Asset price bubbles in financial networks

Asset price bubbles are commonly defined as the difference between the market value of an asset and its fundamental value. In this work, we study the two main stages characterizing the evolution of an asset price bubble: a first phase when the bubble takes place and blows up, and a second period when the burst of the bubble can affect financial institutions, leading to a crisis. Networks play an essential role in our analysis: we study how an investors network influences the evolution of the bubble through trading effects via contagion between investors, and in which way a bubble alters the structure of a banking network due to preferential attachment mechanisms in investments between financial institutions. The first part of the thesis is devoted to the analysis of the first phase of the evolution of a bubble. In our approach, we follow the so called martingale theory of bubbles, recently developed starting from the assumption of absence of arbitrage. However, we slightly move the focus of the analysis: whereas the classic martingale theory of bubble spotlights the fundamental price of an asset, assuming that the market price is exogenously given, we start from a constructive model given by Jarrow et al. [2012], where the fundamental value is exogenously given and the market price can deviate from it due to illiquidity effects. This makes it possible to directly model the fast increase of the market value commonly observed when a bubble takes place. We embed this model in the martingale theory of bubbles, finding a flow of equivalent local martingale measures for the market wealth of the asset such that the fundamental wealth is justified as the expectation of discounted future earnings. In particular, we model the dynamics of the bubble as influenced by a contagion mechanism spreading among investors within a financial network. We show that the spread of contagion strongly depends on some characteristics of the network. In this way, the structure of the network affects the evolution of the bubble, through the impact of the trades on the price due to the illiquidity effects mentioned above. On the other hand, we study the effects on the economy of the burst, and show how a bubble can influence the structure of a banking network. We consider a banking network represented by a system of stochastic differential equations coupled by their drift. We assume a core-periphery structure, and that the banks in the core hold a bubbly asset. The banks in the periphery have not direct access to the bubble, but can take initially advantage from its increase by investing on the banks in the core. Investments are modeled by the weight of the links, which is a function of the robustness of the banks. In this way, a preferential attachment mechanism towards the core takes place during the growth of the bubble. We then investigate how the bubble distort the shape vii of the network, both for finite and infinitely large systems, assuming a non vanishing impact of the core on the periphery. Due to the influence of the bubble, the banks are no longer independent, and the law of large numbers cannot be directly applied at the limit. This results in a term in the drift of the diffusions which does not average out, and that increases systemic risk at the moment of the burst. We test this feature of the model by numerical simulations.