Overview

Credit risk — the risk of loss from a counterparty failing to meet its contractual obligations — is the dominant risk for commercial banks, and one of the largest for insurers with bond portfolios. Unlike market risk, where losses materialise continuously as prices move, credit losses are discrete, often large, and concentrated in economic downturns when correlations also spike. This topic develops the statistical and mathematical machinery for a single obligor: how to model default, price credit-risky bonds and CDS, and understand the structural and reduced-form approaches as complementary perspectives on the same event.

The two modelling philosophies are:

• Structural (firm-value) models: default occurs when asset value $V_t$ falls below a debt threshold $B$. The canonical example is the Merton (1974) model, which makes equity a call option on firm assets and prices credit risk from Black-Scholes. The KMV/EDF industry model operationalises this through distance-to-default.
• Reduced-form (hazard rate) models: the default time $\tau$ is modelled directly as a stopping time with a hazard rate process $\gamma(t)$. Deterministic hazard rates suffice for bond and CDS pricing; stochastic (doubly stochastic, Cox process) hazard rates handle default clustering and the term structure of credit spreads.

By the end of this topic you should be able to:

• define EAD, PD, and LGD and explain why they are jointly distributed, not independent;
• describe credit rating migration as a Markov chain and use empirical transition matrices;
• derive equity and debt prices in the Merton model and compute the Merton default probability and credit spread;
• explain the distance-to-default (DD) and the EDF methodology;
• define the hazard rate $\gamma(t)$ and compute survival probabilities and conditional default probabilities;
• price a default-free bond, a defaultable bond, and a CDS in a deterministic hazard-rate model;
• define a doubly stochastic default time and explain why it extends deterministic hazard rates to stochastic environments;
• describe the CIR square-root diffusion for the hazard rate and state the bond and CDS pricing formula under affine dynamics.

1. Contents
2. The big picture