Overview
The Poisson model is a model that utilizes the mathematics of survival analysis. It uses the Poisson distribution to model the time of default. The time of default is defined to be the time of first event of the Poisson distribution (labeled as {% \tau %}).
Model
Given a static value of the Poisson parameter {% \lambda %}, the probability that no default has occurred by time {% t %} is
{% P = exp(- \lambda t) %}
Note that this equation is the same as a
continuously compounded
discount rate. That is, {% \lambda %} can be interpreted as a
zero volatility spread
over the normal discount rate
that is charged to cover default risk.
Number of Defaults
The Poisson distribution is used to model the number of events that occur in a given time period. It can be applied to modeling the default of a loan, where default is assumed to occur whenever {% N %} (number of events) is greater than or equal to one.
In general, re-interpreting the number {% N %} to be either {% 0 %} or {% 1 %} because typically the probability of {% N \geq 2 %} is vanishing small.
The Poisson distribution has the convenient fact that they can easily be summed. That is given two Poisson distributions, {% L_1 \sim Pois(\lambda_1) %} and {% L_2 \sim Pois(\lambda_2) %} we have
{% L_1 + L_2 \sim Pois(\lambda_1 + \lambda_2) %}
This means that it is relatively easy to create a model of the number of defaults in a portfolio of loans, given
a Poisson model of default of a single loan.
Nonconstant Lambda
When {% \lambda %} is not constant, we have
{% \lambda(t) = \lim_{\Delta t \rightarrow 0} \; \frac{1}{\Delta t} P(t < T \leq t + \Delta t | T > t ) %}
Then the probability of surviving up to time {% T %} is
{% P(\tau > T) = (1-\lambda(dt)dt)(1-\lambda(2dt)dt) ... (1-\lambda(T)dt) %}
{% \displaystyle P(\tau > T) = exp(-\int_0^T \lambda(t)dt) %}
This is the standard relationship between the
survival function
and the
hazard rate.