's payoff is convex (concave), the claim's price is a convex (concave) function of the underlying asset's value. However, when volatility is less specialized, or when the underlying process is discontinuous or non‐Markovian, a call's price can be a decreasing, concave function of the underlying price over some range, increasing with the passage of time, and decreasing in the level of interest rates.