Bus 864 1 Simulating Continuous Time Rating Transitions

This note describes how to simulate state changes in continuous time Markov chains. An important application to credit risk is the evolution of a borrower’s credit rating. Default counts as a rating. Potential CFS contracts have payments contingent not only on time of default, but also on credit rating immediately prior (e.g., through effect on collateral support agreements). Thus a means of simulating the entire process is needed. 1. The continuous time transition generator In discrete time Markov chains, the likelihood of change between n states is described by a n×n transition matrix P ≡ [pij]. Element pij is the probability of transition from state i at time t to state j at time t+ 1. Elements of P are non-negative and each row sums to 1. In continuous time, the process is described by a generator matrix Q. Off-diagonal elements of Q are non-negative and each row sums to 0. The corresponding transition matrix over an interval of length t is etQ. The exponential of a matrix is defined in the same manner as the exponential function of a scalar: by series expansion. That is, etQ = I + tQ+ (tQ)2/2! + (tQ)3/3! +... (1) in which I denotes the identity matrix. As can be seen, setting t = 1 and supposing