Asymptotic subadditivity/superadditivity of Value-at-Risk under tail dependence

This paper presents a new method for discussing the asymptotic subadditivity/superadditivity of Value-at-Risk (VaR) for multiple risks. We consider the asymptotic subadditivity and superadditivity properties of VaR for multiple risks whose copula admits a stable tail dependence function (STDF). For the purpose, a marginal region is defined by the marginal distributions of the multiple risks, and a stochastic order named tail concave order is presented for comparing individual tail risks. We prove that asymptotic subadditivity of VaR holds when individual risks are smaller than regularly varying (RV) random variables with index −1 under the tail concave order. We also provide sufficient conditions for VaR being asymptotically superadditive. For two multiple risks sharing the same copula function and satisfying the tail concave order, a comparison result on the asymptotic subadditivity/superadditivity of VaR is given. Asymptotic diversification ratios for RV and log regularly varying (LRV) margins with specific copula structures are obtained. Empirical analysis on financial data is provided for highlighting our results.     

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.     

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     

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     

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.     

Sharing the value‐at‐risk under distributional ambiguity

This paper considers the problem of risk sharing, where a coalition of homogeneous agents, each bearing a random cost, aggregates their costs, and shares the value‐at‐risk of such a risky position. Due to limited distributional information in practice, the joint distribution of agents' random costs is difficult to acquire. The coalition, being aware of the distributional ambiguity, thus evaluates the worst‐case value‐at‐risk within a commonly agreed ambiguity set of the possible joint distributions. Through the lens of cooperative game theory, we show that this coalitional worst‐case value‐at‐risk is subadditive for the popular ambiguity sets in the distributionally robust optimization literature that are based on (i) convex moments or (ii) Wasserstein distance to some reference distributions. In addition, we propose easy‐to‐compute core allocation schemes to share the worst‐case value‐at‐risk. Our results can be readily extended to sharing the worst‐case conditional value‐at‐risk under distributional ambiguity.     

Model risk in credit risk

We provide sharp analytical upper and lower bounds for value‐at‐risk (VaR) and sharp bounds for expected shortfall (ES) of portfolios of any dimension subject to default risk. To do so, the main methodological contribution of the paper consists in analytically finding the convex hull generators for the class of exchangeable Bernoulli variables with given mean and for the class of exchangeable Bernoulli variables with given mean and correlation in any dimension. Using these analytical results, we first describe all possible dependence structures for default, in the class of finite sequences of exchangeable Bernoulli random variables. We then measure how model risk affects VaR and ES.     

Cross‐commodity analysis and applications to risk management

The understanding of joint asset return distributions is an important ingredient for managing risks of portfolios. Although this is a well‐discussed issue in fixed income and equity markets, it is a challenge for energy commodities. In this study we are concerned with describing the joint return distribution of energy‐related commodities futures, namely power, oil, gas, coal, and carbon. The objective of the study is threefold. First, we conduct a careful analysis of empirical returns and show how the class of multivariate generalized hyperbolic distributions performs in this context. Second, we present how risk measures can be computed for commodity portfolios based on generalized hyperbolic assumptions. And finally, we discuss the implications of our findings for risk management analyzing the exposure of power plants, which represent typical energy portfolios. Our main findings are that risk estimates based on a normal distribution in the context of energy commodities can be statistically improved using generalized hyperbolic distributions. Those distributions are flexible enough to incorporate many characteristics of commodity returns and yield more accurate risk estimates. Our analysis of the market suggests that carbon allowances can be a helpful tool for controlling the risk exposure of a typical energy portfolio representing a power plant. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:197–217, 2009     

Self‐similarity in long‐horizon returns

Asset returns incorporate new information via the effects of independent and possibly identically distributed random shocks. They may also incorporate long memory effects related to the concept of self‐similarity. The two approaches are here combined. In addition, methods are proposed for estimating the contribution of each component and evidence supporting the presence of both components in both the physical and risk‐neutral distributions is presented. Furthermore, it is shown that long‐horizon returns may be nonnormal when there is a self‐similar component. The presence of a self‐similar component also questions positive equity biases over the longer term.     

Risk management with weighted VaR

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.     

Sovereign Ratings and Oil‐Exporting Countries: The Effect of High Oil Prices on Ratings

We investigate how high and rising oil prices in the 2003–2008 period affected the sovereign ratings of oil‐exporting countries, after controlling for fundamentals. Based on a large dataset of countries from Standard and Poor's and Moody's, we find strong statistical evidence of a large ratings premium – nearly two notches – for those oil‐exporting countries with a large share of net oil revenue to gross domestic product, relative to countries with similar economic fundamentals. We have some limited forecast information from the rating agencies and the effect increases when we include this information, providing further evidence that this ratings premium is not driven by expected improvements in fundamentals. This finding has implications for asset prices in oil‐exporting countries and highlights the risk that in the event of a sharp unanticipated drop in oil prices, sovereign rating downgrades of oil‐exporting countries could be sharper than the deterioration in their economic fundamentals.     

