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61.
This paper develops a discrete time version of the continuous time model of Bouchard et al. [J. Control Optim., 2009, 48, 3123–3150], for the problem of finding the minimal initial data for a controlled process to guarantee reaching a controlled target with probability one. An efficient numerical algorithm, based on dynamic programming, is proposed for the quantile hedging of standard call and put options, exotic options and quantile hedging with portfolio constraints. The method is then extended to solve utility indifference pricing, good-deal bounds and expected shortfall problems. 相似文献
62.
The contour maps of the error of historical and parametric estimates of the global minimum risk for large random portfolios optimized under the Expected Shortfall (ES) risk measure are constructed. Similar maps for the VaR of the ES-optimized portfolio are also presented, along with results for the distribution of portfolio weights over the random samples and for the out-of-sample and in-sample estimates for ES. The contour maps allow one to quantitatively determine the sample size (the length of the time series) required by the optimization for a given number of different assets in the portfolio, at a given confidence level and a given level of relative estimation error. The necessary sample sizes invariably turn out to be unrealistically large for any reasonable choice of the number of assets and the confidence level. These results are obtained via analytical calculations based on methods borrowed from the statistical physics of random systems, supported by numerical simulations. 相似文献
63.
The article contributes to the ongoing search for a market risk measure that is both coherent and elicitable. We compare two traditional measures, namely Value-at-Risk and the expected shortfall, with another relatively novel one established on the expectile probability term. Our research is based on five models: Black–Scholes, exponential tempered stable, Heston, Bates and another stochastic volatility model with a tempered stable jump correction. We apply the general Fourier inversion formula to derive closed form formulas for calculating not only the expectile based risk measure but also the Value-at-Risk and the expected shortfall. These models are calibrated by combining nonlinear programming with simulated annealing at a moving window. Additionally, we compare the generated values of the risk measures with the real ones. Last but not least, we modify the expectile based risk measure as well as the expected shortfall by introducing correction coefficients. 相似文献
64.
Johanna F. Ziegel 《Mathematical Finance》2016,26(4):901-918
The risk of a financial position is usually summarized by a risk measure. As this risk measure has to be estimated from historical data, it is important to be able to verify and compare competing estimation procedures. In statistical decision theory, risk measures for which such verification and comparison is possible, are called elicitable. It is known that quantile‐based risk measures such as value at risk are elicitable. In this paper, the existing result of the nonelicitability of expected shortfall is extended to all law‐invariant spectral risk measures unless they reduce to minus the expected value. Hence, it is unclear how to perform forecast verification or comparison. However, the class of elicitable law‐invariant coherent risk measures does not reduce to minus the expected value. We show that it consists of certain expectiles. 相似文献
65.
Pengyu Wei 《Mathematical Finance》2018,28(4):1020-1060
This article studies the optimal portfolio selection of expected utility‐maximizing investors who must also manage their market‐risk exposures. The risk is measured by a so‐called weighted value‐at‐risk (WVaR) risk measure, which is a generalization of both value‐at‐risk (VaR) and expected shortfall (ES). The feasibility, well‐posedness, and existence of the optimal solution are examined. We obtain the optimal solution (when it exists) and show how risk measures change asset allocation patterns. In particular, we characterize three classes of risk measures: the first class will lead to models that do not admit an optimal solution, the second class can give rise to endogenous portfolio insurance, and the third class, which includes VaR and ES, two popular regulatory risk measures, will allow economic agents to engage in “regulatory capital arbitrage,” incurring larger losses when losses occur. 相似文献
66.
67.
The optimized certainty equivalent (OCE) is a decision theoretic criterion based on a utility function, that was first introduced by the authors in 1986. This paper re-examines this fundamental concept, studies and extends its main properties, and puts it in perspective to recent concepts of risk measures. We show that the negative of the OCE naturally provides a wide family of risk measures that fits the axiomatic formalism of convex risk measures. Duality theory is used to reveal the link between the OCE and the φ-divergence functional (a generalization of relative entropy), and allows for deriving various variational formulas for risk measures. Within this interpretation of the OCE, we prove that several risk measures recently analyzed and proposed in the literature (e.g., conditional value of risk, bounded shortfall risk) can be derived as special cases of the OCE by using particular utility functions. We further study the relations between the OCE and other certainty equivalents, providing general conditions under which these can be viewed as coherent/convex risk measures. Throughout the paper several examples illustrate the flexibility and adequacy of the OCE for building risk measures. 相似文献
68.
We adopt Schwartz and Smith's model to calculate risk measures of Brent oil and light sweet crude oil (WTI) futures contracts and Mirantes, Poblacion, and Serna's model to calculate risk measures of natural gas, gasoil, heating oil, RBOB gasoline, PJM Western Hub peak, and off-peak electricity futures contracts. The models generate well in-sample goodness of fit and satisfactory out-of-sample Value-at-Risk and expected shortfall forecasts for all the eight of the analyzed commodities. A simple and flexible estimation method improving upon existing estimation methods is developed. 相似文献
69.
This paper provides a coherent method for scenario aggregation addressing model uncertainty. It is based on divergence minimization from a reference probability measure subject to scenario constraints. An example from regulatory practice motivates the definition of five fundamental criteria that serve as a basis for our method. Standard risk measures, such as value‐at‐risk and expected shortfall, are shown to be robust with respect to minimum divergence scenario aggregation. Various examples illustrate the tractability of our method. 相似文献
70.
In various fields of applications such as capital allocation, sensitivity analysis, and systemic risk evaluation, one often needs to compute or estimate the expectation of a random variable, given that another random variable is equal to its quantile at some prespecified probability level. A primary example of such an application is the Euler capital allocation formula for the quantile (often called the value‐at‐risk), which is of crucial importance in financial risk management. It is well known that classic nonparametric estimation for the above quantile allocation problem has a slower rate of convergence than the standard rate. In this paper, we propose an alternative approach to the quantile allocation problem via adjusting the probability level in connection with an expected shortfall. The asymptotic distribution of the proposed nonparametric estimator of the new capital allocation is derived for dependent data under the setup of a mixing sequence. In order to assess the performance of the proposed nonparametric estimator, AR‐GARCH models are proposed to fit each risk variable, and further, a bootstrap method based on residuals is employed to quantify the estimation uncertainty. A simulation study is conducted to examine the finite sample performance of the proposed inference. Finally, the proposed methodology of quantile capital allocation is illustrated for a financial data set. 相似文献