Stochastic Processes and Financial Mathematics
(part two)

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19.3 Approximating contagion by a Galton-Watson process $\msconly $

Consider the debt contagion model from the previous section. We are interested to discover more about the quantity $\mathscr {L}$ defined in (19.2).

We need to specify which graph $G$ we are using for the banking network. In fact, what we’ll do is use some approximations. Firstly, we think of the cascade defined in Section 19.2 as exploring, edge by edge, a large random graph $G$. Recall the degree distribution $D_G$, defined in (19.1).

1. We’ll imagine that our cascade explores a (large) random graph $G$. As we move through the graph, we map out the effects of the defaults.

In reality, to evaluate the cascade we would need to keep track of which nodes and edges we’ve already visited, but this makes our model too complicated. To make our calculations simpler:

2. We’ll assume that each time we add a new out-edge into $L_t$, its associated in-edge is attached to a previously unseen node.

Each time we move along a defaulted edge into previously unseen node, we do not (by assumption!) encounter any of the nodes/edges previously involved in the cascade. So, following any given defaulted edge results in a random number $G$ of new defaulted edges, independently of all else. This provides an important property: the number of loans $Z_n$ that are marked defaulted at each stage of the cascade is a Galton-Watson process.

This assumption we’ve just made is an approximation – we are essentially approximating $G$ with a randomly sampled tree. We need to be precise about how the sampling is done. At the same time, from Section 4.3 it is clear that we are most interested in knowing the expectation of the offspring distribution of the Galton-Watson process. In other words, we want to know, when we follow an edge of the cascade, the expected number of new edges that become added into the cascade.

Consider what would happen if we sampled an edge, uniformly at random from $G$, and moved along it. We use this as our approximation for what is found when we follow a defaulted edge. Where do we end up? Given a node $v\in G$, the number of in-edges of this node is $\indeg v$. Sampling a random edge is equivalent to sampling a random in-edge, so the chance that we end up at a given node $v$ is

\[\frac {\indeg (v)}{\sum \limits _{u\in V}\indeg (u)}.\]

The chance that this node fails as a result of our discovering it in our cascade (i.e. one of its in-edges defaults) is $\eta _j$, where $j=\deg _{in}(v)$. So the chance that we end up in $v$ and that $v$ is defaulted is

\[\frac {\indeg (v)\eta _{\indeg (v)}}{\sum \limits _{u\in V}\indeg (u)}.\]

When this case occurs, all out-edges of $v$ will become defaulted, which adds $\deg _{out}(v)$ new defaulted edges to our cascade. Hence, the expected number of newly defaulted edges that we discover in our cascade is

\begin{equation} \label {eq:argh} \sum \limits _{v\in V}\outdeg (v)\frac {\indeg (v)\eta _{\indeg (v)}}{\sum \limits _{u\in V}\indeg (u)}. \end{equation}

Let us write $|V|$ for the number of nodes in the graph and $|E|$ for the number of edges. Note that

\[|E|=\sum \limits _{u\in V}\indeg (u)=\sum \limits _{u\in V}\outdeg (u).\]

Also, let us write $p_{j,k}=\P [D_G=(j,k)]$. By definition of $D_G$, the number of nodes with degree $(j,k)$ is $|V|p_{j,k}$. Therefore, $\sum _{v\in V}(\ldots )$ is the same operation as $\sum _{j,k=0}^\infty |V|p_{j,k}(\ldots )$, where $(j,k)$ represents the degree of node $v\in V$, and so we have

\begin{align} \eqref {eq:argh} &=\sum \limits _{j,k=0}^\infty |V|p_{j,k}\frac {jk\eta _j}{|E|}\notag \\ &=\frac {|V|}{|E|}\sum \limits _{j,k=0}^\infty jkp_{j,k}\eta _j.\label {eq:gw_mean_gk} \end{align} This is the expected number of newly defaulted loans that result from any given defaulted loans. If this quantity is strictly greater than one, then our Galton-Watson process has positive probability of tending to $\infty $ – meaning that our cascade of defaults can grow infinitely large. If not, then our Galton-Waton process only ever contains finitely many defaulted loans. Therefore, our (approximate) analysis suggests that we could use the value of (19.4) as a criteria for how resilient our financial network is to debt contagion.

Stochastic Processes and Financial Mathematics (part two)

19.3 Approximating contagion by a Galton-Watson process \(\msconly \)

Stochastic Processes and Financial Mathematics
(part two)