[We introduce a model that captures the main properties that characterize employee stock options (ESO). We discuss the likelihood of early voluntary ESO exercise, and the obligation to exercise immediately if the employee leaves the firm, except if this happens before options are vested, in which case the options are forfeited. We derive an analytic formula for the price of the ESO and in a case study compare it to alternative methods.]

[We determine the minimum cost of superreplicating a nonnegative contingent claim when there are convex constraints on portfolio weights. We show that the optimal cost with constraints is equal to the price of a related claim without constraints. The related claim is a dominating claim, that is, a claim whose payoffs are increased in an appropriate way relative to the original claim. The results hold for a variety of options, including some path-dependent options. Constraints on the gamma of the replicating portfolio, constraints on portfolio amounts, and constraints on the number of shares are also considered.]

We derive a closed-form solution for the optimal portfolio of a nonmyopic utility maximizer who has incomplete information about the alphas or abnormal returns of risky securities. We show that the hedging component induced by learning about the expected return can be a substantial part of the demand. Using our methodology, we perform an "ex ante" empirical exercise, which shows that the utility gains resulting from optimal allocation are substantial in general, especially for long horizons, and an "ex post" empirical exercise, which shows that analysts' recommendations are not very useful.

We study survival, price impact, and portfolio impact in heterogeneous economies. We show that, under the equilibrium risk-neutral measure, long-run price impact is in fact equivalent to survival, whereas long-run portfolio impact is equivalent to survival under an agent-specific, wealth-forward measure. These results allow us to show that price impact and portfolio impact are two independent concepts: a nonsurviving agent with no long-run price impact can have a significant long-run impact on other agents' optimal portfolios.

We study the incentive effects of granting levered or unlevered stock to a risk-averse manager. The stock is granted by risk-neutral shareholders who choose leverage and compensation level. The manager applies costly effort and selects the level of volatility, both of which affect expected return. The results are driven by the attempt of the risk-neutral shareholders to maximize the value of their claims net of the compensation package. We consider a dynamic setting and find that levered stock is optimal for high-type managers, firms with high momentum, large firms, and firms for which additional volatility only implies a modest increase in expected return.

We introduce a model that captures the main properties that characterize employee stock options (ESO). We discuss the likelihood of early voluntary ESO exercise, and the obligation to exercise immediately if the employee leaves the firm, except if this happens before options are vested, in which case the options are forfeited. We derive an analytic formula for the price of the ESO and in a case study compare it to alternative methods. Since the mid-1980s, stock options have been a substantial component of com-pensation packages for employees. For example, in 1999, 94 % of companies in the S&P 500 offered stock options to their top employees (see Murphy, 1999; Hall and Murphy, 2002). In 1995, the Financial Accounting Standards Board (FASB) (with FAS 123) set a standard that required firms to expend stock-based compensation at the moment the compensation was granted (see FASB, 1995). Firms were encour-aged to use the “fair value ” of the stock option to compute the value of the compensation, but were allowed to use the “intrinsic value”—market price of the stock minus strike price. Since employee stock options (ESOs) are typically granted at the money, the intrinsic value is zero, which results in no expense recorded at the time of the grant, and this is probably one of the reasons that helped their popularity.

Constraints on the gamma of the replicating portfolio, constraints on portfolio amounts, and constraints on the number of shares are also considered.

We derive a closed-form solution for the optimal portfolio of a nonmyopic utility maximizer who has incomplete information about the alphas or abnormal returns of risky securities. We show that the hedging component induced by learning about the expected return can be a substantial part of the demand. Using our methodology, we perform an “ex ante” empirical exercise, which shows that the utility gains resulting from optimal allocation are substantial in general, especially for long horizons, and an “ex post” empirical exercise, which shows that analysts’ recommendations are not very useful.

This paper presents an equilibrium model in a pure exchange economy when investors have three possible sources of heterogeneity. Investors may differ in their beliefs, in their level of risk aversion, and in their time preference rate. The authors study the impact of investors’ heterogeneity on equilibrium properties and, in particular, on the consumption shares, the market price of risk, the risk-free rate, the bond prices at different maturities, the stock price and volatility as well as on the stock's cumulative returns, and optimal portfolio strategies. The authors relate the heterogeneous economy with the family of associated homogeneous economies with only one class of investors. Cross-sectional as well as long-run properties are analyzed.