There are two main families of default models.

Structural models, such as the Merton model based on the firm’s assets and liabilities and reduced-form models focusing directly on the timing of default.

We will give here an introduction to reduced-form credit risk models.

## Default Time Modelling

In reduced-form or intensity-based models, the default time 𝛕 is the first time there is a jump in a Poisson process (link).

## Constant Default Intensity Model

We consider an homogeneous Poisson process N(t) with parameter λ.

N(0) = 0

And the probability to have N(t) = k, so k jump between 0 and t has the following expression. N(t) follows a Poisson(λ.t) distribution.

The default probability between 0 and t is the probability that there is at least one jump in the Poisson process before t.

We recognize the cumulative distribution function of an exponential distribution with parameter λ. The default time 𝛕 follows an exponential distribution.

λ is the default intensity. λ x dt gives the probability that there is a default between t and t+dt knowing that there was no default before t. It is the marginal default probability between t and t+dt.

Proof:

## Non-Constant Default Intensity Model

We consider now a non-homogeneous Poisson process N(t) with an intensity (λ_{t}).

N(0) = 0

The default probability has the following expression, it is a function of the integral of the default intensity between 0 and t.

In this case, the marginal default probability between t and t+dt is given by λ_{t} x dt.

As discussed in another article (link), the evolution of the cumulative and marginal default probabilities depend on the credit quality of the borrower.

For an investment grade borrower with a good credit quality, the default risk is low on the short-term. However if the borrower does not default, there are risks that it will not able to maintain the same credit quality over time, that it downgrades over time meaning an increase of the marginal default probability.

For a high yield borrower with a bad credit quality it is the opposite. Its default probability is high on the short-term. But if it survives it is likely that it upgrades meaning a better credit quality over time and a decrease of the marginal default probability.

## Stochastic Default Intensity Model

A Cox process is a Poisson process where the intensity is itself a stochastic process.

In this framework we have:

Duffie and Singleton (1999) uses a Cox-Ingersoll-Ross process for the default intensity:

The survival probability S(t) has the following expression in this case. It is similar to the pricing of a zero coupon bond assuming that the short-term interest rate follows a Cox-Ingersoll-Ross process (link).

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