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

ACTA WASAENSIA 119The condition dE i /dt =0formarketclearingimpliesthen,thattherespectivenumberof shares remains constant through time.Let us now turn to the population dynamics of the different trader types. Like Lux &Marchesi, we follow the synergetics literature in modeling interactions between membersof the investment community in terms of Markov chains. That is, for each traderwe postulate a transition probability to change her state of behaviour, or equivalentlyto move to another subpopulation, which depends only upon the investment communitiescurrent state, as described by the respective numbers of different trader types.Suppose there are L subpopulations (trader types) n 1 ,...,n i ,...,n j ,...,n L and denotethe transition probability to move from subpopulation i to subpopulation j asp ij . The evolution of expected population sizes through time is the described by thequasi-meanvalue equations 137n˙i =L(n j p ji − n i p ij ), i =1,...,L. (5.4)j=1Intuitively, they state that the expected change in population size per time unit ṅ i consistsof expected population inflows from all other states n j p ji minus all expectedpopulation outflows into other states n i p ij . In our case we have L = 5 subpopulations:two chartist populations of size n c1 and n c2 , two fundamentalist populations ofsize n f1 and n f2 , and one bondholder population of sizen B := N − n E , where n E := n c1 + n c2 + n f1 + n f2 (5.5)denotes the number of equity investors. Our task is now to specify the transitionprobabilities p ij according to which traders change from one subgroup to another. Asin Lux, it is assumed that traders change their strategy according to the perceivedprofits of the other strategies compared to their own. The perceived profits or utilitiesF i of fundamentalists holding a position in stock i are modeled asF i = s|p fi − p i |, i =1, 2, (5.6)137 see appendix A7 for a derivation. The triangular brackets t to indicate the expectationoperator applied to the process at time t have been dropped for notational convenience.