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

46 ACTA WASAENSIA3.2.4 Descriptive ModelsWhile the distributions and processes discussed in the preceeding paragraph have beenmotivated by economical considerations, such as consistency with arbitrage-free pricing,price response to information flow, or changing regimes in the economy, there haveadditionally been suggested a large number of data driven models, that came into lifesolely due to the quality of their fit to empirically observerd return data.Smith (1981) suggests the Logistic Distribution with probability densityf(x) =exp[(x − μ)/α]α(1 + exp[(x − μ)/α]) 2 (3.12)where μ ∈ R and α ∈ (0, ∞) are location and scale parameters, respectively. Thelogistic distribution is symmetric and leptokurtic with exponentially declining tails.Bookstaber & McDonald (1987) suggest the Generalized Beta of the second kind (GB2)as unconditional distribution of financial returns. The 4 parameter family of distributionscontains among others the lognormal, the log-Student t, and the log-logisticdistributions as special cases. Higher moments may or may not exist, depending uponthevaluesoftheshapeparameters. TheGB2distributionshavebeengeneralizedbyMcDonald & Xu (1995) to the 5 parameter family of Generalized Beta (GB) distributions,then containing also the generalized beta of the first kind (GB1), which includeamong others the Lévy stable distributions as a special case. Exponentiating the GBdistributions yields the Exponential Generalized Beta (EGB) distributions, which containamong others the exponential power distributions by Box & Tiao (1973).McDonald & Newey (1988) 69 introduce the 3 parameter family of symmetric GeneralizedT (GT) distributions nesting both the Student t and the exponential power distributions.Theodossiou (1998) generalized GT into the 4 parameter family of SkewedGeneralized T (SGT) distributions in order to allow for skewness. As said above, neitherthe EGB nor the SGT family of distributions, like any other distribution discussedin this section, have a foundation in economic theory. Their usefulness is rather due69 see also the discussion in Butler, McDonald, Nelson & White (1990).