# A Probability Metrics Approach to Financial Risk Measures by Svetlozar T. Rachev, Stoyan V. Stoyanov, Frank J. Fabozzi

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

*A chance Metrics method of monetary threat Measures* relates the sector of chance metrics and danger measures to each other and applies them to finance for the 1st time.

- Helps to reply to the query: which hazard degree is better for a given problem?
- Finds new family among latest periods of hazard measures
- Describes purposes in finance and extends them the place possible
- Presents the idea of chance metrics in a extra available shape which might be acceptable for non-specialists within the field
- Applications comprise optimum portfolio selection, probability thought, and numerical equipment in finance
- Topics requiring extra mathematical rigor and aspect are integrated in technical appendices to chapters

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**Extra resources for A Probability Metrics Approach to Financial Risk Measures**

**Example text**

1. s. Then LH (X, Y) = E H(d(Z, V)) is a p. distance in X(U). Clearly, LH is finite in the subspace of all X with finite moment E H(d(X, a)) for some a ∈ U. The Kruglov’s distance Kr(X, Y) := Kr(FX , FY ) is a p. semidistance in X(R). 4 DEFINITIONS OF PROBABILITY DISTANCES AND METRICS Examples of p. 4) L0 (X, Y) := E I{X, Y} := Pr(X, Y). 2) are p. semimetrics in X(R). 1. 1 is free of the choice of the initial probability space, and depends only on the structure of the metric space U. The main reason for considering not arbitrary but separable metric spaces (U, d) is that we need the measurability of the metric d in order to connect the metric structure of U with that of X(U).

Sometimes, for notational convenience, we will use X Y instead of PX PY without changing the assumption that we are comparing the probability distributions. A preference relation or a preference order of an economic agent on the set of all lotteries X is a relation concerning the ordering of the elements of X, which satisfies certain axioms called the axioms of choice. A more detailed description of the axioms of choice is provided in the appendix to this chapter. A numerical representation of a preference order is a real-valued function U defined on the set of lotteries, U : X → R, such that PX PY if and only if U(PX ) ≥ U(PY ), PX PY ⇐⇒ U(PX ) ≥ U(PY ).

The probability distributions are regarded as objective: that is, the theory is consistent with the classical view that, in some sense, the randomness is inherent in Nature and all individuals observe the same probability distribution of a given random variable. In 1954, a decade after the pioneering von Neumann–Morgenstern theory was published, a new theory of decision making under uncertainty appeared. It was based on the concept that probabilities are not objective, rather they are subjective and are a numerical expression of the decision maker’s beliefs that a given outcome will occur.