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By Svetlozar T. Rachev, Stoyan V. Stoyanov, Visit Amazon's Frank J. Fabozzi Page, search results, Learn about Author Central, Frank J. Fabozzi,

ISBN-10: 047005316X

ISBN-13: 9780470053164

This groundbreaking e-book extends conventional techniques of threat size and portfolio optimization via combining distributional types with chance or functionality measures into one framework. all through those pages, the specialist authors clarify the basics of likelihood metrics, define new methods to portfolio optimization, and talk about numerous crucial possibility measures. utilizing a variety of examples, they illustrate more than a few purposes to optimum portfolio selection and chance conception, in addition to functions to the realm of computational finance that could be worthwhile to monetary engineers.

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Extra info for Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures

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4) H= . ..  ..   . .   . 2 ∂ 2 f (x) ∂ 2 f (x) . . ∂ ∂xf (x) 2 ∂xn ∂x1 ∂xn ∂x2 n 40 ADVANCED STOCHASTIC MODELS which is called the Hessian matrix or just the Hessian. The Hessian is a symmetric matrix because the order of differentiation is insignificant, ∂ 2 f (x) ∂ 2 f (x) = . ∂xi ∂xj ∂xj ∂xi The additional condition is known as the second-order condition. We will not provide the second-order condition for functions of n-dimensional arguments because it is rather technical and goes beyond the scope of the book.

If you know the correlation of these two random variables, this does not imply that you know the dependence structure between the asset prices itself because the asset prices (P and Q for asset X and Y, respectively) are obtained by Pt = P0 exp(X) and Qt = Q0 exp(Y), where P0 and Q0 denote the corresponding asset prices at time 0. The asset prices are strictly increasing functions of the return but the correlation structure is not maintained by this transformation. This observation implies that the return could be uncorrelated whereas the prices are strongly correlated and vice versa.

This example demonstrates that the first-order conditions are generally insufficient to characterize the points of local extrema. 2 The top plot shows a function f (x1 , x2 ) with two local maxima. The bottom plot shows the contour lines of f (x1 , x2 ) together with the gradient evaluated at a grid of points. The middle black point shows the position of the saddle point between the two local maxima. of local minimum or maximum is given through the matrix of second derivatives,   ∂ 2 f (x) ∂ 2 f (x) ∂ 2 f (x) .

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