## Hazard Rates

The relationship between the hazard rate $h(t)$, survival probability $S(t,T)$ and the forward hazard rate $h(t,T)$ is

Where $\Bbb{E}_t$ represents the risk-neutral expectation conditional on information at time $t$ and $h(t)=h(t,t)$.

Initial survival probabilities are represented as a log survival probability curve $I(t)$ defined as where $I(0)=0$ and $I(u)\ge I(t)$ for $u \ge t$.

## HWHazardRateModel

The Hull-White instantaneous hazard rate process is modeled as:

All symbols defined as per Hull White 1 factor for interest rates. The final form of the model is

Where:

• $B(t) = \frac{1-e^{-\alpha t}}{\alpha}$, $Y(t) \sim N(0, \frac{1-e^{-2 \alpha t}}{2\alpha})$
• $A(t,T)=\sigma^2 B(T-t)\Big(B(T-t)\frac{B(2t)}{2}+B(t)^2\Big)$