Before revision:  This includes European call, put, and digital options, forward start options, and forward start optionson the underlying return. We derive sharp analytical and numerical bounds on the series truncation errors. We illustrate the performance by numerical examples, which show that our approach offers a viable alternative to Fourier transform techniques .
After revision:  This includes European call, put, and digital options, forward start options, and can be applied to discretely monitored Asian options. In a numerical analysis we find that the price approximations become accurate within short CPU time .