Employee Reload Options: Pricing, Hedging, and Optimal Exercise

Reload options, call options granting new options on exercise,are popularly used in compensation. Although the compound optionfeature may seem complicated, there is a distribution-free dominantpolicy of exercising reload options whenever they are in themoney. The optimal policy implies general formulas for numericalvaluation. Simpler formulas for valuation and hedging followfrom Black–Scholes assumptions with or without continuousdividends. Time vesting affects the optimal policy, but numericalresults indicate that it is nearly optimal to exercise in themoney whenever feasible. The results suggest that reload optionsproduce similar incentives as employee stock options and sharegrants.     

Duality,interest rates,and the theory of present value

This paper uses duality theory to simplify and extend previous work which characterized technologies which have present value decreasing in interest rates. For “unbounded” interest rate sets, value nonincreasing is equivalent to truncation never being desirable. For bounded interest rate sets, the result is true if truncation is replaced by limited truncation, appropriately defined. Similarly, for equal forward rates, undesirability of geometric truncation characterizes nonincreasing value. Another result shows that if truncation is undesirable at an internal rate of return, then the internal rate of return is well behaved, even if value is not decreasing everywhere.     

Portfolio turnpikes

Profolio turnpike theorems show that if preferences at largewealth levels are similar to power utility, then the investmentstrategy converges to the power utility strategy as the horizonincreases. We state and prove two simple and general portfolioturnpike theorems. Unlike existing literature, our main resultdoes not assume independence of returns and depends only ondiscounting of future cash flows. We also provide a critiqueof portfolio turnpike results, based on the observations that(1) the time required for convergence is often too large tobe relevant, and (2) there is no convergence for consumptionwithdrawal problems.     

Present values and internal rates of return

This paper resolves the question of finding the complete set of conditions under which the present value of a return stream decreases as interest rates rise. Previous work has shown that if a return stream can be costlessly abandoned, or truncated, at any date prior to its natural termination, then its value is monotone. We show that truncation is the same as monotonicity. First, monotonicity is equivalent to the value function being identical to the truncation set value function. Second, monotonicity is equivalent to the truncation set being contained within the convex hull, augmented by free (value) disposal.     

Inefficient dynamic portfolio strategies or how to throw away a million dollars in the stock market

A number of portfolio strategies followed by practitioners aredominated because they are incompletely diversified over time.The payoff distribution pricing model is used to compute thecost of following undiversified strategies. Simple numericalexamples illustrate the technique, and computer-generated examplesprovide realistic estimates of the cost of some typical policies,using reasonable parameter values. The cost can be substantialand should not be ignored by practitioners. A section on generalizationsmodels and other general models of returns.     

Adoption externalities as public goods

A public good problem arises naturally in situations characterized by positive externalities or negative externalities. For positive externalities, agents furnish a public good by taking an action. Similarly, for negative externalities, agents furnish a public good by not taking an action. We examine these externalities in a binary choice problem, i.e. adoption externalities. In each case, there is a potential role for government intervention, even when equilibrium transfers are not allowed, since one equilibrium Pareto dominates all others. We analyze the positive and normative features of equilibria in these models, and we explore the possibility of useful government intervention.     

Increases in risk aversion and the distribution of portfolio payoffs

Oliver Hart proved the impossibility of deriving general comparative static properties in portfolio weights. Instead, we derive new comparative statics for the distribution of payoffs: A is less risk averse than B iff A?s payoff is always distributed as B?s payoff plus a non-negative random variable plus conditional-mean-zero noise. If either agent has nonincreasing absolute risk aversion, the non-negative part can be chosen to be constant. The main result also holds in some incomplete markets with two assets or two-fund separation, and in multiple periods for a mixture of payoff distributions over time (but not at every point in time).     

An explicit bound on individual assets' deviations from APT pricing in a finite economy

Ross's Arbitrage Pricing Theory (APT) is a tractible and reasonable alternative to the mean-variance model. Nonetheless, understanding of the theory has been obscured by the complexity of the sequence economy models used for motivation. By contrast, we give a simple and direct derivation of the APT in a finite economy. Using an explicit bound on the deviations from APT prices across assets, a coarse calculation shows that theoretical deviations from APT pricing are negligible in our economy.     

Empty promises and arbitrage

Analysis of absence of arbitrage normally ignores payoffs instates to which the agent assigns zero probability. We extendthe fundamental theorem of asset pricing to the case of 'noempty promises' in which the agent cannot promise arbitrarilylarge payments in some states. There is a superpositive pricingrule that can assign positive price to claims in zero probabilitystates important to the market as well as assigning positiveprices to claims in the states of positive probability. Withcontinuous information arrival, no empty promises can be enforcedby shutting down the agent's subsequent investments once wealthhits zero.     

Author index for volume 26

The main result in revealed preference theory is that under fairly weak assumptions (strict concavity and monotonicity of the utility function), an agent's preferences can be recovered from the entire demand function. This paper addresses the possibility of recovering preferences from less extensive information, given additional maintained restrictions on preferences. Specifically, we explore whether an agent's utility function can be recovered from preferences over nominal gambles (when money is risky in real terms) or from demand for assets whose nominal returns contain common inflation or other risk. The recovery is constructive to the extent that there is a good numerical technique for doing Fourier transforms and inversions, and other numerical computations can be done.