Typical characteristics of electricity day-ahead prices at the European Power Exchange (EPEX) are very high volatility and a large number of extreme price changes. In this paper, we look at hourly spot prices at the German electricity market and apply extreme value theory (EVT) to investigate the tails of the price change distribution. Our results show the importance of delimiting price spikes and modeling them separately from the core of the price distribution. In particular, we get a realistic fit of the generalized Pareto distribution (GPD) to autoregressive-generalized autoregressive conditional heteroscedasticity (AR-GARCH) filtered price change series; based on this model, accurate forecasts of extreme price quantiles are obtained. Generally, our results suggest that EVT is of interest for both risk managers and portfolio managers in the highly volatile electricity market.