By Svetlozar T. Rachev, Stoyan V. Stoyanov, Frank J. Fabozzi CFA

This groundbreaking ebook extends conventional methods of probability size and portfolio optimization by way of combining distributional versions with hazard or functionality measures into one framework. all through those pages, the specialist authors clarify the basics of likelihood metrics, define new ways to portfolio optimization, and speak about various crucial threat measures. utilizing various examples, they illustrate more than a few purposes to optimum portfolio selection and probability idea, in addition to purposes to the realm of computational finance that could be invaluable 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)**

**Sample text**

In such cases, the analysis may require us to resort to general arguments based on certain general inequalities from the theory of probability. In this section, we give an account of such inequalities and provide illustration where possible. 1 Chebyshev’s Inequality Chebyshev’s inequality provides a way to estimate the approximate probability of deviation of a random variable from its mean. Its most simple form concerns positive random variables. Suppose that X is a positive random variable, X > 0.

It turns out that all points of local extrema of the objective function are characterized by a zero gradient. As a result, the points yielding the local extrema of the objective function are among the solutions of the system of equations, ∂f (x) =0 ∂x1 ... ∂f (x) = 0. 3) is often referred to as representing the first-order condition for the objective function extrema. However, it is only a necessary condition; that is, if the gradient is zero at a given point in the n-dimensional space, then this point may or may not be a point of a local extremum for the function.

We discuss the stable distribution in Chapter 4. 3 Exponential Distribution The exponential distribution is popular, for example, in queuing theory when we want to model the time we have to wait until a certain event takes place. Examples include the time until the next client enters the store, the time until a certain company defaults or the time until some machine has a defect. 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.