Central banks have already started to look at climate-related risks in the context of financial stability. Should they also take the carbon intensity of assets into account in the context of monetary policy? The guiding principle in the implementation of monetary policy has been 'market neutrality', whereby the central bank buys a proportion of the market portfolio of available corporate and bank bonds (in addition to government bonds). But this implies a carbon bias, because capital-intensive companies tend to be more carbon intensive. The author first reviews the legal mandate of the Eurosystem. While the primary objective is price stability, the Treaty on European Union allows the greening of monetary policy as a secondary objective. He proposes a tilting approach to steer or tilt the allocation of the Eurosystem's assets and collateral towards low-carbon sectors, which would reduce the cost of capital for these sectors relative to high-carbon sectors. This allocation policy must be designed so it does not affect the effective implementation of monetary policy. The working of the tilting approach is calibrated with data on European corporate and bank bonds. We find that a modest tilting approach could reduce carbon emissions in the corporate and bank bond portfolio by 44 per cent and lower the cost of capital of low carbon companies by 4 basis points. Our findings also suggest that such a low carbon allocation can be done without undue interference with the transmission mechanism of monetary policy. Price stability, the primary objective, is, and should remain, the priority of the Eurosystem.