A dynamical systems approach to systemic risk

In the last years, many physical works have achieved important results in interdisciplinary fields, such as finance, applying the methodologies of dynamical systems theory, networks theory, and statistical mechanics. A topic of growing interest concerns the study of the dynamics of financial systems, and specifically the study of their stability. Typically, this problem is approached in the field of complex systems. In fact, financial markets exhibit several of the properties that characterize complex systems. They are open systems in which many subunits, financial institutions, interact nonlinearly in the presence of feedbacks. The aim of the thesis concerns the development of a theoretical model based on empirical evidences, in order to explain the intrinsic features of the dynamics of financial systems. Particularly, we focus on the problem of systemic risk and systemic financial stability, a problem of renewed interest after the financial crisis of 2007-2009. In financial markets, systemic risk is in effect an emergent phenomenon and can be seen as a phase transition from stability to instability. Specifically, systemic risk refers to a financial instability that affects initially a micro-region of the system, but it has bad consequences at the macro level, inducing potentially a catastrophic collapse of the entire financial market. Although systemic financial instability is usually triggered by a stochastic event, there are many empirical evidences, which lead to consider systemic risk as related to the network structure and dynamical properties of financial systems. A financial market can be seen as a bipartite network characterized by financial institutions (representing one type of nodes) owning a portfolio (i.e. creating links between nodes) of some risky investment assets (the other type of nodes), whose values evolve stochastically in time. Each investment asset is characterized by a risk, that is the diffusion rate of its price. In order to maximize their profit minimizing the risk, financial institutions in creating a portfolio, choose the optimal values of diversification (number of assets in the portfolio) and financial leverage (the ratio between invested capital and initial equity). The major financial institutions operating in real financial markets, adopt a strategy based on the periodical rebalancing of portfolio, in order to maintain fixed the chosen (target) leverage. The portfolio rebalancing induces feedback effects on asset prices, because in financial markets the demand (supply) for an asset tends to put as upward (downward) pressure on its price. The larger are leverage and diversification, the larger is the impact of rebalancing feedback effects. This mechanism is in effect a positive feedback. Since the positive feedback has the effect of amplifying asset price movements, it increases the risk of assets in financial markets and, as a consequence, has the potential of leading to the breaking of systemic financial stability. In this thesis, we propose a dynamical systems approach to systemic risk, in order to answer the following question: What is the relationship between the role of feedback effects, defining the dynamical behavior of financial institutions at the micro level, and the emerging macro consequences on systemic financial stability? Specifically, we focus on how financial institutions respond to an increase of risk due to the positive feedback. In solving the problem of portfolio optimization, financial institutions form expectations about future asset risks through estimates of observed past risks. We consider two types of expectations scheme: naive expectations, i.e. when financial institutions forecast future risk to be equal to the last observed one, and adaptive expectations, i.e. when financial institutions forecast future risk to be equal to a weighted average, with geometrically declining weights, of all past risks. According to the forecasting strategy, they define periodically the optimal portfolio configuration. In our dynamical approach, the key point is that the portfolio choices based on the expectations scheme adopted by financial institutions together with the impact on risk due to feedback effects arising from the target leverage strategy, drive the dynamics of financial market. Depending on diversification costs, under naive expectations, at a given threshold the positive feedback triggers the appearance of financial cycles characterized by a sequence of speculative periods and non-speculative ones. During financial cycles, we can notice how the financial leverage switches from aggressive configurations (speculative periods) to cautious ones (non- speculative periods). The financial leverage cycles reflect the occurrence of periods characterized by a macro-component of risk, due to an higher impact of feedback effects, followed by periods in which feedback effects do not affect importantly the asset risk. When financial cycles appear, the amplitude of cycles can be interpreted as a measure of systemic risk in financial market. Furthermore, under adaptive expectations, the financial system exhibits a dynamical transition from a periodic cyclical behavior to (deterministic) chaos. In the chaotic regime, in addition to the occurrence of highly risky periods identified by a very large macro-component of risk, the chaotic dynamical behavior of financial system is characterized by positive entropy, suggesting how much an improvement of expectations scheme by financial institutions may be hard due to missing information about financial market dynamics. Although feedback effects have been recognized as an important source of systemic risk in financial markets, this thesis represents an original dynamical systems approach to the considered problem. Particularly, our work focuses on the possible dynamical outcomes displayed by a financial system due to rebalancing feedback, and not only on the consequences on asset prices and risks due to the feedback effects. We believe that our original results, which especially suggest the possibility that chaos may occur in financial markets, indicate that our model deserves attention. Furthermore, a result of this type bypasses the specific dynamical model under consideration, since the occurrence of chaos in financial markets may be the consequence of universal aspects related to the nonlinearity of the feedback effects. Finally, from the point of view of financial market policy, we believe that our original results deserve attention because they highlight how the dynamical properties of financial markets may drastically change when market conditions change. Specifically, a decrease of diversification costs in the presence of strong feedback effects may lead to the breaking of systemic financial stability