By Eckhard Platen, David Heath

ISBN-10: 3540262121

ISBN-13: 9783540262121

A framework for monetary industry modeling, the benchmark strategy extends past commonplace danger impartial pricing idea. It allows a unified remedy of portfolio optimization, by-product pricing, built-in chance administration and assurance probability modeling. This e-book provides the required mathematical instruments, through an intensive creation to monetary modeling lower than the benchmark method, explaining a number of quantitative equipment for the reasonable pricing and hedging of derivatives.

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26) The SGH density is a four parameter density. The parameter µ is a location parameter. The two shape parameters for its tails are λ and α ¯ = α δ, deﬁned so that they are invariant under scale transformations. The other parameters contribute to the scaling of the density. 27) c = 2 ¯ ⎩ δ Kλ+1 (α) otherwise. 2 Continuous Random Variables and Distributions 19 It can be shown that as λ → ± ∞ and/or α ¯ → ∞ the SGH density asymptotically approaches the Gaussian density. To illustrate certain typical SGH densities we shall describe four special cases of the SGH density in the sequel.

Note that the kurtosis is not ﬁnite for a Student t distribution with degrees of freedom not greater than four. The hyperbolic distribution yields only a kurtosis of six, which limits its applicability as a log-return distribution because a much higher kurtosis is typically observed for log-returns. It is an empirical stylized fact, which we shall document later on, that the probability densities of log-returns of stock indices, stock prices and exchange rates have much thicker tails than that of a Gaussian density, which means they are leptokurtic.

Xn (x1 , x2 , . . , xn ) = Nn,D (x) = P (Xi < xi , i ∈ {1, 2, . . 50) for x = (x1 , x2 , . . , xn ) ∈ n . We say that the n-dimensional vector x ∼ Nn (0, D) is Gaussian distributed with mean vector µ = (0, . . , 0) and covariance matrix D. Let X ∼ Nn (0, D) and b = (b1 , b2 , . . , bn ) ∈ n be an n-dimensional deterministic vector, then the scalar random variable Z=b X is Gaussian with Z ∼ N (0, b D b). 52) Y = B X ∼ Nm (0, B D B ), where the mean vector is a vector of zeros and the covariance matrix BDB is an m × m matrix.

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