We analyze the price dynamics of European allowances and international carbon credits in the second phase of the European carbon market. We develop and use a model combining fundamental drivers associated with the demand for quotas by installations and risk-return considerations related to the financial nature of carbon permits. We estimate it with autoregressive conditional heteroskedasticity models. Although carbon permits present some characteristics of financial assets, we find that an increased volatility is not associated with an increased return. The price of allowances and credits are explained by similar factors. However, whereas the corresponding returns present comparable dynamics, the long-term relationships between the price of these two types of permits and their drivers differ significantly. While the price of allowances is demand-driven, we suggest the existence of a supply-side effect for credits, and explain it by the flexibility in the related market. The impact of the European economic activity is less visible on credits than on allowances. The price elasticity of allowances with regards to the coal and gas prices is negative in time periods of low economic activity while it is positive in the rest of the time. We suggest an explanation for this dynamics difference.