# CREDIT INTENSITY MODELS

Credit intensity models serves for modeling times to default or times to migration events. Used for default, they are known as the reduced form model, since they do not rely on conceptual foundations, as the "structural model of default," described in Chapter 42, does. The intensity model serves for modeling time elapsed between rare events. They have been implemented for market pricing for capturing "jumps" in market prices, and they apply as well both to defaults and migrations in the credit risk universe.

## Default Intensity Models

Credit intensity, or hazard, models are continuous time models. Over the small interval *At,* the intensity is the probability of default over the period, conditional on no prior default:

The default intensity per unit of time is *X(t).* In continuous time the small interval *At* tends towards zero and is replaced by the infinitesimal *dt.* The model is identical to the so-called "rare event process" or "jump process." Remember that the number of "jumps" or rare events is a count of discrete occurrences taking place during a given time interval. The number of rare events follows a Poisson distribution using *X* as the time intensity of rare events.

In pricing theory, jumps in asset prices are modeled by using stochastic processes for the asset return that depend both a diffusion term of variance *dt* and a Poisson process for adding "jumps," which are large variations that occur discretely. In credit risk, such model of rare events serves for modeling time to default rather that a discrete default probability. The default intensity is the number of defaults per unit of time. It is measured consistently with the time unit. For example, if the unit is one year, the time intensity should be measured accordingly. Default intensity models are also the foundation of the pricing of credit default swaps. The fixed leg of a CDS is made of a recurring premium, whether the variable leg is contingent on the first default event, making credit insurers pay the loss under default to the buyer of protection.

In order to demonstrate the connection between the jump model and the time to default, we can use as a first proxy a constant time intensity of default. In practice, the *X(t)* are piece-wise linear, that is constant over discrete time periods but variable across time periods. The time to default follows an exponential distribution. If *X* is a constant default rate, the probability of survival or of no default from 0 to *t* is (1 - *XAtf* over *n* time intervals of length *At — tin.* The survival probability is then compounded over the *n* small intervals.

When *n* increases, the expression (1 + *xlrif* tends towards exp(x):

Accordingly, the survival probability is an exponential function of time *t:*