BERND PAPE Asset Allocation, Multivariate Position Based Trading ...

78 ACTA WASAENSIA& Winker (2003) and for an extended version with asymmetric transition probabilitiesby Alfarano, Lux & Wagner (2005). Another advantage is that it may help to discerntrue scaling, that is, the emergence of power laws over all orders of magnitude,from spurious multiscaling only in the pre-asymptotic regime. Alfarano & Lux (2006)consider a simplified variant of the noise trader infection model by Lux & Marchesi(2000) to be discussed in the next section, simple enough to be formulated as an ergodicMarkov chain with an analytical accessible limit distribution. While the true process isa stationary stochastic volatility process with finite third and fourth moments and exponentiallydeclining autocorrelation in volatility, the authors are able to demonstrateapparent power law scaling in volatility and tail indices near 3 in the pre-asymptoticregime of a few thousand observations, a common sample size in empirical investigationsof financial returns. The authors attribute this apparent scaling reminiscentof a stochastic volatility model with apparent multiscaling by LeBaron (2001) and ashort memory model with apparent long memory by Granger & Teräsvirta (1999), toswitches between high and low volatility regimes as suggested e.g. by Stǎricǎ & Mikosch(2000); Diebold & Inoue (2001) and Mikosch & Stǎricǎ (2004), here due to temporarydominance of either chartists or fundamentalists in the market.Consider finally the herding model by Lux (1995) as an introduction to the next section.He considers an investment community of n + +n − =2N speculators with n + optimists(buyers) and n − pessimists (sellers). The configuration of the investment communityis then uniquely specified in terms of the state variablen := 1 2 (n + − n − ) with − N ≤ n ≤ N. (4.17)Lux models the population dynamics as a Markov process, in which P (n; t) denotesthe probability of finding the investment community in state n at time t, applying themaster equation approach. Because n is a discrete variable, the master equation (4.2)reduces todP(n; t)= [w(n|n )P (n ; t) − w(n |n)P (n; t)] , (4.18)dtn where w(n |n) denotes the per unit time transition probability from state n to n .Changes in the configuration of the investment community are governed by switchesof individual agents between the optimist and pessimist subgroups. Denote the state