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

42 ACTA WASAENSIAThis approach has been generalized by Kon (1984) to allow for a discrete mixtureof an arbitrary number of normals, where variation in the mean generates skewness,whereas variation in the variance of the components generates excess kurtosis in theresulting probability distribution. Kon motivates the varying parameters in the mixtureof distributions by changing regimes in the underlying economy, rather than ordinaryand extraordinary information events.3.2.3 Subordinated Normal Model andTime Changed Brownian MotionMandelbrot & Taylor (1967) motivate the use of the Lévy stable distribution for describingthe increments in the random walk of logarithmic prices Z(t) ≡ ln P t witha non-uniform distribution of trading activity over calendar time t. In order to takethis irregularity of transactions into account, they suggest to introduce a randomizedoperational time T (t) measuring the volume or number of transactions up to physicaltime t. If T (t) is assumed to follow a Lévy stable distribution with characteristicexponent α S < 1, and increments in X(v), representing price reactions measured innumbers of transactions, are assumed assumed to be iid normal distributed; then itcan be shown that the price reaction in calendar time t measured as increments of theprocess Z(t) =X(T (t)) are Lévy stable distributed with α S < 1 despite the normaldistribtution of the price reaction conditional on trading volume.This is a special case of the Subordinated Normal Model for logarithmic stock prices,in which transformed calendar time T (t) is subordinated to Brownian motion. 59 Astochasticprocess {X(T (t))} is called Subordinated to the process {X(t)}, iftheDirectingProcess T (t) is strincly increasing and has stationary independent increments. 60 .Subordinated Brownian Motion is particularly interesting for modelling stock pricessince any arbitrage-free price process may be written as time-changed Brownian motionB(T (t)). 61 The chronometer T (t) need however not necessarily be a subordinator, that59 see Westerfield (1977).60 see e.g. Feller (1966: pp. 333—336).61 see e.g. Ané & Geman (2000).