Anthropogenic greenhouse gas emissions are changing the energy balance of our planet. Various climatic feedbacks make the resulting warming over the next decades and centuries highly uncertain. We quantify how this uncertainty changes the optimal carbon tax in a stochastic dynamic programming implementation of an integrated assessment model of climate change. We derive a general analytic formula for the "risk premium" governing the resulting climate policy. The formula generalizes simple precautionary savings analysis to more complex economic interactions and it builds the economic intuition for policy making under uncertainty. It clarifies the distinct roles of risk aversion, prudence, characteristics of the damage formulation, and future policy response. We show that an optimal response to uncertainty substantially reduces the risk premium.