We consider a continuous-time framework featuring a central bank, private agents, and a financial market. The central bank's objective is to maximize a functional, which measures the classical trade-off between output and inflation over time plus income from the sales of inflation-indexed bonds minus payments for the liabilities that the inflation-indexed bonds produce at maturity. Private agents are assumed to have adaptive expectations. The financial market is modeled in continuous-time Black–Scholes–Merton style and financial agents are averse against inflation risk, attaching an inflation risk premium to nominal bonds. Following this route, we explain demand for inflation-indexed securities on the financial market from a no-arbitrage assumption and derive pricing formulas for inflation-linked bonds and calls, which lead to a supply-demand equilibrium. Furthermore, we study the consequences that the sales of inflation-indexed securities have on the observed inflation rate and price level. Similar to the study of Walsh, we find that the inflationary bias is significantly reduced, and hence that markets for inflation-indexed bonds provide a mechanism to reduce inflationary bias and increase central bank's credibility.