# An extended likelihood framework for modelling discretely observed credit rating transitions

*M. Pfeuffer*,
*L. Möstel* and
*M. Fischer*

*Quantitative Finance*, 2019, vol. 19, issue 1, 93-104

**Abstract:**
The estimation of the parameters of a continuous-time Markov chain from discrete-time observations, also known as the embedding problem for Markov chains, plays in particular an important role for the modeling of credit rating transitions. This missing data problem boils down to a latent variable setting and thus, maximum likelihood estimation is usually conducted using the expectation-maximization (EM) algorithm. We illustrate that the EM algorithm is likely to get stuck in local maxima of the likelihood function in this specific problem setting and adapt a stochastic approximation simulated annealing scheme (SASEM) as well as a genetic algorithm (GA) to combat this issue. Above that, our main contribution is to extend our method GA by a rejection sampling scheme, which allows one to derive stochastic monotone maximum likelihood estimates in order to obtain proper (non-crossing) multi-year probabilities of default. We advocate the use of this procedure as direct constrained optimization (of the likelihood function) will not be numerically stable due to the large number of side conditions. Furthermore, the monotonicity constraint enables one to combine structural knowledge of the ordinality of credit ratings with real-life data into a statistical estimator, which has a stabilizing effect on far off-diagonal generator matrix elements. We illustrate our methods by Standard and Poor’s credit rating data as well as a simulation study and benchmark our novel procedure against an already existing smoothing algorithm.

**Date:** 2019

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