This paper studies the volatility implications of anticipated cost-push shocks (i.e. news shocks) in a New Keynesian model with hybrid price setting both under optimal unrestricted and discretionary monetary policy with flexible inflation targeting. If the degree of backward-looking price setting behavior is sufficiently small (large), anticipated cost-push shocks lead in both policy regimes to a higher (lower) volatility in the output gap and in the central bank's loss than an unanticipated shock of the same size. This inversion of the volatility effects of news shocks follows from the inverse relation between the price-setting behavior and the optimal monetary policy. Under a fully microfounded hybrid New Keynesian Phillips curve with price indexation, this inversion of volatility results is not possible since the Phillips curve remains hybrid even in the limit case of full price indexation.