Stochastic Hazard Rates and Affine Credit Models

A deterministic hazard function prices bonds and CDS correctly at calibration, but then credit spreads cannot move — they are locked to the fixed $\gamma^\mathbb{Q}(t)$ curve. This means the model cannot price credit spread options, explain why observed CDS spreads fluctuate over time (Figure 10.6 in QRM shows ratios of 1.5 to 3.5 over two years for a single firm), or capture the correlation between spreads and the business cycle. The remedy is to let the hazard rate itself be a stochastic process $(\gamma_t)$.

When the default time $\tau$ is doubly stochastic (a Cox process), the pricing machinery extends cleanly: conditional on the full path of $(\gamma_t)$, the firm behaves as if it has a deterministic hazard rate equal to that path. The key formula is that pricing a credit-risky claim reduces to pricing a default-free claim in a market with an adjusted short rate $R_t = r_t + \gamma_t$ — the "credit tax" on the discount factor. This separation makes the model tractable: you only need an effective model for $R_t = r_t + \gamma_t$.

Affine diffusion models are the workhorse for this. When $r_t$ and $\gamma_t$ are affine functions of some Markov state vector $(\Psi_t)$ driven by affine SDEs, the bond price takes the exponential-affine form $\exp(\alpha(T-t) + \beta(T-t)\Psi_t)$ — all the maturity dependence is in two deterministic functions $\alpha$ and $\beta$ satisfying an ODE (the Riccati equations). The CIR square-root diffusion gives a closed-form solution to these ODEs, making it the canonical hazard-rate process for single-name credit risk.

Learning objectives. By the end of this note you should be able to: