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

38 ACTA WASAENSIAspeculative prices, which assumes only E( t ) = 0 rather than requiring t to be drawnfrom a fixed distribution.Samuelson (1965, 1973) shows that the martingale model for stock prices discountedat the risk-free rate is consistent with an arbitrage-free market, which prices the stockat Fundamental Value, that is the expected present value of all future dividends. Thelatter hypothesis, that prices evolve as if market participants used the true probabilitydistribution of events in making their predictions, has first been advanced by Muth(1961) and denoted by him as Rational Expectations.3.2 Modelling the Unconditional Return Distribution3.2.1 Infinite Variance HypothesisSignificant excess kurtosis as well as strong time variationinvarianceofreturnsledMandelbrot (1963) to argue for the use of Lévy Stable Distributions in the descriptionof financial returns. The general class of Lévy stable distributions introduced by Lévy(1925) lacks any closed form solution, but may be describted by its characteristicfunction ϕ X (u) =E(e iXu )as:⎧⎨iδu − γ|u| α S1 − iβ uln ϕ X (u) =tan πα |u| 2 S if α S =1,⎩iδu − γ|u| 1+iβ 2 uln |u| if απ |u| S =1,(3.3)with location parameter δ ∈ (−∞, ∞), skewness index β ∈ (−∞, ∞), scale parameterγ ∈ (0, ∞) determining the width, and characteristic exponent α S ∈ (0, 2] determiningthe shape of the distribution. The normal distribution corresponds to the special caseα S = 2. In most other cases, the distribution function and density of X can only beobtained by numerically evaluating the inverse Fourier transform of (3.3).Allnon-normalLévy stable distributions are leptokurtic and have hyperbolically decliningtails with tail index α identical to their characteristic exponent α S , which impliesthat they have infinite variance (see section 2.4). Apparent variation in the variance