By Svetlozar T. Rachev, Stoyan V. Stoyanov, Frank J. Fabozzi CFA
This groundbreaking publication extends conventional ways of chance size and portfolio optimization by means of combining distributional versions with probability 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 numerous crucial hazard measures. utilizing a variety of examples, they illustrate quite a number functions to optimum portfolio selection and danger conception, in addition to functions to the world of computational finance which may be worthy 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)
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. In a similar way, the upper Fr´echet bound, sometimes referred to as the maximal copula, represents perfect positive dependence between the two random variables. 8 SUMMARY We considered a number of concepts from probability theory that will be used in later chapters in this book.
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. We only state it for two-dimensional functions.
The daily return of the S&P 500 index over the last two years), but we don’t know the distribution which generates these returns. Consequently, we are not able to apply our knowledge about the calculation of statistical moments. But, having the observations r1 , . . , rk , we can try to estimate the true moments out of the sample. The estimates are sometimes called sample moments to stress the fact that they are obtained out of a sample of observations. The idea is simple. The empirical analogue for the mean of a random variable is the average of the observations: EX ≈ 11 Formally, 1 k k ri .