By Eckhard Platen, David Heath

ISBN-10: 3540262121

ISBN-13: 9783540262121

A framework for monetary marketplace modeling, the benchmark technique extends past usual probability impartial pricing idea. It allows a unified therapy of portfolio optimization, by-product pricing, built-in probability administration and coverage hazard modeling. This ebook provides the required mathematical instruments, by way of an intensive advent to monetary modeling lower than the benchmark strategy, explaining quite a few quantitative equipment for the reasonable pricing and hedging of derivatives.

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**Example text**

28) Let X = (X1 , X2 , . . , Xn ) and Y = (Y1 , Y2 , . . , Ym ) with n, m ∈ N denote two random vectors. 29) i,j=1 The matrix Cov(X) = Cov(X, X) is called the autocovariance matrix of the vector X. 4 Joint Distributions and Random Vectors 45 If X = (X1 , X2 , . . , Xn ) is an n-dimensional vector, A = [Ai,j ]n,m i,j=1 a deterministic n × m matrix and b = (b1 , b2 , . . 30) Cov(Y , Y ) = A Cov(X, X) A . 31) and For example, if X is a vector of n independent random variables with variance Var(Xi ) = 1, i ∈ {1, 2, .

A characteristic function φX (θ) uniquely determines the density function fX (x) of a continuous random variable X. 83) see Lukacs (1960). We mentioned at the end of Sect. 2 that the SGH distribution covers logreturn distributions for a number of important asset price models. 84) for θ ∈ . Recall that Kλ is the modiﬁed Bessel function of the third kind with index λ. If one searches in probability or statistics textbooks and encyclopedias, then the characteristic function of the Student t distribution is notably absent or erroneous.

This is also reﬂected in its limiting kurtosis of three as n → ∞. In Fig. 24), in dependence on the two shape parameters λ and α. ¯ 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.

### A Benchmark Approach to Quantitative Finance by Eckhard Platen, David Heath

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