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 \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 \lambda as a function of time \inline \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 [t,t+dt] is

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

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

p ( \tau < dt\ |\ \tau > 0 ) = \lambda 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 2dt and similar terms,

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

and in general

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

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

we see that

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

and its cumulative density function is

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

A few comments on this result. Firstly, note that the form of \inline p(\tau=t) is the Poisson distribution for \inline n=1, which makes a lot of sense since this was what we started with! It implies a mean survival time of \inline \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.

The survival/default probabilities for a firm in the exponential default model with various values of  decay parameter
The survival/default probabilities for a firm in the exponential default model with various values of decay parameter

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 p(\tau=t) is as follows:

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

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

p(\tau=t) = -{\partial S \over \partial t}

combining this we have

\lambda = -{1\over S}\cdot {\partial S \over \partial t}

and integrating gives

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

S(t) = e^{-\lambda t}

Fitting the initial discount curve in a stochastic rates model

I’ve introduced the Vasicek stochastic rates model in an earlier post, and here I’m going to introduce a development of it called the Hull-White (or sometimes Hull-White extended Vasicek) model.

The rates are modelled by a mean-reverting stochastic process

    \[dr = \bigl( \theta(t) - a r(t) \bigr) dt + \sigma dW_t\]

which is similar to the Vasicek model, except that the \theta(t) term is now allowed to vary with time (in general a and \sigma are too, but I’ll ignore those issues for today).

The freedom to set theta as a deterministic function of time allows us to calibrate the model to match the initial discount curve, which means at least initially our model will give the right price for things like FRAs. Calibrating models to match market prices is one of the main things quants do. Of course, the market doesn’t really obey our model. this means that, in time, the market prices and the prices predicted by our model will drift apart, and the model will need to be re-calibrated. But the better a model captures the various risk factors in the market, the less often this procedure will be needed.

Using the trick

    \[d \bigl( e^{at} r(t)\bigr) = e^{at} \bigl(dr + ar(t)dt \bigr)\]

to re-express the equation and integrating gives

    \[r(t) = r_0 e^{-at} + e^{-at}\int^t_0 \theta(u) e^{au} du + \sigma e^{-at} \int^t_0 e^{au} dW_u\]

where r_0 is the rate at t=0. The observable quantities from the discount curve are the initial discount factors (or equivalently the initial forward rates) P(0,t), where

    \[P(0,t) = {\mathbb E}\bigl[ e^{-\int^t_0 r(u) du} \bigr]\]

The rate r_0 is normally distributed, so the integral \int^t_0 r(u) du must be too. This is because an integral is essentially a sum, and a sum of normal distributions is also normally distributed. Applying the Ito isometry as discussed before, the expectation of this variable will come wholly from the deterministic terms and the variance will come entirly from the stochastic terms, giving

    \begin{align*} {\mathbb E} \bigl[ \int^t_0 r(u) du \bigr] &= r_0 B(0,t) + \int^t_0 e^{-au} (\int^u_0 \theta(s) e^{as} ds ) du \nonumber \\ {\mathbb V}\bigl[ \int^t_0 r(u) du \bigr] &= \sigma^2 \int^t_0 B(u,t)^2 du \nonumber \end{align*}

where throughout

    \[B(t,t') = {1 \over a}\bigl( 1 - e^{-a(t'-t)}\bigr)\]

and since

    \[{\mathbb E} \bigl[ e^x | x \sim {\mathbb N}(\mu,\sigma^2) \bigr] = e^{\mu + {1\over 2}\sigma^2}\]

we have

    \[P(0,t) = \exp \Bigl\{ -r_0 B(0,t) - \int^t_0 e^{-au} \bigl( \int_0^u \theta(s) e^{as} ds + {1 \over 2} \sigma^2 B(u,t)^2 \bigr) du \Bigr\}\]

Two differentiations of this expression give

    \[-{\partial \over \partial t} \ln P(0,t) = r_0 e^{-at} + e^{-at} \int^t_0 \theta(u) e^{au} du + {1\over 2} \sigma^2 B(0,t)^2\]

    \[-{\partial^2 \over \partial t^2} \ln P(0,t) = -ar_0 e^{-at} + \theta(t) - ae^{-at} \int^t_0 \theta(u) e^{au} du + \sigma^2 e^{-at} B(0,t)\]

and combining these equations gives an expression for \theta(t) that exactly fits the initial discount curve for the given currency

    \[\theta(t) = -{\partial^2 \over \partial t^2} \ln P(0,t) - a{\partial \over \partial t} \ln P(0,t) + {\sigma^2\over 2a}( 1 - e^{-2at} )\]

and since -{\partial \over \partial t} \ln P(0,t) = f(0,t) is simply the initial market observed forward rate to each time horizon coming from the discount curve, this can be compactly expressed as

    \[\theta(t) = {\partial \over \partial t} f(0,t) + a f(0,t) + {\sigma^2 \over 2a} ( 1 - e^{-2at} )\]

Today we’ve seen how a simple extension to the ‘basic’ Vasicek model allows us to match the initial discount curve seen in the market. Allowing the volatility parameter to vary will allow us to match market prices of other products such as swaptions (an option to enter into a swap), which I’ll discuss another time. But we’re gradually building up a suite of simple models that we can combine later to model much more complicated environments.