Prediction markets, in which contract prices are used to forecast future events, are increasingly applied to various domains ranging from political contests to scientific breakthroughs. However, the dynamics of such markets are not well understood. Here, we study the return dynamics of the oldest, most data-rich prediction markets, the Iowa Electronic Presidential Election "winner-takesall" markets. As with other financial markets, we find uncorrelated returns, power-law decaying volatility correlations, and, usually, power-law decaying distributions of returns. However, unlike other financial markets, we find conditional diverging volatilities as the contract settlement date approaches. We propose a dynamic binary option model that captures all features of the empirical data and can potentially provide a tool with which one may extract true information events from a price time series.