Read e-book online A Probability Metrics Approach to Financial Risk Measures PDF

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

ISBN-10: 1405183691

ISBN-13: 9781405183697

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

  • Helps to respond to the query: which possibility degree is better for a given problem?
  • Finds new family members among latest periods of danger measures
  • Describes functions in finance and extends them the place possible
  • Presents the idea of likelihood metrics in a extra obtainable shape which might be acceptable for non-specialists within the field
  • Applications contain optimum portfolio selection, chance idea, and numerical tools in finance
  • Topics requiring extra mathematical rigor and element are incorporated in technical appendices to chapters

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Additional resources for A Probability Metrics Approach to Financial Risk Measures

Sample text

45 CHAPTER 3 CHOICE UNDER UNCERTAINTY The solutions of Bernoulli and Cramer are not completely satisfactory because the lottery can be changed in such a way that the fair value becomes infinite even with their choice of utility functions. Nevertheless, their attempt to solve the problem uses concepts which were later developed into theories of decision making under uncertainty. 2 The von Neumann–Morgenstern expected utility theory The St Petersburg Paradox shows that the naive approach to calculate the fair value of a lottery can lead to counter-intuitive results.

Define sets C1 , C2 , . . by C2n−1 = An and C2n = Bn n (n = 1, 2, . . ) and f : U → R by f (x) = ∞ n=1 2ICn (x)/3 . Then f is a 33 CHAPTER 2 PROBABILITY DISTANCES AND METRICS Borel-isomorphism of (U, d) onto f (U) ⊆ K, where K is the Cantor set, ∞ ˛n /3n : ˛n take value 0 or 2 . K := n=1 Define the metric e by e(x, y) = |f (x) − f (y)|, so that (U, e) is isometric with f (U) ⊆ K. Then An = f −1 {x ∈ K; x(n) = 2}, where x(n) is the nth digit in the ternary expansion of x ∈ K. Thus, An is clopen in (U, e), as required.

Another mainstream utility theory describing choices under uncertainty is the state-preference approach of Kenneth Arrow and G´erard Debreu. The basic principle is that the choice under uncertainty is reduced to a choice problem without uncertainty by considering state-contingent bundles of commodities. The agent’s preferences are defined over bundles in all states of the world and the notion of randomness is almost ignored. This construction is quite different from the theories of von Neumann–Morgenstern and Savage because preferences are not defined over lotteries.

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


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