A nonlinear structural model for volatility clustering

A simple nonlinear structural model of endogenous belief heterogeneity is proposed. News about fundamentals is an IID random process, but nevertheless volatility clustering occurs as an endogenous phenomenon caused by the interaction between different types of traders, fundamentalists and technical analysts. The belief types are driven by adaptive, evolutionary dynamics according to the success of the prediction strategies as measured by accumulated realized profits, conditioned upon price deviations from the rational expectations fundamental price. Asset prices switch irregularly between two different regimes --periods of small price fluctuations and periods of large price changes triggered by random news and reinforced by technical trading -- thus, creating time varying volatility similar to that observed in real financial data.

Nonlocal onset of instability in an asset pricing model with heterogeneous agents

Empirical time series of financial market data, like day-to-day stock returns, exhibit the phenomenon that although usually tomorrow's price is unpredictable, the absolute value of the price change is correlated with the magnitude of past price changes; though the corresponding correlation coefficients are not very large, they are significantly different from zero. This phenomenon is known as `volatility clustering' in the financial liturature. In this note a micro-economic model of volatility clustering, introduced by Gaunersdorfer and Hommes, will be analysed. The deterministic skeleton of the model has a Chenciner bifurcation, and hence periodic points and invariant quasi-periodic circles coexisting with the `fundamental' equilibrium. Adding noise in form of stochastic supply shocks, volatility clustering is generated by the system jumping between the bases of attraction of the fundamental equilibrium (low volatility), and that of the non-fundamental attractor (high volatility).

Bifurcation Routes to Volatility Clustering under Evolutionary Learning

A simple asset pricing model with two types of adaptively learning traders, fundamentalists and technical analysts, is studied. Fractions of these trader types, which are both boundedly rational, change over time according to evolutionary learning, with technical analysts conditioning their forecasting rule upon deviations from a benchmark fundamental. Volatility clustering arises endogenously in this model. Two mechanisms are proposed as an explanation. The first is coexistence of a stable steady state and a stable limit cycle, which arise as a consequence of a so-called Chenciner bifurcation of the system. The second is intermittency and associated bifurcation routes to strange attractors. Both phenomena are persistent and occur generically. Simple economic intuition why these phenomena arise in nonlinear multi-agent evolutionary systems is provided.

The Dynamics of Asymmetric Games

A game dynamical analysis of a simple asymmetric game (two roles with two alternatives each) shows that an interesting class of "semi-stable" heteroclinic cycles leading to a highly unpredictable behavior can occur in a robust way. Biological examples related to conflicts over ownership and parental investment are analyzed

Asset Price Dynamics in a Model of Investors Operating on Different Time Horizons

We present a dynamic asset pricing model based on a heterogenous class of traders. These traders are homogenous in the sense that they are fundamentalists who base their investment decisions on an exogenoulsy given fundamental value. They are heterogenous in the sense that each trader is working with a different frequency of the underlying price data. As a result we have a system of interacting investors who together influence the market price. We derive a system that characterizes out-of-equilibrium dynamics of prices in this market which is structurally equivalent to the Nosé-Hoover thermostat equation in non-equilibrium thermodynamics. We explore the time series properties of these prices and find that they exhibit fat tails of returns distributions, volatility clustering and power laws.Series: Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science

Time Averages for Heteroclinic Attractors

In the neighborhoods of attracting heteroclinic cycles, the time averages fail to converge for almost all initial conditions, but spiral closer and closer to the boundary of a polygon. This is shown by using a Poincare-section argument

Bifurcation Routes to Volatility Clustering under Evolutionary Learning

A simple asset pricing model with two types of boundedly rational traders, fundamentalists and chartists, is studied. Fractions of trader types change over time according to evolutionary learning, with chartists conditioning their forecasting rule upon deviations from a benchmark fundamental. Volatility clustering arises endogenously and two generic mechanisms are proposed as an explanation: (1) coexistence of a stable steady state and a stable limit cycle, due to a so-called Chenciner bifurcation of the system and (2) intermittency and associated bifurcation routes to strange attractors. Economic intuition as to why these phenomena arise in nonlinear multi-agent evolutionary systems is provided

No-regret Dynamics and Fictitious Play

Potential based no-regret dynamics are shown to be related to fictitious play. Roughly, these are epsilon-best reply dynamics where epsilon is the maximal regret, which vanishes with time. This allows for alternative and sometimes much shorter proofs of known results on convergence of no-regret dynamics to the set of Nash equilibria

Evolutionary Dynamics May Eliminate All Strategies Used in Correlated Equilibria

We show on a 4x4 example that many dynamics may eliminate all strategies used in correlated equilibria, and this for an open set of games. This holds for the best-response dynamics, the Brown-von Neumann-Nash dynamics and any monotonic or weakly sign-preserving dynamics satisfying some standard regularity conditions. For the replicator dynamics and the best-response dynamics, elimination of all strategies used in correlated equilibrium is shown to be robust to the addition of mixed strategies as new pure strategies