Applications of hidden Markov chains to credit risk modelling.

摘要

We propose that the credit rating evolution can be described by a Markov chain but that we do not observe this Markov chain directly. Rather, it is hidden in "noisy" observations represented by the posted credit ratings. We consider the discrete time model with a Markov Chain observed in martingale noise (Hidden Markov Model). By introducing a new probability measure we are able to obtain unnormalized, recursive estimates for the state of the Markov chain governing the credit rating evolution. We use the so-called EM (Expectation Maximization) algorithm to estimate the parameters of the model, namely probabilities of migration between "true"' credit quality states and probabilities of observing a particular rating given the "true" credit worthiness of the issuer. The model is then applied to a data set of credit ratings obtained from the Standard and Poor's COMPUSTAT database. We also consider a Kalman filtering model for estimating the dynamics of credit quality aimed to overcome some of the challenges posed by the nature of available credit rating data.; Finally, we introduce an intensity-based credit migration model of default risk. We take default to be an unpredictable event governed by a hazard process defined in terms of intensity. The value of a zero-recovery defaultable zero-coupon bond is then its value if it were risk-free, adjusted by the probability of no default before maturity. This probability is calculated explicitly in terms of intensity and the issuer's credit quality. We suppose that the latter is governed by a Markov chain and distinguish two cases. First we take the issuer's credit rating to represent the "true" credit quality and then extend the model to value zero-recovery defaultable bonds when "true" credit quality is not observed directly but only through noisy observations given by posted ratings. We also consider valuation of defaultable bonds when a fraction of face value is paid at the time of default.