Computation And Asymptotic Properties Of
Estimated Coherent Risk Measures
Author(s)
D. J. Miller & M. Kim
Abstract
Recent advances in mathematical finance established linkages among several key
concepts related to coherence, distorted risk measures, and information theory. The
purpose of this paper is to extend these theoretical results for empirical applications
in computational finance. First, we use a concentrated (dual) entropy approach
to derive a computational algorithm for estimating the parameters of a distorted
probability model associated with a coherent risk measure for a given sample
of observed data. Second, we derive the asymptotic sampling properties of the
estimated model parameters, which may be used to conduct classical hypothesis
tests or to form other statistical inferences based on the estimated coherent risk
measure. Third, we note that researchers may also require an estimate of the net
loss distribution, and we propose an information theoretic procedure for jointly
estimating the net loss probability model and the distorted probability distribution
associated with a particular coherent risk measure.
Keywords:coherence, distorted risk measure, entropy, extremum estimator, information
theory.
1 Introduction
Risk measures based on the distribution of potential asset losses or returns are
widely used in empirical finance, and prominent applications of these tools include
determining insurance premia, option prices, margin deposits for hedged and
speculative positions in futuresmarkets, and capital reserve requirements for banks
and other firms (e.g., see Wirch and Hardy [1]). To be specific, suppose an asset
has risk or net loss represented by random variable X with cumulative distribution
function (CDF) F(x) = P(X ≤ x). A risk measure is a mapping ρ(X) : R →R+
Keywords
coherence, distorted risk measure, entropy, extremum estimator, information
theory.
Related Book
Other papers in this volume
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