Financial markets (and more generally the real economy) display a wide range of important nonlinearities. This paper focuses on stock returns, which are skewed left—generating crashes—and have volatility that moves over time, is itself skewed, is strongly related to the level of prices, and displays long memory. This paper shows that such behavior is actually almost inevitable when prices are formed by investors acquiring information about the true, but latent, value of stocks. It studies a general model of filtering in which agents receive signals about the fundamental value of the stock market and dynamically update their beliefs (potentially with biases). When those beliefs are non-normal and investors believe crashes can happen, prices generically display the range of nonlinearities observed in the data. While the model does not explain where crashes come from, it shows that investors believing that prices can crash is sufficient to generate the rich higher-order dynamics observed empirically. In a simple calibration with i.i.d. shocks to fundamentals, the model fits well quantitatively, and regression-based tests support the model’s mechanism.