Value‐at‐Risk bounds with two‐sided dependence information

Value‐at‐Risk (VaR) bounds for aggregated risks have been derived in the literature in settings where, besides the marginal distributions of the individual risk factors, one‐sided bounds for the joint distribution or the copula of the risks are available. In applications, it turns out that these improved standard bounds on VaR tend to be too wide to be relevant for practical applications, especially when the number of risk factors is large or when the dependence restriction is not strong enough. In this paper, we develop a method to compute VaR bounds when besides the marginal distributions of the risk factors, two‐sided dependence information in form of an upper and a lower bound on the copula of the risk factors is available. The method is based on a relaxation of the exact dual bounds that we derive by means of the Monge–Kantorovich transportation duality. In several applications, we illustrate that two‐sided dependence information typically leads to strongly improved bounds on the VaR of aggregations.     

股市流动性与收益的Copula尾部相关性分析

将Copula函数引入风险管理,可以更加准确地反映变量间的相关结构,尤其是尾部相关特征,而混合Copula函数的使用又可以捕捉到不同的相关结构。应用Copula函数对中国股市收益和流动性在尾部的相关关系的实证研究结果表明:在中国证券市场上,收益和流动性存在同时发生极值的一致性,但拒绝了收益与流动性的尾部对称相关结构,上尾的相关系数大于下尾的相关系数。     

IMPLIED VOLATILITY IN THE HULL–WHITE MODEL

We study the implied volatility K ↦ I ( K ) in the Hull–White model of option pricing, and obtain asymptotic formulas for this function as the strike price K tends to infinity or zero. We also prove that the function I is convex near zero and concave near infinity, and characterize the behavior of the first two derivatives of this function.     

Option prices and risk‐neutral densities for currency cross rates

The theoretical relationship between the risk‐neutral density (RND) of the euro/ pound cross rate and the bivariate RND of the dollar/euro and the dollar/pound rates is derived; the required bivariate RND is defined by the dollar‐rate marginal RNDs and a copula function. The cross‐rate RND can be used by banks, international businesses, and central bankers to assess market expectations, to measure risks, and to value options, without relying on over‐the‐counter markets, which may be either non‐existent or illiquid. Empirical comparisons are made between cross‐rate RNDs estimated from several data sets. Five one‐parameter copula functions are evaluated and it is found that the Gaussian copula is the only one‐parameter copula function that is ranked highly in all of the comparisons we have made. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:324–360, 2010     

Improved estimation of portfolio value‐at‐risk under copula models with mixed marginals

Portfolio value‐at‐risk (PVAR) is widely used in practice, but recent criticisms have focused on risks arising from biased PVAR estimates due to model specification errors and other problems. The PVAR estimation method proposed in this article combines generalized Pareto distribution tails with the empirical density function to model the marginal distributions for each asset in the portfolio, and a copula model is used to form a joint distribution from the fitted marginals. The copula–mixed distribution (CMX) approach converges in probability to the true marginal return distribution but is based on weaker assumptions that may be appropriate for the returns data found in practice. CMX is used to estimate the joint distribution of log returns for the Taiwan Stock Exchange (TSE) index and the associated futures contracts on SGX and TAIFEX. The PVAR estimates for various hedge portfolios are computed from the fitted CMX model, and backtesting diagnostics indicate that CMX outperforms the alternative PVAR estimators. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:997–1018, 2006     

Commodity tail risks

In this study, we investigate the cross-section of option-implied tail risks in commodity markets. In contrast to findings from equity markets, left and right tail risks implied by option markets are both large. Commodity-specific variables exert the largest influence on tail risk, while there is no evidence of systematic commodity factors that are linked to tail risk. Additionally, we find strong links to the equity markets, but also comovements to macroeconomic factors. Left or right tail risks are largely independent of variance risk premiums. Finally, both left and right tail risks are priced in the cross-section of commodity futures returns.     

LARGE DEVIATIONS IN MULTIFACTOR PORTFOLIO CREDIT RISK

The measurement of portfolio credit risk focuses on rare but significant large-loss events. This paper investigates rare event asymptotics for the loss distribution in the widely used Gaussian copula model of portfolio credit risk. We establish logarithmic limits for the tail of the loss distribution in two limiting regimes. The first limit examines the tail of the loss distribution at increasingly high loss thresholds; the second limiting regime is based on letting the individual loss probabilities decrease toward zero. Both limits are also based on letting the size of the portfolio increase. Our analysis reveals a qualitative distinction between the two cases: in the rare-default regime, the tail of the loss distribution decreases exponentially, but in the large-threshold regime the decay is consistent with a power law. This indicates that the dependence between defaults imposed by the Gaussian copula is qualitatively different for portfolios of high-quality and lower-quality credits.     

