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We compare two approaches to the coherent risk contribution: the directional risk contribution is defined as where ρ is a coherent risk measure; the linear risk contribution ρl(X; Y) is defined through a set of axioms, one of which is the linearity in X . The linear risk contribution exists and is unique for any ρ from the Weighted V@R class. We provide the representation for both risk contributions in the general setting as well as in some examples, including the MINV@R risk measure defined as where X1, … , XN are independent copies of X . 相似文献
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Alexander S. Cherny 《Finance and Stochastics》2007,11(4):537-569
The aim of the paper is to provide as explicit as possible expressions for upper/lower prices and for superhedging/subhedging
strategies based on discrete-time coherent risk measures. This is done on three levels of generality. For a general infinite-dimensional
model, we prove the fundamental theorem of asset pricing. For a general multidimensional model, we provide expressions for
prices and hedges. For a wide class of models, in particular, including GARCH, we give more concrete formulas, a sufficient
condition for the uniqueness of a hedging strategy, and a numerical algorithm.
相似文献
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A drawdown constraint forces the current wealth to remain above a given function of its maximum to date. We consider the portfolio optimisation problem of maximising the long-term growth rate of the expected utility of wealth subject to a drawdown constraint, as in the original setup of Grossman and Zhou (Math. Finance 3:241–276, 1993). We work in an abstract semimartingale financial market model with a general class of utility functions and drawdown constraints. We solve the problem by showing that it is in fact equivalent to an unconstrained problem with a suitably modified utility function. Both the value function and the optimal investment policy for the drawdown problem are given explicitly in terms of their counterparts in the unconstrained problem. 相似文献
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We define the capital allocation and the risk contribution for discrete-time coherent risk measures and provide several equivalent representations of these objects. The formulations and the proofs are based on two instruments introduced in the paper: a probabilistic notion of the extreme system and a geometric notion of the generator . These notions are also of interest on their own and are important for other applications of coherent risk measures. All the concepts and results are illustrated by JP Morgan's Risk Metrics model. 相似文献
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We prove that on an atomless probability space, every dilatation monotone convex risk measure is law invariant. This result,
combined with the known ones, shows the equivalence between dilatation monotonicity and important properties of convex risk
measures such as law invariance and second-order stochastic monotonicity.
We would like to thank Johannes Leitner for helpful discussions. The second author made contributions to this paper while
being affiliated to Heriot-Watt University and would like to express special thanks to Mark Owen, whose project (EPSRC grant
no. GR/S80202/01) supported this research. 相似文献
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We propose a methodology for estimating the risk of portfolios that exhibit nonlinear dependence on the risk driving factors
and have scarce observations, which is typical for portfolios of investments in hedge funds. The methodology consists of two
steps: first, regressing the portfolio return on nonlinear functions of each single risk driving factor and second, merging
together the obtained estimates taking into account the dependence between different factors. Performing the second step leads
us to a certain probabilistic problem, for which we propose an analytic and computationally feasible solution for the case
where the joint law of the factors is a Gaussian copula. A typical practical application can be to estimate the risk of a
hedge fund or a portfolio of hedge funds. As a theoretical consequence of our results, we propose a new definition of the
factor risk, i.e., the risk of a portfolio brought by a given factor. 相似文献
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Alexander S. Cherny 《Mathematical Finance》2010,20(4):571-595
We solve the risk‐reward optimization problem in the discrete‐time setting, the reward being measured as the expected Profit and Loss and the risk being measured by a dynamic coherent risk measure. 相似文献
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A. S. Cherny 《Finance and Stochastics》2006,10(3):367-393
The paper deals with the study of a coherent risk measure, which we call Weighted V@R. It is a risk measure of the form
where μ is a probability measure on [0,1] and TV@R stands for Tail V@R. After investigating some basic properties of this risk measure, we apply the obtained results to the financial problems of pricing, optimization, and capital allocation. It turns out that, under some regularity conditions on μ, Weighted V@R possesses some nice properties that are not shared by Tail V@R. To put it briefly, Weighted V@R is “smoother” than Tail V@R. This allows one to say that Weighted V@R is one of the most important classes (or maybe the most important class) of coherent risk measures. 相似文献
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