Stochastic Processes and Financial Mathematics
(part two)
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19.4 Modelling discussion on financial networks \(\msconly \)
The analysis in Section 19.3 was essentially suggested in 2010 by Gai and Kapadia1. It was one of the
earliest attempts at modelling debt contagion, and was published shortly after the financial crisis of 2008.
Typically, in the world of stochastic modelling, the first models of any new phenomenon are both simple and inaccurate; they provide a starting point for extension and refinement. With this in mind, let us discuss
the shortcomings of the analysis in Section 19.3, and what might (and, in some cases, has) be done to improve it.
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• The assumption \((\dagger )\) claims that a bank is equally dependent on each of its creditors. In practice, some loans are bigger than others, and some banks are better
able to absorb defaults than others. For example, if it was the case that larger loans tended to be between larger (and, consequently, more strongly connected) banks, the model would not capture the effect.
To correct this we’d want to understand the correlations between the size of a banks own balance sheet and the number of creditors/debtors it is connected too.
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• The approximation used to turn the cascade into a Galton-Watson process results in us only ever visiting each bank once. This means that, in our approximation, each bank
only ever sees one of its creditors default. As a result, we ignore the possibility that, once one of bank \(A\)s creditors defaults it becomes very likely that other creditors of \(A\) will also default.
For example, the real banking network could contain a core of strongly connected large banks all of whom lend large amounts to each other – and the effect of contagion essentially depends on how much it injures
this central network. (In fact, in several cases where data on banking networks is available, this is now believed to be the case.)
One obvious question to ask is ‘why not simply take the values from the real network and simulate a cascade of debt contagion on it?’. This does get done to some extent, with more complex models, but it is far
from a complete answer to the problem. Regulators are interested in how you can modify the network (i.e. restructure the graph or change the rules) to become more resilient, and this demands a deeper
understanding of the problem than simulations can typically provide. Also, simply running simulations and trusting them is very exposed to the shortcomings of the model, which itself may work well in some
situations and badly in others. In practice, simulations and theoretical study are combined to provide insight into how resilient banking networks are.
Similar (but not identical) methodology is used to model the spread of disease, the vulnerability of computer networks to hacking attacks, the propagation of news across social networks, and many other scenarios.