"Technical Analysis in Financial Markets"

B. Wealth invested according to belief h

Choice probability

The probability that agent j chooses belief h is determined by the discrete choice model in (6.15). The return of belief h in period t is equal to: r^F+y_t-1^h(r_t^P-r^F). Therefore the fitness measure is defined as

F_j,t^h=r^F+y_t-1^h(r_t^P-r^F) - C_j^h +η_j F_j,t-1^h; (52)

As in the BH model it is assumed that for all agents β_j=β, η_j=η and C_j^h=C^h. However, we adjust the probabilities with which each belief is chosen by introducing a lower bound m^h on the probabilities as motivated by Westerhoff (2002):

q_t^h=

exp(β F_t-1^h)

∑

k=1

exp(β F_t-1^k)

;

q_t^h=m^h+(1-

∑

h=1

m^h) q_t^h,

where m^h≥ 0 ∀ h and 0 ≤ ∑_h=1^H m^h ≤ 1. For example, Taylor and Allen (1990, 1992) found in questionnaire surveys that a small group of traders always uses technical or fundamental analysis and do not switch beliefs³. We define X_j,t^h=1 if agent j chooses belief h at time t and X_j,t^h=0 if agent j chooses a belief other than belief h. Because X_1,t^h, ..., X_N,t^h are iid with E(X_j,t^h)=q_t^h and limited variance V(X_j,t^h)=q_t^h (1-q_t^h), the fraction of agents who choose belief h converges in probability to q_t^h:

∑

j=1

X_j,t^h

→

q_t^h,

as the number of agents goes to infinity. Furthermore we did assume that all agents have the same risk aversion parameter a_j=a, so that agents who follow the same belief have the same demand. Hence in the end we assume that agents are only heterogeneous in the beliefs they can choose from.

Wealth assigned to belief h by agent j

The wealth invested according to belief h by agent j at time t is equal to W_j,t^h=X_j,t^h W_j,t. From this it follows that the total wealth assigned to belief h by all agents is equal to

278