This note deals with the stability properties of an economy where the central bank is concerned with stock market developments. We introduce a Taylor rule reacting to stock price growth rates along with inflation and output gap in a New-Keynesian setup. We explore the performance of this rule from the vantage of equilibrium uniqueness. We show that this reaction function is isomorphic to a rule with an interest rate smoothing term, whose magnitude increases in the degree of aggressiveness towards asset prices growth. As shown by Bullard and Mitra (2007, Determinacy, learnability, and monetary policy inertia, Journal of Money, Credit and Banking 39, 1177-1212) this feature of monetary policy inertia can help at alleviating problems of indeterminacy.