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

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98 ACTA WASAENSIATable 3. Kurtosis estimates over the full sample of 20,000 observations.Simulation I Simulation II Lux&MarchesiParameter Set I 147.47 27.59 135.73Parameter Set II 43.61 23.46 16.10Parameter Set III 147.95 34.65 27.11Parameter Set IV 22.51 96.44 37.74presented in appendix A2. Making use of the symmetry of returns, the positive andnegative tails have been merged into absolute returns in order to provide a betterstatistical basis for the estimation by means of a larger sample size. All return serieshave been seperated into 10 subsamples of 2,000 observations each, in order to facilitatecomparison with the simulation results by Lux & Marchesi (2000). The table reportsthe smallest, median, and highest tail index estimate within the ten subsamples for thecommonly chosen tail regions of the largest 2.5%, 5%, and 10% absolute returns. Theresults agree both with Lux & Marchesi and the empirical findings in financial marketsin that the median estimates hoover around in the range 2 to 4 with decreasing α-estimate for increasing tail size.A less precise measure of fat-tailedness is given by the sample kurtosis, 130 presented intable 3. Here the agreement within the simulations and with Lux & Marchesi is onlyqualitative in that all simulations generate at least double digit estimates, indicatingstrong fat-tailedness. The numerical differences of the estimates between the differentsimulation runs are however often large. This is not surprising, given that tail indicesbelow 4 imply that the kurtosis of the process is not defined. 131 We may thereforenotexpectthesamplekurtosistoconvergetoanyspecific number in such processes.This is illustrated in figure 5, which shows the kurtosis estimate of the unconditionalreturn distribution as a function of increasing sample size. It is seen that the kurtosisestimate contains sudden jumps after which–even though initially leveling off–it doesnot converge to any stationary level. This finding is in harmony with the behaviour ofthe kurtosis estimator in empirical financial data. 132 Comparison with the return seriesin figures 1 and 2 reveals that the sudden jumps in the kurtosis estimate are caused by130 see section 2.3.131 see section 2.4.132 see e.g. Cont (2001).
ACTA WASAENSIA 99200Parameter Set I (Simulation I)40Parameter Set I (Simulation II)sample kurtosis15010050sample kurtosis30201000 0.5 1sample size1.5 2x 10 4Parameter Set II (Simulation I)6000 0.5 1sample size1.5 2x 10 4Parameter Set II (Simulation II)50sample kurtosis504030201000 0.5 1sample size1.5 2x 10 4Parameter Set III (Simulation I)400sample kurtosis4030201000 0.5 1sample size1.5 2x 10 4Parameter Set III (Simulation II)40sample kurtosis300200100sample kurtosis30201000 0.5 1sample size1.5 2x 10 4Parameter Set IV (Simulation I)3000 0.5 1sample size1.5 2x 10 4Parameter Set IV (Simulation II)15025sample kurtosis201510sample kurtosis10050500 0.5 1 1.5 2sample sizex 10 400 0.5 1 1.5 2sample sizex 10 4Figure 5. Sample kurtosis estimates for increasing sample size.
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ACTA WASAENSIA 52.10.1 Cross-Sectio
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ACTA WASAENSIA 7A5 Matlabcodeforerr
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10 ACTA WASAENSIAIn chapter 5 I sha
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12 ACTA WASAENSIA2 Statistical Prop
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14 ACTA WASAENSIA2.2 Absence of Ser
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ACTA WASAENSIA 17The survival or ta
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ACTA WASAENSIA 19& Scheinkman (1987
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ACTA WASAENSIA 212.6 Long Range Dep
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ACTA WASAENSIA 23A particular class
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26 ACTA WASAENSIAMultiscaling may t
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28 ACTA WASAENSIAHansen (1982) to c
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30 ACTA WASAENSIAThe leverage hypot
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32 ACTA WASAENSIAIn order to aviod
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34 ACTA WASAENSIAon a risk-adjusted
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36 ACTA WASAENSIAplanations for the
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38 ACTA WASAENSIAspeculative prices
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40 ACTA WASAENSIAindex α is howeve
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42 ACTA WASAENSIAThis approach has
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44 ACTA WASAENSIAimplying exponenti
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46 ACTA WASAENSIA3.2.4 Descriptive
Page 48 and 49: 48 ACTA WASAENSIAdivisible version
Page 50 and 51: 50 ACTA WASAENSIAto check their ade
