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

ACTA WASAENSIA 97Table 2. Median estimates of the tail index over ten samples of 2,000 observationseach and the range of estimates for common choices of the tail region.Parameter 2.5%tail 5%tail 10%tailSet I min median max min median max min median maxSimulation I 1.86 3.34 4.80 1.60 2.57 3.15 1.58 2.27 2.66Simulation II 2.12 3.23 3.59 1.97 2.51 3.56 1.73 2.17 2.65Lux&Marchesi 1.61 2.04 4.50 1.51 2.11 2.64 1.26 1.93 2.44Parameter 2.5%tail 5%tail 10%tailSet II min median max min median max min median maxSimulation I 2.18 2.61 4.44 2.21 2.32 3.09 1.76 2.19 2.67Simulation II 2.34 3.43 4.08 2.12 2.96 3.46 2.11 2.35 2.95Lux&Marchesi 2.28 2.82 3.73 2.00 2.52 3.17 1.55 2.18 2.36Parameter 2.5%tail 5%tail 10%tailSet III min median max min median max min median maxSimulation I 2.02 3.34 4.58 1.81 2.93 4.43 1.46 2.28 5.09Simulation II 1.58 3.21 4.81 1.25 2.81 4.12 1.72 2.67 3.33Lux&Marchesi 2.41 4.63 6.82 2.33 3.48 8.60 1.80 2.86 4.84Parameter 2.5%tail 5%tail 10%tailSet IV min median max min median max min median maxSimulation I 2.92 3.70 7.75 1.97 2.90 4.13 1.94 2.83 3.85Simulation II 2.07 3.25 5.06 1.95 2.98 3.74 1.65 2.33 3.24Lux&Marchesi 2.11 3.08 4.06 2.13 2.46 7.68 1.65 1.97 3.18illustrates the difference between the stochastic dynamics of the state variables x, p andz, and the dynamics of their expected values t , t and t , whose dynamics isdescribed by the deterministic differential equations (4.51). If (4.51) would describe thedynamics of the state variables themselves, the case z>z max could not occur as long asz was initialized below this threshold. On the other hand, occurence of z>z max wouldnecessarily lead to market instability. Occasional violation of the stability conditionz