By Svetlozar T. Rachev, Stoyan V. Stoyanov, Frank J. Fabozzi CFA
This groundbreaking publication extends conventional ways of probability size and portfolio optimization by means of combining distributional types with threat or functionality measures into one framework. all through those pages, the professional authors clarify the basics of likelihood metrics, define new techniques to portfolio optimization, and speak about various crucial chance measures. utilizing a variety of examples, they illustrate more than a few purposes to optimum portfolio selection and probability thought, in addition to functions to the realm of computational finance which may be helpful to monetary engineers.
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Additional resources for Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures (Frank J. Fabozzi Series)
In this discussion, we consider only the copulas with a density. 7 The copula density of a two-dimensional normal distribution. 8. 6. 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.
Mikosch, T. (2006). ‘‘Copulas—tales and facts,’’ Extremes 9: 3–20. Patton, A. J. (2002). Application of copular theory in financial econometrics, Doctoral Dissertation, Economics, University of California, San Diego. Working paper, London School of Economics. ¨ Ruschendorf, L. (2004). ‘‘Comparison of multivariate risks and positive dependence,’’ Journal of Applied Probability 41(2): 391–406. Shiryaev, A. N. (1996). Probability, New York: Springer. Sklar, A. (1959). ‘‘Fonctions de r´epartition a` n dimensions et leurs marges,’’ Publications de l’Institut de Statistique de l’Universit´e de Paris 8: 229–231.
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. We apply Fr´echetHoeffding inequality in the two-dimensional case in Chapter 3 when discussing minimal probability metrics. Since copulas are essentially probability distributions defined on the unit hypercube, Fr´echet-Hoeffding inequality holds for them as well.
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