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

44 ACTA WASAENSIAimplying exponential decay of the tails.Hsu (1979b) shows that subordinating Brownian motion to a directing process withexponentially distributed increments, results in a process with increments followingthe double exponential distribution. This approach may be generalized 66 to yield theExponential Power Distributions (EPD) by Box & Tiao (1973) with probability densityf(x) =k β φ −1 exp − 1 x − μ 2 (3.9)2/(1+β)φwhere k β is a normalizing constant, φ ∈ (0, ∞) is a scale parameter, μ ∈ R is a locationparameter, and β ∈ (−1, 1] is a parameter affecting the shape of the distribution. TheEPD are leptokurtic for 0 < β ≤ 1, but have tails with either finite endpoints orexponential decline 67 .Madan & Senata (1990) model the variance in driftless Brownian motion to follow agamma distribution, and call the resulting process Variance Gamma (VG). There isno analytical expression available for the probability density of the VG distribution,but it has a very simple characteristic function for the unit period returnϕ X (u) =1+ 1 2 υσ2 u 2 −1/υ(3.10)with scale parameter σ 2 ∈ (0, ∞) determining the variance, and shape parameterυ ∈ (0, ∞) determining the kurtosis of the returns. One attractive feature of the VGmodel is that, unlike the Student t distribution, it is closed under convolution, therebyallowing returns measured at varying time intervals to be described by members of thesame family of distributions. The VG model has later been generalized by Madan,Carr & Chang (1998) in order to allow for skewness in returns. It has finite momentsof all orders and exponentially declining tails despite its leptokurtosis.Unlike Brownian motion, which is a continuous process of unbounded variation, VGis a pure jump process of bounded variation 68 . Carr, Geman, Madan & Yor (2002)66 see Hsu (1980, 1982).67 see Hsu (1980) and Box & Tiao (1973: pages 156—160).68 A function f :[0,T] → R is of bounded variation if ni=1 |f(t i) − f(t i−1 )| < ∞ for all possiblepartitions 0 = t 0