By Svetlozar T. Rachev, Stoyan V. Stoyanov, Frank J. Fabozzi CFA
This groundbreaking publication extends conventional methods of threat size and portfolio optimization by means of combining distributional types with danger or functionality measures into one framework. all through those pages, the professional authors clarify the basics of likelihood metrics, define new ways to portfolio optimization, and talk about quite a few crucial danger measures. utilizing various examples, they illustrate more than a few functions to optimum portfolio selection and threat thought, in addition to purposes to the realm of computational finance that could be precious to monetary engineers.
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Extra resources for Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures (Frank J. Fabozzi Series)
Maximizing the objective function is the same as minimizing the negative of the objective function and then changing the sign of the minimal value, maxn f (x) = − minn [−f (x)]. 1. As a consequence, problems for maximization can be stated in terms of function minimization and vice versa. 1 Minima and Maxima of a Differentiable Function If the second derivatives of the objective function exist, then its local maxima and minima, often called generically local extrema, can be characterized. 38 ADVANCED STOCHASTIC MODELS Denote by ∇f (x) the vector of the first partial derivatives of the objective function evaluated at x, ∇f (x) = ∂f (x) ∂f (x) .
5) where > 0. 5) can be used to estimate the probability of observing a large observation by means of the mathematical expectation and the level . Chebyshev’s inequality is rough as demonstrated geometrically in the following way. 9 Chebyshev’s inequality, a geometric illustration. The area of the rectangle in the upper-left corner is smaller than the shaded area. which means that it equals the area closed between the distribution function and the upper limit of the distribution function. 9 as the shaded area above the distribution function.
In this case, it has a simpler form because the marginal distributions are uniform. The lower and the upper Fr´echet bounds equal W(u1 , . . , un ) = max(u1 + · · · + un + 1 − n, 0) and M(u1 , . . , un ) = min(u1 , . . , un ) respectively. Fr´echet-Hoeffding inequality is given by W(u1 , . . , un ) ≤ C(u1 , . . , un ) ≤ M(u1 , . . , un ). In the two-dimensional case, the inequality reduces to max(u1 + u2 − 1, 0) ≤ C(u1 , u2 ) ≤ min(u1 , u2 ). In the two-dimensional case only, the lower Fr´echet bound, sometimes referred to as the minimal copula, represents perfect negative dependence between the two random variables.
Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures (Frank J. Fabozzi Series) by Svetlozar T. Rachev, Stoyan V. Stoyanov, Frank J. Fabozzi CFA