PORTFOLIO OPTIMIZATION AND STOCHASTIC VOLATILITY ASYMPTOTICS

We study the Merton portfolio optimization problem in the presence of stochastic volatility using asymptotic approximations when the volatility process is characterized by its timescales of fluctuation. This approach is tractable because it treats the incomplete markets problem as a perturbation around the complete market constant volatility problem for the value function, which is well understood. When volatility is fast mean‐reverting, this is a singular perturbation problem for a nonlinear Hamilton–Jacobi–Bellman partial differential equation, while when volatility is slowly varying, it is a regular perturbation. These analyses can be combined for multifactor multiscale stochastic volatility models. The asymptotics shares remarkable similarities with the linear option pricing problem, which follows from some new properties of the Merton risk tolerance function. We give examples in the family of mixture of power utilities and also use our asymptotic analysis to suggest a “practical” strategy that does not require tracking the fast‐moving volatility. In this paper, we present formal derivations of asymptotic approximations, and we provide a convergence proof in the case of power utility and single‐factor stochastic volatility. We assess our approximation in a particular case where there is an explicit solution.     

The world stock markets under geopolitical risks: Dependence structure

This paper reveals joint stochastic behaviours of the world’s stock markets and geopolitical risk by a copula approach for the 37 world’s stock markets over the period of June 1997 to December 2017. The various bivariate copulas show the different degrees of tail dependences and rank correlations. The differences between overall geopolitical risk index and action‐related geopolitical risk index lie in the higher tail dependence with overall geopolitical risk index, the dominancy of concordant movements of stock market indexes with overall geopolitical risk and that of discordant movements of stock market indexes with action‐related geopolitical risk index. The results illustrate that action‐related geopolitical risk is more often adversely related to the world’s stock market performances with less tail dependence.     

Asymptotics for small nonlinear price impact: A PDE approach to the multidimensional case

We provide an asymptotic expansion of the value function of a multidimensional utility maximization problem from consumption with small nonlinear price impact. In our model, cross‐impacts between assets are allowed. In the limit for small price impact, we determine the asymptotic expansion of the value function around its frictionless version. The leading order correction is characterized by a nonlinear second‐order PDE related to an ergodic control problem and a linear parabolic PDE. We illustrate our result on a multivariate geometric Brownian motion price model.     

Coherent Measures of Risk

In this paper we study both market risks and nonmarket risks, without complete markets assumption, and discuss methods of measurement of these risks. We present and justify a set of four desirable properties for measures of risk, and call the measures satisfying these properties "coherent." We examine the measures of risk provided and the related actions required by SPAN, by the SEC/NASD rules, and by quantile-based methods. We demonstrate the universality of scenario-based methods for providing coherent measures. We offer suggestions concerning the SEC method. We also suggest a method to repair the failure of subadditivity of quantile-based methods.     

AN ANALYTICAL SOLUTION FOR THE TWO‐SIDED PARISIAN STOPPING TIME,ITS ASYMPTOTICS,AND THE PRICING OF PARISIAN OPTIONS

In this paper, we obtain a recursive formula for the density of the two‐sided Parisian stopping time. This formula does not require any numerical inversion of Laplace transforms, and is similar to the formula obtained for the one‐sided Parisian stopping time derived in Dassios and Lim. However, when we study the tails of the two distributions, we find that the two‐sided stopping time has an exponential tail, while the one‐sided stopping time has a heavier tail. We derive an asymptotic result for the tail of the two‐sided stopping time distribution and propose an alternative method of approximating the price of the two‐sided Parisian option.     

Maximum utility portfolio construction in the forward freight agreement markets: Evidence from a multivariate skewed t copula

We form portfolios consisting of diverse quarterly forward freight agreement (FFA) contracts to maximize the market participant's expected utility. The empirical findings indicate that individual FFA returns display clear autocorrelation, seasonality, fat tail, and heteroscedasticity. The multivariate positively skewed t copula is suggested for constructing maximum utility FFA portfolios, implying that the constituent FFA returns exhibit higher correlations when they rise together. The out-of-sample trading strategy performance metrics and various robustness checks further indicate that the aforementioned copula performs best and robustly for all portfolios. These findings provide profound methodological and managerial implications for market participants to improve risk management.     

Dependence of stock and commodity futures markets in China: Implications for portfolio investment

We examine the recent trends in dependence structure between the fast-growing commodity markets and the stock markets in China. We address this issue by using copula functions that allow for measuring both average and tail dependence. Our results provide evidence of low and positive correlations between these markets, suggesting that commodity futures are a desirable asset class for portfolio diversification. By comparing the market risks of alternative portfolio strategies, we show that Chinese investors can take advantage of commodity futures during different times to realize risk diversification and downside risk reduction benefits.     

