Ambiguous Volatility and Asset Pricing in Continuous Time

This paper formulates a model of utility for a continuous time framework that captures the decision-maker’s concern with ambiguity about both volatility and drift. Corresponding extensions of some basic results in asset pricing theory are presented. First, we derive arbitrage-free pricing rules based on hedging arguments. Ambiguous volatility implies market incompleteness that rules out perfect hedging. Consequently, hedging arguments determine prices only up to intervals. However, sharper predictions can be obtained by assuming preference maximization and equilibrium. Thus we apply the model of utility to a representative agent endowment economy to study equilibrium asset returns. A version of the C-CAPM is derived and the e¤ects of ambiguous volatility are described. Key words: ambiguity, option pricing, recursive utility, G-Brownian motion, robust stochastic volatility, sentiment, overcon…dence, optimism Department of Economics, Boston University, [email protected] and School of Mathematics, Shandong University, [email protected]. We gratefully acknowledge the …nancial support of the National Science Foundation (awards SES-0917740 and 1216339), the National Basic Research Program of China (Program 973, award 2007CB814901) and the National Natural Science Foundation of China (award 10871118). We have bene…ted also from very helpful comments by Pietro Veronesi and two referees, and from discussions with Shige Peng, Zengjing Chen, Mingshang Hu, Gur Huberman, Julien Hugonnier, Hiro Kaido, Jin Ma, Semyon Malamud, Guihai Zhao and especially Jianfeng Zhang. Epstein is grateful also for the generous hospitality of EIEF and CIRANO where much of this work was completed during extended visits. The contents of this paper appeared originally under the title “Ambiguous volatility, possibility and utility in continuous time,”…rst posted March 5, 2011. That paper now focuses on a mathematically rigorous treatment of the model of utility and excludes the application to asset pricing.