NONLOCAL ONSET OF INSTABILITY IN AN ASSET PRICING MODEL WITH HETEROGENEOUS AGENTS

Empirical time series of nancial market data, like day-to-day stock returns, ex-hibit 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 coecients are not very large, they are signicantly dierent from zero. This phenomenon is known as `volatility clus-tering ' in the nancial liturature. In this note a micro-economic model of volatility clustering, introduced by Gaunersdorfer and Hommes7, will be analysed. The de-terministic skeleton of the model has a Chenciner bifurcation, and hence periodic points and invariant quasi-periodic circles coexisting with the `fundamental ' equi-librium. Adding noise in form of stochastic supply shocks, volatility clustering is generated by the system jumping between the bases of attraction of the fundamen-tal equilibrium (low volatility), and that of the non-fundamental attractor (high volatility). 1. The model We build a variant of the heterogeneous adaptive beliefs model of Brock and Hommes 2, see also 4;5, investigated by Gaunersdorfer and Hommes 7 and Gaunersdorfer et al.8 There are economic agents which trade one kind of risky asset (a stock) and one risk-less asset (a bond) on a nancial market. Assets are assumed to be innitely lived and perfectly divisible; moreover, short-selling is allowed (i.e. it is possible to buy negative quantities of them). Information. The risk-less assets live one period; they are bought at a xed price 1 per unit, and pay an amount R = 1 + r> 0 in the next pe-riod. Risky asset are traded at the market price pt, and pay each period a dividend yt: one risky asset bought at time t at price pt yields divi-dend yt+1 and can be sold at price pt+1 next period. In this model, th