Robust risk aggregation with neural networks

We consider settings in which the distribution of a multivariate random variable is partly ambiguous. We assume the ambiguity lies on the level of the dependence structure, and that the marginal distributions are known. Furthermore, a current best guess for the distribution, called reference measure, is available. We work with the set of distributions that are both close to the given reference measure in a transportation distance (e.g., the Wasserstein distance), and additionally have the correct marginal structure. The goal is to find upper and lower bounds for integrals of interest with respect to distributions in this set. The described problem appears naturally in the context of risk aggregation. When aggregating different risks, the marginal distributions of these risks are known and the task is to quantify their joint effect on a given system. This is typically done by applying a meaningful risk measure to the sum of the individual risks. For this purpose, the stochastic interdependencies between the risks need to be specified. In practice, the models of this dependence structure are however subject to relatively high model ambiguity. The contribution of this paper is twofold: First, we derive a dual representation of the considered problem and prove that strong duality holds. Second, we propose a generally applicable and computationally feasible method, which relies on neural networks, in order to numerically solve the derived dual problem. The latter method is tested on a number of toy examples, before it is finally applied to perform robust risk aggregation in a real‐world instance.     

Optimal portfolio under fractional stochastic environment

Rough stochastic volatility models have attracted a lot of attention recently, in particular for the linear option pricing problem. In this paper, starting with power utilities, we propose to use a martingale distortion representation of the optimal value function for the nonlinear asset allocation problem in a (non‐Markovian) fractional stochastic environment (for all values of the Hurst index ). We rigorously establish a first‐order approximation of the optimal value, when the return and volatility of the underlying asset are functions of a stationary slowly varying fractional Ornstein–Uhlenbeck process. We prove that this approximation can be also generated by a fixed zeroth‐ order trading strategy providing an explicit strategy which is asymptotically optimal in all admissible controls. Furthermore, we extend the discussion to general utility functions, and obtain the asymptotic optimality of this fixed strategy in a specific family of admissible strategies.     

Principal Component Value at Risk

Value at risk (VaR) is an industrial standard for monitoring financial risk in an investment portfolio. It measures potential losses within a given confidence interval. The implementation, calculation, and interpretation of VaR contains a wealth of mathematical issues that are not fully understood. In this paper we present a methodology for an approximation to value at risk that is based on the principal components of a sensitivity‐adjusted covariance matrix. The result is an explicit expression in terms of portfolio deltas, gammas, and the variance/covariance matrix. It can be viewed as a nonlinear extension of the linear model given by the delta‐normal VaR or RiskMetrics (J.P. Morgan, 1996).     

t分布下期货套期保值VaR风险的敏感度分析

使用风险价值的方法度量期货套期保值的风险,并且分析套期保值风险价值的敏感性。在t分布下,分别导出空头和多头套期保值风险价值关于期货量的一阶、二阶变化率,并解释其经济意义,这样就为套期者根据套期保值风险价值的敏感程度增减期货量提供指导。     

贝叶斯下的市场风险资产损失计量

目前VaR模型是国际上通行的综合衡量和管理风险的计量技术,但当分布是非正态分布或是不连续的时候VaR就没有稳定性。在应用VaR以及CVaR的基本原理,利用具有尖峰后尾特征的对数正态分布并结合贝叶斯风险,对参数通过先验分布和后验分布进行修正得到市场风险资产CVaR以及VaR的计算公式,对其进行比较分析,将其与pareto分布下的市场风险VaR值进行比较。研究表明:对数正态分布下市场风险资产的损失率比损失率呈pareto分布的风险损失显现出更好的拟合特征,能够描述市场风险资产的随机性最大损失,以及CvaR计量下的市场风险资产损失比VaR计量下的有更好的尾部特征,更加符合风险的充分性。     

Optimal futures hedging for energy commodities: An application of the GAS model

This paper applies generalized autoregressive score-driven (GAS) models to futures hedging of crude oil and natural gas. For both commodities, the GAS framework captures the marginal distributions of spot and futures returns and corresponding dynamic copula correlations. We compare within-sample and out-of-sample hedging effectiveness of GAS models against constant ordinary least square (OLS) strategy and time-varying copula-based GARCH models in terms of volatility reduction and Value at Risk reduction. We show that the constant OLS hedge ratio is not inherently inferior to the time-varying alternatives. Nonetheless, GAS models tend to exhibit better hedging effectiveness than other strategies, particularly for natural gas.