A simple financial market model with chartists and fundamentalists: market entry levels and discontinuities

We present a simple financial market model with interacting chartists and fundamentalists. Since some of these speculators only become active when a certain misalignment level has been crossed, the dynamics are driven by a discontinuous piecewise linear map. The model endogenously generates bubbles and crashes and excess volatility for a broad range of parameter values - and thus explains some key phenomena of financial markets. Moreover, we provide a complete analytical study of the model's dynamical system. One of its surprising features is that model simulations may appear to be chaotic, although only regular dynamics can emerge.financial market crisis; bull and bear market dynamics; discontinuous piecewise linear maps; border-collision bifurcations; period adding scheme.

The Role of Constraints in a Segregation Model: The Symmetric Case

In this paper we study the effects of constraints on the dynamics of an adaptive segregation model introduced by Bischi and Merlone (2011). The model is described by a two dimensional piecewise smooth dynamical system in discrete time. It models the dynamics of entry and exit of two populations into a system, whose members have a limited tolerance about the presence of individuals of the other group. The constraints are given by the upper limits for the number of individuals of a population that are allowed to enter the system. They represent possible exogenous controls imposed by an authority in order to regulate the system. Using analytical, geometric and numerical methods, we investigate the border collision bifurcations generated by these constraints assuming that the two groups have similar characteristics and have the same level of tolerance toward the members of the other group. We also discuss the policy implications of the constraints to avoid segregation

Codimension-two border collision bifurcation in a two-class growth model with optimal saving and switch in behavior

We consider a two-class growth model with optimal saving and switch in behavior. The dynamics of this model is described by a two-dimensional (2D) discontinuous map. We obtain stability conditions of the border and interior fixed points (known as Solow and Pasinetti equilibria, respectively) and investigate bifurcation structures observed in the parameter space of this map, associated with its attracting cycles and chaotic attractors. In particular, we show that on the x-axis, which is invariant, the map is reduced to a 1D piecewise increasing discontinuous map, and prove the existence of a corresponding period adding bifurcation structure issuing from a codimension-two border collision bifurcation point. Then, we describe how this structure evolves when the related attracting cycles on the x-axis lose their transverse stability via a transcritical bifurcation and the corresponding interior cycles appear. In particular, we show that the observed bifurcation structure, being associated with the 2D discontinuous map, is characterized by multistability, that is impossible in the case of a standard period adding bifurcation structure

Critical Transitions in financial models: Bifurcation- and noise-induced phenomena

A so-called Critical Transition occurs when a small change in the input of a system leads to a large and rapid response. One class of Critical Transitions can be related to the phenomenon known in the theory of dynamical systems as a bifurcation, where a small parameter perturbation leads to a change in the set of attractors of the system. Another class of Critical Transitions are those induced by noisy increments, where the system switches randomly between coexisting attractors. In this thesis we study bifurcation- and noise-induced Critical Transitions applied to a variety of models in finance and economy. Firstly, we focus on a simple model for the bubbles and crashes observed in stock prices. The bubbles appear for certain values of the sensitivity of the price based on past prices, however, not always as a Critical Transition. Incorporating noise to the system gives rise to additional log-periodic structures which precede a crash. Based on the centre manifold theory we introduce a method for predicting when a bubble in this system can collapse. The second part of this thesis discusses traders' opinion dynamics captured by a recent model which is designed as an extension of a mean-field Ising model. It turns out that for a particular strength of contrarian attitudes, the traders behave chaotically. We present several scenarios of transitions through bifurcation curves giving the scenarios a market interpretation. Lastly, we propose a dynamical model where noise-induced transitions in a double-well potential stand for a company shifting from a healthy state to a defaulted state. The model aims to simulate a simple economy with multiple interconnected companies. We introduce several ways to model the coupling between agents and compare one of the introduced models with an already existing doubly-stochastic model. The main objective is to capture joint defaults of companies in a continuous-time dynamical system and to build a framework for further studies on systemic and individual risk

discrete time dynamic oligopolies with adjustment constraints

A classical $n$-firm oligopoly is considered first with linear demand and cost functions which has a unique equilibrium. We then assume that the output levels of the firms are bounded in a sense that they are unwilling to make small changes, the output levels are bounded from above, and if the optimal output level is very small then the firms quit producing, which are realistic assumptions in real economies. In the first part of the paper, the best responses of the firms are determined and the existence of infinitely many equilibria is verified. The second part of the paper examines the global dynamics of the duopoly version of the game. In particular we study the stability of the system, the bifurcations which can occur and the basins of attraction of the existing attracting sets, as a function of the speed of adjustment parameter

Modeling house price dynamics with heterogeneous speculators

This paper investigates the impact of speculative behavior on house price dynamics. Speculative demand for housing is modeled using a heterogeneous agent approach, whereas ‘real’ demand and housing supply are represented in a standard way. Together, real and speculative forces determine excess demand in each period and house price adjustments. Three alternative models are proposed, capturing in different ways the interplay between fundamental trading rules and extrapolative trading rules, resulting in a 2D, a 3D, and a 4D nonlinear discretetime dynamical system, respectively. While the destabilizing effect of speculative behavior on the model’s steady state is proven in general, the three specific cases illustrate a variety of situations that can bring about endogenous dynamics, with lasting and significant price swings around the ‘fundamental ’ price, as we have seen in many real markets

Dynamical Systems on Networks: A Tutorial

We give a tutorial for the study of dynamical systems on networks. We focus especially on "simple" situations that are tractable analytically, because they can be very insightful and provide useful springboards for the study of more complicated scenarios. We briefly motivate why examining dynamical systems on networks is interesting and important, and we then give several fascinating examples and discuss some theoretical results. We also briefly discuss dynamical systems on dynamical (i.e., time-dependent) networks, overview software implementations, and give an outlook on the field.Comment: 39 pages, 1 figure, submitted, more examples and discussion than original version, some reorganization and also more pointers to interesting direction

Complexity, Emergent Systems and Complex Biological Systems:\ud Complex Systems Theory and Biodynamics. [Edited book by I.C. Baianu, with listed contributors (2011)]

An overview is presented of System dynamics, the study of the behaviour of complex systems, Dynamical system in mathematics Dynamic programming in computer science and control theory, Complex systems biology, Neurodynamics and Psychodynamics.\u

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