SERIES WP 0622 Risk bubbles and market instability

We discuss a simple model of correlated assets capturing the feedback effects induced by portfolio investment in the covariance dynamics. This model predicts an instability when the volume of investment exceeds a critical value. Close to the critical point the model exhibits dynamical correlations very similar to those observed in real markets. Maximum likelihood estimates of the model’s parameter for empirical data indeed confirms this conclusion. We show that this picture is confirmed by the empirical analysis for different choices of the time horizon. Globalization and the informatics revolution have expanded the reach and density of financial interactions to an unprecedented extent. At the same time, the complexity of financial instruments has increased in complexity at a much faster pace than our understanding of their consequences on the collective behavior. Somewhat at odds with this, most financial instruments are designed under the assumption that market prices are exogenous processes, i.e. that individuals have no impact at all on the market. Such a price taking assumption might be a good approximation for the individual investor. But, if a particular strategy is used by many traders it will likely have a sizeable impact on the market. Actually, the Minority Game shows that the global behavior of interacting systems such as a financial markets may change dramatically by the addition of a single agent (see 4.5.3 [1]). It also shows that the properties of financial markets would drastically change if agents accounted, even approximately, for their impact on it (see 5.2.3 in [1]). This occurs because, in the schematic picture of Minority Games, markets have a tendency to react to trading strategies in such a way as to reduce their profitability: The more speculators enter the market, the less this is predictable and the more its state is susceptible to perturbations. The point where the market becomes information-efficient is also 1 Corresponding author. E-mail: marsili@ictp.it Preprint submitted to Elsevier Science 12 January 2006 the point where its susceptibility diverges, implying that the market reacts with a finite response even to infinitesimal perturbations. Interestingly, close to this point Minority Game models exhibit anomalous fluctuations phenomena [2] which bear some resemblance with the stylized facts observed in real markets[3]. We have recently shown [4] that this picture extends, qualitatively, to multiasset markets described by a very simple model. Considerations about risk are essential in this extension 2 . One of the key functions of financial markets is indeed that of allowing companies to “trade” their risk for return, by spreading it across financial investors. Investors on their side, diversify (i.e. spread) their strategies across stocks so as to minimize risk, as postulated by portfolio optimization theory [5]. Ref. [4] discusses how the market is affected, when many agents spread their risk on the market following portfolio theory. From the point of view of risk, the key quantity of interest is the covariance matrix of asset returns. Much attention has been recently put into the characterization of the statistical properties of the covariance matrix Ĉ (at the daily scale in most cases): The key findings are that • The bulk of the eigenvalue distribution of Ĉ is dominated by noise and it is well described by random matrix theory [6]. • The largest eigenvalue Λ – the so-called market mode – is very well separated from the other eigenvalues. • The few large eigenvalues smaller than Λ, which leak out of the noise background contain significant economic information. Indeed the taxonomy of assets built [7–10] from financial correlations alone bears remarkable similarity with a classification in economic sectors. • market correlations also exhibit highly non-trivial dynamics: correlations “build up” as the time scale on which returns are measured increases, and they saturate for returns on the scale of some days [11,12]. Furthermore, financial correlations are persistent over time [13] and they follow recurrent patterns [10]. Fig. 1a shows the time dependence of the largest eigenvalue of (the exponentially averaged) correlation matrix of daily returns for Toronto Stock Exchange [14]. Similar behaviour has been reported earlier [15] for different markets. A closer look [4] at the data in Fig. 1 shows that fluctuations in the largest eigenvalue Λ(t+1)−Λ(t) are broadly distributed, suggesting that Fig. 1a can hardly be explained entirely as the effect of few exogenous shocks. Ref. [4] formulates the hypothesis that such non-trivial dynamics arises as a 2 Most models of single asset markets focus on speculation and expected return. Their generalization to many assets induces no nontrivial dynamics of correlation across assets.