I am doing some reading on the derivation of credit rating migration/transition matrices and probability of default term structures. I understand that a homogeneous Markov chain can be either discrete-time or continuous-time.
In the discrete-time case, the one year transition matrix can be derived using the cohort method which involves computing the proportion of observed migrations from the beginning of a year to the end of a year.
In the continuous-time case, we need to derive a generator matrix G. I am not sure I understood it correctly, but the generator matrix needs to be derived from an observed transition matrix, correct? If yes, what is the best way of obtaining the observed transition matrix? Simply the cohort method (and afterwards some smoothing etcetera to ensure that the generator matrix exists) or should I take into account all rating migrations during the one year time interval?