• chaos
• complex systems
• dynamical systems
• econophysics
• nonlinear dynamics
• systemic risk

In this thesis, a special instance of a system consisting of several interacting degrees of freedom (or "agents") is studied. In particular, the agents will be banks, and their interaction will happen through them investing in the same assets. This system may exhibit a kind of emergent behavior known as systemic risk, which, in a financial context, refers to the event in which multiple institutions fail together. This may result in a large-scale crisis, whose impact on the economy shall not be underestimated; indeed, the largest banks have asset sizes of the order of several trillions of US dollars, which is more than most countries’ GDP.
As mentioned, systemic risk may be due to banks interacting by investing in the same asset; shall the latter severely lose value, all the banks may, in principle, simultaneously fail. This will be the main mechanism for systemic risk considered in this thesis.
It turns out that the dynamics of the assets’ returns is endogenously determined by the portfolio choices of the banks, so that this kind of systemic risk is crucially dependent on banks’ leverages, the evolution of which will therefore be the main object of study. The leverage may be considered as the state variable associated with each agent constituting the system, and it is defined as the ratio between the capital invested by a bank and the capital really owned by it when its debts are paid back. As the name itself suggests, a large leverage means an amplification of resources invested, which is made possible by the bank borrowing money to invest (a "pumping" of money from the creditors allows the bank to amplify its invested resources) and which translates into an amplification of gains (but also, possibly, losses).
To maximize gains, banks want to be as leveraged as possible, but they have to face constraints imposed by institutions on the maximum leverage attainable. This results in banks adopting a target leverage strategy, which can cause, due to finite liquidity of the market, a positive feedback between leverages and asset prices, possibly leading to instabilities (very much like what happens in physical systems with positive feedback) and thus to an increased systemic risk.
In most previous works considering the impact of this kind of feedback on the dynamical properties of the leverages, either a single bank was considered or multiple banks and multiple assets but with all banks being in some sense equivalent: in particular, their leverages were at all times equal. In this thesis, starting from some of these previous works, models of increasing generality for the evolution of the leverages are developed to deal with more general settings, up to the one in which several banks of different sizes arbitrarily choose the assets in which to invest among several (possibly distinct) ones.
Having obtained the equations for the evolution of the leverages, numerical simulations are conducted to explore their behavior, then trying to interpret them both intuitively and analytically. Whenever feasible, analytical results concerning in particular topological aspects of the dynamical systems obtained are provided.
Next, the connection between the dynamics of the leverages and the riskiness of the configuration of the system is explored further, studying how systemic risk (of the kind above) varies as parameters are varied. To give some examples of the obtained results, it is observed and proven that in case two groups of banks are present, one with a much larger "size" than the other, then the choices of the larger group alone determine the level of risk of the whole system. Also, when for example two banks and two assets are considered, it is observed that the smaller bank is in a riskier situation the more the larger bank distributes its capital evenly among the two assets.
In developing the models, specific assumptions will be made about the system under consideration; of course, other choices would lead to different, and possibly more accurate, models. Nevertheless, it is hoped that the results obtained might serve as starting points for further generalizations.
In the last part, another mechanism of systemic risk is included in the description, namely, the one determined by fire sales: an initial failure of a single bank will cause a loss of value of the assets it was investing in, in turn determining a loss of value in the portfolios of banks with open positions in those assets, which may lead to them failing as well. In particular, the case in which the initial failure triggering the "contagion" is caused by an endogenous drop in the value of some assets is developed, coming to define some “observables” to serve as proxies of the riskiness of the configuration.
As intuitively expected, the larger the leverages and the portfolio overlaps, the larger the risk of contagion spreading.
Throughout the thesis, several connections with physics will emerge, coming both from analogies between the models here obtained and the ones which may describe physical systems and from the nature of the method and approach employed, which is familiar to the physicist.