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Options can induce risk taking for arbitrary preferences.

Article Excerpt
Summary. It is widely believed that call options induce risk-taking behavior. However, Ross (2004) challenges this intuition by demonstrating the impossibility of inducing managers with arbitrary preferences to always act as if they were less risk averse. If preferences and price distributions are unknown, risk-taking behavior cannot be always induced by an option contract. Here, we prove a new result showing that, with no information about preferences and some knowledge about prices, one can write a call option that makes all managers prefer riskier projects to safer ones. This points out that in order to design options that induce risk taking it is sufficient to have information about price distributions.

Keywords and Phrases: Stochastic dominance, Stock options, Risk taking.

JEL Classification Numbers: D81, G00, J33, M21.

1 Introduction

Regardless of the reasons why firms grant stock options to their executives, it is widely believed that a consequence of this practice is an increase in managerial willingness to take risks. Intuitively, it is reasonable to expect that compensation schedules based on call options would induce risk taking, since call options display convex payoff structures and their values increase with the volatility of the underlying stock--see Smith and Stulz (1985) and DeFusco, Johnson, and Zorn (1990). However, as noticed by Ross (2004), this argument ignores the fact that option-based compensation schemes move the evaluation of a given lottery to a locus in the domain where utility functions might be more concave. Thus, the convex shape of call payoffs could be offset by locally higher degrees of risk aversion. (3) In fact, Ross (2004) proves an impossibility theorem. He shows that, among the class of strictly convex compensation schedules, there is no single schedule that would make all strictly concave utility functions uniformly display lower risk aversion everywhere in their domains. Based on this result, Ross (2004) states that "the common folklore that giving options to agents will make them more willing to take risks is false."

In this paper, we ask what the conditions are under which call options may induce risk taking. We prove a new result that is helpful in understanding the relationship between options and risk taking. If F and G are two cumulative distribution functions and F second-order stochastically dominates G (i.e. G is unambiguously riskier than F), then there is a censoring point such that the distribution G censored at this point first-order stochastically dominates the distribution F censored at the same point.

Because call options pay only if the stock price surpasses a threshold level (the strike price), the payoff distribution implied by a call option is a linear transformation of the original stock-price distribution censored at the strike price. According to our result, by choosing a proper censoring point (i.e., an appropriate strike price), one can transform a project that is unambiguously riskier than others into a lottery that first-order stochastically dominates all others. Consequently, executives facing compensation schedules based on call options with such strike prices will always prefer to implement riskier projects, regardless of the specific functional form of their utility functions. We also show that this result is robust to scenarios in which one has limited information about the price probability distributions implied by each project.

These findings are complementary to Ross's (2004) in the following sense. Given Ross's impossibility result, one needs to further restrict the environment in order to get some possibilities. By restricting the class of admissible utility functions, Ross (2004) is able to find convex compensation schedules that would make any agent displaying decreasing absolute risk aversion become less risk averse. Our possibility theorem takes an alternative route. We do not restrict the class of utility functions; in fact, utilities here can be concave, convex, or neither. We show that if the distribution...

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