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

ISBN-10: 0470253606

ISBN-13: 9780470253601

This groundbreaking ebook extends conventional methods of possibility dimension and portfolio optimization via combining distributional types with possibility or functionality measures into one framework. all through those pages, the specialist authors clarify the basics of chance metrics, define new methods to portfolio optimization, and talk about a number of crucial chance measures. utilizing a number of examples, they illustrate quite a number functions to optimum portfolio selection and chance concept, in addition to functions to the world of computational finance that could be invaluable to monetary engineers.

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

**Sample text**

As it is used to model waiting times, the exponential distribution is concentrated on the positive real numbers and the density function f and the cumulative distribution function F of an exponentially distributed random variable τ possess the following form: fτ (x) = 1 − βx e , x>0 β and x Fτ (x) = 1 − e− β , x > 0. 4. The reader might think of independence as nointerference between the random variables. 11 Concepts of Probability In credit risk modeling, the parameter λ = 1/β has a natural interpretation as hazard rate or default intensity.

Denote by W(y1 , . . , yn ) = max(FY1 (y1 ) + · · · + FYn (yn ) + 1 − n, 0) 32 ADVANCED STOCHASTIC MODELS and by M(y1 , . . , yn ) = min(FY1 (y1 ), . . , FYn (yn )), in which FYi (yi ) stands for the distribution function of the i-th marginal. The following inequality is known as Fr´echet-Hoeffding inequality, W(y1 , . . , yn ) ≤ FY (y1 , . . , yn ) ≤ M(y1 , . . , yn ). 6) The quantities W(y1 , . . , yn ) and M(y1 , . . , yn ) are also called the Fr´echet lower bound and the Fr´echet upper bound.

4) reveals that, if the random variable Y has independent components, then the density of the corresponding copula, denoted by c0 , is a constant in the unit hypercube, c0 (u1 , . . , un ) = 1 and the copula C0 has the following simple form, C0 (u1 , . . , un ) = u1 . . un . This copula characterizes stochastic independence. Now let us consider a density c of some copula C. 4) is a ratio of two positive quantities because the density function can only take nonnegative values. For each value of the vector of arguments y = (y1 , .

### Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures by Svetlozar T. Rachev, Stoyan V. Stoyanov, Visit Amazon's Frank J. Fabozzi Page, search results, Learn about Author Central, Frank J. Fabozzi,

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