Reduced-Form Models: Hazard Rates and Risk-Neutral Default Probabilities

1. Nonzero short-term credit spreads (because $\gamma(0) > 0$ means the firm can default imminently).
2. Direct calibration to CDS market quotes.
3. Tractable pricing formulas for bonds and CDS.

The hazard rate $\gamma(t)$ is the instantaneous probability of default at time $t$, given survival to $t$. The survival function takes the exponential form $\bar F(t) = \exp(-\int_0^t \gamma(s)\,ds)$. For pricing purposes, we work under a risk-neutral measure $\mathbb{Q}$ with risk-neutral hazard function $\gamma^\mathbb{Q}(t)$, which differs from the physical hazard by a risk premium. A key practical result: for a constant hazard rate and short maturities, the CDS spread satisfies $x^* \approx \gamma^\mathbb{Q} \cdot \delta$ — the spread is approximately the product of the risk-neutral default intensity and the LGD.

• define the cumulative hazard function $\Gamma(t)$ and hazard function $\gamma(t)$;
• derive the survival probability formula $\bar F(t) = \exp(-\int_0^t \gamma(s)\,ds)$;
• compute conditional survival probabilities using (10.28);
• price a default-free zero-coupon bond and a defaultable bond under the RT recovery model;
• state the premium and default payment legs of a CDS and derive the fair spread formula;
• use the approximation $\gamma^\mathbb{Q} \approx x^*/\delta$ to back out the implied hazard rate from CDS quotes;
• explain the ratio $q/p$ between risk-neutral and physical default probabilities.

1. 1. Hazard rate models
2. 1. Bond pricing under the RT recovery model
3. 1. CDS pricing and calibration
4. 1. Risk-neutral vs physical default probabilities