## Credit Default Swaps Pt I: A Default Model for Firms

In today’s post I’m going to discuss a simple model for default of a firm, and in Part II I’ll discuss the price of insuring against losses caused by the default. As usual, the model I discuss today will be a vast over-simplification of reality, but it will serve as a building block for development. Indeed, there are many people in the credit derivatives industry who take these things to a much higher level of complexity.

Modelling the default of a firm is an interesting challenge. Some models are based around ratings agencies, giving firms a certain grade and treating transitions between grades as a Markov chain (similar to a problem I discussed before). I’m going to start with a simpler exponential default model. This says that the event of a given firm defaulting on all of its liabilities is a random event obeying a Poisson process. That is, it is characterised by a single parameter $\inline&space;\lambda$ which gives the likelihood of default in a given time period. We make a simplifying assumption that this is independent of all previous time periods, so default CAN’T be foreseen (this may be a weakness of the model, but perhaps not… discuss!).

Also, the firm can’t default more than once, so the process stops after default. A generalisation of the model will treat $\inline&space;\lambda$ as a function of time $\inline&space;\lambda(t)$ and even potentially a stochastic variable, but we won’t think about that for now.

Mathematically, the probability of default time tau occurring in the small window
$\inline&space;[t,t+dt]$ is

$\lim_{dt&space;\to&space;0}&space;p(&space;\tau&space;<&space;t&space;+&space;dt\&space;|\&space;\tau&space;>&space;t&space;)&space;=&space;\lambda&space;dt$

If we start at $\inline&space;t=0$ with the firm not yet having defaulted, this tells us that

$p&space;(&space;\tau&space;<&space;dt\&space;|\&space;\tau&space;>&space;0&space;)&space;=&space;\lambda&space;dt$

and in a second and third narrow window, using the independence of separate time windows in the Poisson process and taking a physicist’s view on the meaning of $\inline&space;2dt$ and similar terms,

$\begin{matrix}&space;p&space;(&space;dt&space;<&space;\tau&space;<&space;2dt&space;)&space;&&space;=&space;&&space;p&space;(&space;\tau&space;<&space;2dt\&space;|\&space;\tau&space;>&space;dt&space;)\cdot&space;p(\tau&space;>&space;dt&space;)&space;\\&space;&=&&space;\lambda&space;dt&space;\cdot&space;(1-&space;\lambda&space;dt)&space;\end{matrix}$$\begin{matrix}&space;p&space;(\&space;2dt&space;<&space;\tau&space;<&space;3dt\&space;)&space;&=&&space;\lambda&space;dt&space;\cdot&space;\bigl(1&space;-&space;\lambda&space;dt&space;\cdot(1&space;-&space;\lambda&space;dt)&space;-&space;\lambda&space;dt&space;\bigr)\\&space;&=&&space;\lambda&space;dt&space;\cdot&space;(1&space;-&space;2\lambda&space;dt&space;-&space;\lambda^2&space;dt^2)\\&space;&=&&space;\lambda&space;dt&space;\cdot&space;(1&space;-&space;\lambda&space;dt)^2\end{matrix}$

and in general

$p&space;(\&space;t&space;<&space;\tau&space;<&space;t+dt\&space;)&space;=&space;(1-&space;\lambda&space;dt)^n\cdot&space;\lambda&space;dt$where $\inline&space;n&space;=&space;{t&space;\over&space;dt}}$ and we must take the limit that $\inline&space;dt&space;\to&space;0$, which we can do by making use of the identity

$\lim_{a\to&space;\infty}&space;(1&space;+&space;{x&space;\over&space;a})^a&space;=&space;e^x$

we see that

$\begin{matrix}&space;\lim_{dt&space;\to&space;0}p&space;(\&space;t&space;<&space;\tau&space;<&space;t+dt\&space;)&space;&&space;=&space;&&space;\lambda&space;e^{-\lambda&space;t}\\&space;&&space;=&space;&&space;p(\tau&space;=&space;t)&space;\end{matrix}$

and its cumulative density function is

${\rm&space;F}(t)&space;=&space;\int_0^t&space;p(\tau&space;=&space;u)&space;du&space;=&space;1&space;-&space;e^{-\lambda&space;t}$which gives the probability that default has happened before time $\inline&space;t$ (the survival function $\inline&space;{\rm&space;S}(t)&space;=&space;1&space;-&space;{\rm&space;F}(t)$ gives the reverse – the chance the firm is still alive at time $\inline&space;t$) *another derivation of $\inline&space;p(\tau=t)$ is given at the bottom

A few comments on this result. Firstly, note that the form of $\inline&space;p(\tau=t)$ is the Poisson distribution for $\inline&space;n=1$, which makes a lot of sense since this was what we started with! It implies a mean survival time of $\inline&space;\lambda^{-1}$, which gives us some intuition about the physical meaning of lambda. The CDF (and consequently the survival function) are just exponential decays, I’ve plotted them below.

Having characterised the default probability of the firm, in Part II I will think about how it affects the price of products that they issue.

*Another derivation of $\inline&space;p(\tau=t)$ is as follows:

$\lambda&space;dt&space;=&space;p&space;(&space;\tau&space;<&space;t+dt\&space;|\&space;\tau&space;>&space;t&space;)&space;=&space;{p(t&space;<&space;\tau&space;<&space;t+dt)\over&space;p(\tau&space;>&space;t)}$

the first of these terms is approximately $\inline&space;p(\tau=t)dt$, while the second is simply the survival function $\inline&space;S(t)$, which by definition obeys

$p(\tau=t)&space;=&space;-{\partial&space;S&space;\over&space;\partial&space;t}$

combining this we have

$\lambda&space;=&space;-{1\over&space;S}\cdot&space;{\partial&space;S&space;\over&space;\partial&space;t}$

and integrating gives

$-\lambda&space;t&space;=&space;\ln&space;{S(t)&space;\over&space;S(0)}$from which we get the result above, that

$S(t)&space;=&space;e^{-\lambda&space;t}$