Why Does Implied Risk Aversion Smile?

Abstract

Implied risk aversion estimates reported in the literature are strongly U-shaped. This article explores different potential explanations for these ‘‘smile’’ patterns: (i) preference aggregation, both with and without stochastic volatility and jumps in returns, (ii) misestimation of investors’ beliefs caused by stochastic volatility, jumps, or a Peso problem, and (iii) heterogeneous beliefs. The results reveal that preference aggregation and misestimation of investors’ beliefs caused by stochastic volatility and jumps are unlikely to be the explanation for the smile. Although a Peso problem can account for the smile, the required probability of a market crash is unrealistically large. Heterogeneous beliefs cause sizable distortions in implied risk aversion, but the degree of heterogeneity required to explain the smile is implausibly large. (JEL: G12, G13) In a representative agent economy, equilibrium asset prices reflect the agent’s preferences and beliefs. Rubinstein (1994) showed that any two of the following imply the third: (i) the representative agent’s preferences, (ii) his subjective probability assessments, and (iii) the state-price density. Therefore, essentially any state-price density can be reconciled with the distribution of asset prices by using an appropriate set of preferences for the representative agent. Building on this insight, a number of papers have derived estimates of the representative agent’s risk aversion from the state-price density and the subjective probability density. Ao¨t-Sahalia and Lo (2000), Jackwerth (2000), and Rosenberg and Engle (2002) report implied risk aversion estimates that have similar shapes, with some differences. They all found that implied risk aversion varies strongly across S&P 500 index values and is U-shaped around the futures price. Jackwerth (2000) and Rosenberg and Engle (2002) even obtained negative risk aversion estimates for a range of index values centered around the futures price. The aim of this article is to explore different potential explanations for the implied risk aversion smile by investigating the properties of implied