Stochastic Models

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Part I

Introduction

Credit risk refers to the probability that the counterparty may not meet its contractual obligations. Bond rating is a good indicator of credit risk of a sovereign issuing the bonds. Thus it is important to analyse the possible future rating movements of a sovereign due to credit events.

1. Probability of Default Rating

To model the credit risks of bonds over time, a continuous time Markov model is implemented to calculate the probability of credit migration. In using a continuous time Markov model, the Kolmogorov forward equations are used:

where Pij(t) is the transition probability from rating i to rating j at time t, and rkj refers to the transition rates from rating k to rating j based on the annual transition rates matrix estimated. The above equation can be rewritten in matrix form, where P(t) is the matrix of transition probabilities over time t, and R is the annual transition rate matrix estimated.

A possible solution to the above equation is, where P(0) is the matrix of initial transition probabilities. It is assumed that the sovereign will remain in their current rating initially (at t=0), so