This paper examines whether tasking central banks with leaning against asset booms can conflict with their existing mandates to stabilize goods prices and output. The paper embeds the Harrison and Kreps (1978) model of speculative booms in a monetary model based on Rocheteau, Weill, and Wong (2018). In the model, a speculation shock that generates an asset boom is associated with higher output but a lower price level, unlike aggregate demand shocks that raise both output and prices. This creates a trilemma for central banks in that contemporaneous monetary policy cannot simultaneously stabilize output, the price level, and real asset prices. Stabilizing all three requires alternative policies.