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

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

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

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

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2 SOME EXAMPLES OF PROBABILITY METRICS band. e. it does not deviate from the benchmark more than in an upward or downward direction. As the width of the performance band decreases, the probability P(|X − Y| > ε) increases because the portfolio returns will be more often outside a smaller band. The metric K(X, Y) calculates the width of a performance band such that the probability of the event that the portfolio return is outside the performance band is smaller than half of it. 7 Lp -metric The Lp -metric is Lp (X, Y) := {E |X − Y|p }1/p p≥1 X, Y ∈ Xp .

Proof. See Hausdorff (1949), section 29, and Kuratowski (1969), sections 21 and 23. 4. Let S = [0, 1] and let be the usual metric on S. Let R be the set of all finite complex-valued Borel measures m on S such that the Fourier transform m(t) = 1 exp(iut)m(du) 0 vanishes at t = ±∞. Let M be the class of sets E ∈ C(S) such that there is some m ∈ R concentrated on E. Then M is an analytic, non-Borel subset of (C(S), r ), see Kaufman (1984). m. s. in terms of their Borel structure. 5. A measurable space M with -algebra M is standard if there is a topology T on M such that (M, T) is a compact metric space and the Borel -algebra generated by T coincides with M.

We need a few preliminaries. 7. (see Loeve (1963), p. 99, and Dudley (1989), p. 82). If ( , A, Pr) is a probability space, we say that A ∈ A is an atom if Pr(A) > 0 and Pr(B) = 0 or Pr(A) for each measurable B ⊆ A. A probability space is non-atomic if it has no atoms. 1. (Berkes and Phillip (1979)). s. (U, d) and suppose that ( , A, Pr) is a non-atomic probability space. Then there is a U-valued random variable X with distribution L(X) = v. Proof. Denote by d∗ the following metric on U 2 : d∗ (x, y) := d(x1 , x2 ) + d(y1 , y2 ) for x = (x1 , y1 ) and y = (x2 , y2 ).

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