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

ACTA WASAENSIA 89by Lux & Marchesi, I shall use this higher precision of 500 micro steps per unit timeinterval throughout.Furthermore, in the system of differential equations (4.51) the price p is the onlynon-stochastic variable in the sense that it does not describe the expected value of astochastic process but has been arrived at in a purely deterministic manner according toequation (4.50) (albeit with inputs derived from a stochastic process). Lux & Marchesiwish to generate p in an anologous manner to x and z by formulating the followingstochastic process with expected time change t given by (4.50): They split theprice unit (1 dollar, say) into 100 elementary units (cents) and consider the probabilityof the price to move from one elementary unit to the next within ∆t. For that purpose,a small noise term μ ∼ N (0, σ 2 ) is added to the excess demand ED t at time t andthe transition probabilities to move one cent up (π p+ )ordown(π p− ) during the timeinterval ∆t are modelled asπ p+ =100max[0, β(ED t + μ)]∆t, π p− = −100 min[0, β(ED t + μ)]∆t, (4.54)such that the expected price change t between t and t + ∆t becomes t =0.01 t − 0.01 t = βED t ∆t (4.55)and (4.50) may be interpreted as t = βED t . (4.56)The binary price adjustment rule (4.54) leaves only the possibilities ∆p = −0.01,0,and+0.01 as possible inputs for ṗ ≈ ∆p/∆t as an approximation for the time derivative ofp in the equations of motion (4.51) from the simulated prices between t and t − ∆t. Ifollow Lux & Marchesi in calculating ṗ from the longer time interval [t−0.2,t)inordertoallowforabroadersetofvalues. Ialsofollowtheminsettingalowerboundof4out of N = 500 agents in any trader subpopulation in order to avoid occurence of theabsorbing states z =0andz = 1 in the simulations, that is the stationary equilibriaof type (ii) and (iii) in the differential equation system (4.51).The simulations run then as follows. Initially, the trading price is set to its fundamentalvalue p = p f , and the traders are randomly distributed over the subpopulations n + ,