It has been shown that maximum distance profile (MDP) convolutional codes have optimal recovery rate for windows of a certain length, when transmitting over an erasure channel. In addition, the subclass of complete MDP convolutional codes has the ability to reduce the waiting time during decoding. Since so far general constructions of these codes have only been provided over fields of very large size, there arises the question about the necessary field size such that these codes could exist. In this paper, we derive upper bounds on the necessary field size for the existence of MDP and complete MDP convolutional codes and show that these bounds improve the already existing ones. For some special choices of the code parameters, we are even able to give the exact minimum field size. Moreover, we derive lower bounds for the probability that a random code is MDP respective complete MDP.