Fat-tailed Distribution under the Smooth Ambiguity Model

We study the ambiguity-adjusted return distribution induced by an investor with smooth ambiguity preferences à la Klibanoff et al. (2005), who faces uncertainty about the variance of asset returns. The variance uncertainty is modeled using a gamma distribution, a second-order prior over the family of normally distributed returns. Our main results present a density distortion that exponentially tilts this prior into an ambiguity-adjusted gamma distribution, characterized by its distorted rate parameter and shape parameter. A smaller distorted rate parameter implies greater weight on high-variance returns. This paper derives the ambiguity-adjusted return distribution as a symmetric variance–gamma distribution reflecting the investor’s risk and ambiguity aversion. The ambiguity-averse investor assigns a variance–gamma distribution with a higher likelihood of extreme returns, while the ambiguity-neutral investor assigns a distribution more peaked around the mean. We obtain the ambiguity-adjusted return variance as an increasing function of risk and ambiguity aversion. An empirical comparison is performed to calibrate the ambiguity aversion parameter of an in- vestor investing in gold.