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Page 54 and 55: 54 ACTA WASAENSIAfeedback of the co
Page 56 and 57: 56 ACTA WASAENSIAarriving at the fo
Page 58 and 59: 58 ACTA WASAENSIAat iteration k com
Page 60 and 61: 60 ACTA WASAENSIAmultipliers in the
Page 62 and 63: 62 ACTA WASAENSIALux & Ausloos (200
Page 64 and 65: 64 ACTA WASAENSIAmarkets dominated
Page 66 and 67: 66 ACTA WASAENSIAThe “representat
Page 68 and 69: 68 ACTA WASAENSIA(1983) 114 motivat
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Page 72 and 73: 72 ACTA WASAENSIASethi (1996) exten
Page 74 and 75: 74 ACTA WASAENSIASummation over all
Page 76 and 77: 76 ACTA WASAENSIAthe microscopic un
Page 78 and 79: 78 ACTA WASAENSIA& Winker (2003) an
Page 80 and 81: 80 ACTA WASAENSIAWe wish to obtain
Page 82 and 83: 82 ACTA WASAENSIAindividual markets
Page 84 and 85: 84 ACTA WASAENSIAreturn series. I l
Page 86 and 87: 86 ACTA WASAENSIASwitches between c
Page 88 and 89: 88 ACTA WASAENSIAfollowing necessar
Page 90 and 91: 90 ACTA WASAENSIATable 1. Parameter
Page 92 and 93: 92 ACTA WASAENSIAin stepwise search
Page 94 and 95: 94 ACTA WASAENSIA0.2Logreturns Para
Page 96 and 97: 96 ACTA WASAENSIA0.8Chartist Index
Page 100 and 101: 100 ACTA WASAENSIAthe most extreme
Page 102 and 103: 102 ACTA WASAENSIATable 4. Estimate
Page 104 and 105: 104 ACTA WASAENSIATable 8. Results
Page 106 and 107: 106 ACTA WASAENSIATable 9. Results
Page 108 and 109: 108 ACTA WASAENSIATable 11. Paramet
Page 110 and 111: 110 ACTA WASAENSIA1 x 105 Aggr. Inv
Page 112 and 113: 112 ACTA WASAENSIA200Aggr. Inventor
Page 114 and 115: 114 ACTA WASAENSIA5 x 105 Aggr. Wea
Page 116 and 117: 116 ACTA WASAENSIAOverall, we can a
Page 118 and 119: 118 ACTA WASAENSIAby Lux (1998). Th
Page 120 and 121: 120 ACTA WASAENSIAwhere s is a disc
Page 122 and 123: 122 ACTA WASAENSIAprocessing valuat
Page 124 and 125: 124 ACTA WASAENSIAProof. See append
Page 126 and 127: 126 ACTA WASAENSIA2.5Logarithmice T
Page 128 and 129: 128 ACTA WASAENSIA0.4Logreturns Ass
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Page 132 and 133: 132 ACTA WASAENSIATable 17. Probabi
Page 134 and 135: 134 ACTA WASAENSIATable 18. Median
Page 136 and 137: 136 ACTA WASAENSIAIn order to test
Page 138 and 139: 138 ACTA WASAENSIAReferencesAbhyank
Page 140 and 141: 140 ACTA WASAENSIABaillie, R. T. (1
Page 142 and 143: 142 ACTA WASAENSIABlack, F. & M. Sc
Page 144 and 145: 144 ACTA WASAENSIAButler, R. J., J.
Page 146 and 147: 146 ACTA WASAENSIACont, R. (2001).
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148 ACTA WASAENSIAEmbrechts, P., C.
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150 ACTA WASAENSIAFielitz, B. D. (1
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152 ACTA WASAENSIAGhysels, E., A. C
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154 ACTA WASAENSIAHeston, S. L. (19
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156 ACTA WASAENSIAKaldor, N. (1939)
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158 ACTA WASAENSIALiesenfeld, R. (1
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160 ACTA WASAENSIALye, J. N. & V. L
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162 ACTA WASAENSIAMikosch, T. (2003
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164 ACTA WASAENSIAPindyck, R. S. (1
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166 ACTA WASAENSIASethi, R. (1996).
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168 ACTA WASAENSIATesfatsion, L. &
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170 ACTA WASAENSIAAAppendixA1Matlab
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172 ACTA WASAENSIA87 % a3 = 1; %imp
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174 ACTA WASAENSIA181 nmout = max([
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176 ACTA WASAENSIA275276 figure;277
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178 ACTA WASAENSIA45 end464748 % In
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180 ACTA WASAENSIA454647 % Check wh
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182 ACTA WASAENSIAA4Matlab code for
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184 ACTA WASAENSIA888990 % create P
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186 ACTA WASAENSIA4344 nobs = 500;
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188 ACTA WASAENSIAA6Matlab code for
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190 ACTA WASAENSIA9192 for t = 1:T
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192 ACTA WASAENSIA185 fcash = fcash
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194 ACTA WASAENSIA279280281 %Output
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196 ACTA WASAENSIA(4.20), and o(τ
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198 ACTA WASAENSIAas t = n k P ˙
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200 ACTA WASAENSIA nBn˙f2 = v B n
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202 ACTA WASAENSIA f˙1−c1 = v B
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204 ACTA WASAENSIAlocal stability o
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