This paper studies the volatility implications of anticipated cost-push shocks (i.e. news shocks) in a New Keynesian model under optimal unrestricted monetary policy with forward-looking rational expectations (RE) and backward-looking boundedly rational expectations (BRE). If the degree of backward-looking price setting behavior is sufficiently small (large), anticipated cost-push shocks lead to a higher (lower) volatility in the output gap and in the central bank's loss than an unanticipated shock of the same size. The inversion of the volatility effects of news shocks between rational and boundedly rational expectations follows from the inverse relation between the price-setting behavior and the optimal monetary policy. By contrast, if the central bank does not optimize and follows a standard Taylor-type rule and the price setters are purely (forward-) backward-looking, the volatility of the economy is (increasing with) independent of the anticipation horizon. The volatility results for the inflation rate are ambiguous.