Most modern energy markets trade electricity in advance for technical reasons. Thus, market participants must commit to delivering or consuming a certain amount of energy before the actual delivery. In Germany, two markets with daily auctions coexist. In the day-ahead auction market, the energy is traded in 60-minute time slots, which are further partitioned into 15-minute time slots for the intraday auction market. Because of the slow ramp-ups of nuclear and fossil power plants, these price-makers trade mostly in the day-ahead market. Only the residual energy is traded in the intraday market, where the market prices fluctuate substantially more. These fluctuations as well as the expected price difference between these markets can be exploited by fast ramping energy storage systems. We address the decision problem of an owner of an energy storage who trades on both markets, taking ramping times into account. Because the state variable of our dynamic programming formulation includes all features of our high-dimensional electricity price forecast, this problem cannot be solved to optimality. Instead, we use approximate dynamic programming. In a numerical study based on real-world data, we benchmark the algorithm against an adapted state-of-the-art approach from literature and an expectation model with a receding horizon. Furthermore, we investigate the influence of the price forecast on expected profit and demonstrate that it is essential for the dynamic program to capture the high dimensionality of the price forecast to compete with the expectation model, which does not suffer from the curses of dimensionality.