Credit Default Swaps (CDS) and Credit Derivatives › Default Probabilities and Hazard Rates

To price CDS contracts rigorously, we need a framework for modeling default. The standard approach uses hazard rates (also called default intensities) to characterize the instantaneous probability of default. This framework is the foundation for credit derivatives pricing, allowing us to move from intuitive concepts to precise values.

The hazard rate $\lambda(t)$ represents the instantaneous conditional probability of default at time $t$, given survival up to that point. This concept may seem abstract at first, but it captures something fundamentally important: the intensity of default risk at each moment, measured as a rate rather than a probability. For a small time interval $\Delta t$:

$$P(\text{default in } [t, t+\Delta t] \mid \text{survival to } t) \approx \lambda(t) \Delta t$$

where:

• $\lambda(t)$: instantaneous hazard rate (default intensity) at time $t$
• $\Delta t$: small time interval
• $t$: current time

Think of the hazard rate as measuring the "default pressure" at each instant. A hazard rate of 0.02 means that, conditional on having survived until now, the probability of defaulting in the next small time interval $\Delta t$ is approximately $0.02 \times \Delta t$. If $\Delta t$ is one year, that corresponds to a 2% conditional probability; if $\Delta t$ is one day (roughly 1/365 of a year), the conditional probability is approximately 0.0055%.

The survival probability $Q(t)$, which is the probability of no default by time $t$, is related to the hazard rate by:

$$Q(t) = \exp\left(-\int_0^t \lambda(u) \, du\right)$$

where:

• $Q(t)$: probability of surviving (no default) until time $t$
• $\lambda(u)$: hazard rate at time $u$
• $\exp(\cdot)$: exponential function
• $t$: time horizon

This formula emerges from the mathematical requirement that survival requires avoiding default at every instant from time 0 to time $t$. The integral accumulates the hazard rate over time, and the exponential function converts this accumulated hazard into a probability. The negative sign in the exponent ensures that higher cumulative hazard translates to lower survival probability, as intuition demands.

For a constant hazard rate $\lambda$, this simplifies to:

$$Q(t) = e^{-\lambda t}$$

where:

• $Q(t)$: survival probability to time $t$
• $\lambda$: constant hazard rate
• $t$: time in years

This exponential survival function is analogous to radioactive decay: at any moment, there is a constant probability rate of default, regardless of how long the entity has survived. Just as a radioactive atom has no memory of how long it has existed, the constant hazard rate model assumes that a company's instantaneous default risk depends only on its current state, not on how long it has been solvent. While this "memoryless" property is a simplification of reality, it provides tractable mathematics and is a reasonable first approximation for many applications.

The cumulative default probability by time $t$ is simply one minus the survival probability. This relationship follows directly from the fact that an entity must either survive or default; there is no third possibility.

$$\begin{aligned} P(\text{default by } t) &= 1 - Q(t) && \text{(complement of survival)} \\ &= 1 - e^{-\lambda t} && \text{(substitute exponential model)} \end{aligned}$$

where:

• $Q(t)$: survival probability
• $\lambda$: constant hazard rate
• $t$: time horizon

For short time horizons and small hazard rates, this approximates to:

$$P(\text{default by } t) \approx \lambda t$$

where:

• $\lambda$: hazard rate
• $t$: time horizon

This linear approximation works because the exponential function $e^{-x} \approx 1 - x$ when $x$ is small. For a hazard rate of 2% and a one-year horizon, the exact default probability is $1 - e^{-0.02} = 0.0198$, while the approximation gives $0.02 \times 1 = 0.02$. The difference is negligible for practical purposes. However, as time extends or hazard rates increase, the approximation breaks down because it fails to account for the compounding effect: the longer the time horizon, the more the cumulative default probability accelerates relative to the linear approximation.