Friedrich Wagner

Stability of money: phase transitions in an Ising economy

The stability of money value is an important requisite for a functioning economy, yet it critically depends on the actions of participants in the market themselves. Here we model the value of money as a dynamical variable that results from trading between agents. The basic trading scenario can be recast into an Ising-type spin model and is studied on the hierarchical network structure of a Cayley tree. We solve this model analytically and observe a phase transition between a one-state phase, always allowing for a stable money value, and a two-state phase, where an unstable (inflationary) phase occurs. The onset of inflation is discontinuous and follows a first-order phase transition. The stable phase provides a parameter region where money value is robust and can be stabilized without fine tuning.

Volatility Cluster and Herding

Stock markets can be characterized by fat tails in the volatility distribution, clustering of volatilities and slow decay of their time correlations. For an explanation models with several mechanisms and consequently many parameters as the Lux–Marchesi model have been used. We show that a simple herding model with only four parameters leads to a quantitative description of the data. As a new type of data we describe the volatility cluster by the waiting time distribution, which can be used successfully to distinguish between different models.

Phase transition in the S&P stock market

We analyze the returns of stocks contained in the Standard & Poor’s 500 index from 1987 until 2011. We use covariance matrices of the firms’ returns determined in a time windows of several years. We find that the eigenvector belonging to the leading eigenvalue (the market) exhibits a phase transition. The market is in an ordered state from 1995 to 2005 and in a disordered state after 2005. We can relate this transition to an order parameter derived from the stocks’ beta and the trading volume. This order parameter can also be interpreted within an agent-based model.

Application of Zhangs square root law and herding to financial markets

We apply an asymmetric version of Kirmans herding model to volatile financial markets. In the relation between returns and agent concentration we use the square root law proposed by Zhang. This can be derived by extending the idea of a critical mean field theory suggested by Plerou et al. We show that this model is equivalent to the so called 32 model of stochastic volatility. The description of the unconditional distribution for the absolute returns is in good agreement with the DAX independent whether one uses the square root or a conventional linear relation. Only the statistic of extreme events prefers the former. The description of the autocorrelations are in much better agreement for the square root law. The volatility clusters are described by a scaling law for the distribution of returns conditional to the value at the previous day in good agreement with the data.

A nonparametric approach tothe noise density in stochastic volatility models

We propose a nonparametric method to determine the functional form of the noise density in discrete-time stochastic volatility models of financial returns. Our approach suggests that the assumption of Gaussian noise is often adequate, but we do observe deviations from Gaussian noise for some assets, for instance gold.

Transitions in the stock markets of the US, UK and Germany

In an analysis of the US, the UK and German stock market, we find a change in the behaviour based on the stocks’ beta values. In the years 1995–2006, trades of stocks with high beta and large volume were concentrated in the IT and technology sector, whereas in 2006–2012 those trades are dominated by stocks from the financial sector. We show that an agent-based model can reproduce such a transition. We further show that the initial impulse for the transition might stem from the increase of high-frequency trading at that time.