A copula‐based regime‐switching GARCH model for optimal futures hedging

The article develops a regime‐switching Gumbel–Clayton (RSGC) copula GARCH model for optimal futures hedging. There are three major contributions of RSGC. First, the dependence of spot and futures return series in RSGC is modeled using switching copula instead of assuming bivariate normality. Second, RSGC adopts an independent switching Generalized Autoregressive Conditional Heteroscedasticity (GARCH) process to avoid the path‐dependency problem. Third, based on the assumption of independent switching, a formula is derived for calculating the minimum variance hedge ratio. Empirical investigation in agricultural commodity markets reveals that RSGC provides good out‐of‐sample hedging effectiveness, illustrating importance of modeling regime shift and asymmetric dependence for futures hedging. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:946–972, 2009     

Estimation and testing of portfolio Value‐at‐Risk based on L‐comoment matrices

This study employs L‐comoments introduced by Serfling and Xiao (2007) into portfolio Value‐at‐Risk estimation through two models: the Cornish–Fisher expansion (Draper, N. R. & Tierney, D. E., 1973) and modified VaR (Zangari, P., 1996). Backtesting outcomes indicate that modified VaR outperforms and L‐comoments give better estimates of portfolio skewness and excess kurtosis than do classical central moments in modeling heavy‐tailed distributions. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:897–908, 2010     

Optimal Currency Composition for China's Foreign Reserves: A Copula Approach

This paper investigates the optimal currency composition for a country's foreign reserves. In the context of China, we examine the asymmetric, fat‐tail and complex dependence structure in distributions of currency returns. A skewed, fat‐tailed and pair‐copula construction is then built to capture features of higher moments. In a D‐vine copula approach, we show that under the disappointment aversion effect, the central bank in our model can achieve sizeable gains in expected economic value from switching from the mean‐variance to copula modelling. We find that this approach will lead to an optimal currency composition that allows China to have more space for international currency diversification while maintaining the leading position of the US dollar in the currency shares of China's reserves.     

Multi level categorical data fusion using partially fused data

Data fusion poses challenging methodological issues for inferring the joint distribution of two random variables when the information available is mainly confined to the marginal distributions. When the variables are categorical, the challenges are even more severe. Applications of categorical data fusion are of top importance in marketing, especially in advertising. A great deal of categorical data fusion methods are confined to binary variables. In this paper we develop an innovative approach to categorical data fusion that extends previous methodologies and applies to categorical variables with any number of levels. We introduce a new concept of “evident dependence” that describes a variety of patterns of joint distributions given the marginals. Using information from partially fused data, our method smoothly accommodates a Bayesian approach based on mixtures of joint distributions constructed using evident dependence. The approach is illustrated using data from the advertising industry.     

Quantitative impact of correlation errors on basket options with time‐varying correlations

This study examines the quantitative impact of correlation errors on basket options with time‐varying correlations and the risk measures (conditional) value‐at‐risk (VaR) in the framework of Basel II. The results show that risk measure misestimation due to correlation errors are the largest and most asymmetric for the at‐the‐money and out‐of‐the‐money basket option. Delta hedging of the basket option reduces risk but increases size and asymmetry effects substantially. Finally, the square‐root‐of‐time rule for VaR does not adjust adequately to correlation errors and consistently underestimates risk measures, which could lead to the VaR exceedance clustering observed during the recent financial crisis. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark     

Profit Shifting in Two‐Sided Markets

We investigate how multinational two‐sided platform firms set their prices on intra‐firm transactions. Two‐sided platform firms derive income from two customer groups that are connected through at least one positive network externality from one group to the other. A main finding is that, even in the absence of taxation, transfer prices deviate from marginal cost of production. A second result of the paper is that it is inherently difficult to establish arm’s length prices in two‐sided markets. Finally, we find that differences in national tax rates may be welfare enhancing, despite the use of (abusive) transfer prices as a profit‐shifting device.     

BOUNDING WRONG‐WAY RISK IN CVA CALCULATION

A credit valuation adjustment (CVA) is an adjustment applied to the value of a derivative contract or a portfolio of derivatives to account for counterparty credit risk. Measuring CVA requires combining models of market and credit risk to estimate a counterparty's risk of default together with the market value of exposure to the counterparty at default. Wrong‐way risk refers to the possibility that a counterparty's likelihood of default increases with the market value of the exposure. We develop a method for bounding wrong‐way risk, holding fixed marginal models for market and credit risk and varying the dependence between them. Given simulated paths of the two models, a linear program computes the worst‐case CVA. We analyze properties of the solution and prove convergence of the estimated bound as the number of paths increases. The worst case can be overly pessimistic, so we extend the procedure by constraining the deviation of the joint model from a baseline reference model. Measuring the deviation through relative entropy leads to a tractable convex optimization problem that can be solved through the iterative proportional fitting procedure. Here, too, we prove convergence of the resulting estimate of the penalized worst‐case CVA and the joint distribution that attains it. We consider extensions with additional constraints and illustrate the method with examples.     

The information content of implied volatility in agricultural commodity markets

In this article we compare the incremental information content of lagged implied volatility to GARCH models of conditional volatility for a collection of agricultural commodities traded on the New York Board of Trade. We also assess the relevance of the additional information provided by the implied volatility in a risk management framework. It is first shown that past squared returns only marginally improve the information content provided by the lagged implied volatility. Secondly, value‐at‐risk (VaR) models that rely exclusively on lagged implied volatility perform as well as VaR models where the conditional variance is modelled according to GARCH type processes. These results indicate that the implied volatility for options on futures contracts in agricultural commodity markets provides relevant volatility information that can be used as an input to VaR models. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:441–454, 2003     

A STRUCTURAL RISK‐NEUTRAL MODEL FOR PRICING AND HEDGING POWER DERIVATIVES

We develop a structural risk‐neutral model for energy market modifying along several directions the approach introduced in Aïd et al. In particular, a scarcity function is introduced to allow important deviations of the spot price from the marginal fuel price, producing price spikes. We focus on pricing and hedging electricity derivatives. The hedging instruments are forward contracts on fuels and electricity. The presence of production capacities and electricity demand makes such a market incomplete. We follow a local risk minimization approach to price and hedge energy derivatives. Despite the richness of information included in the spot model, we obtain closed‐form formulae for futures prices and semiexplicit formulae for spread options and European options on electricity forward contracts. An analysis of the electricity price risk premium is provided showing the contribution of demand and capacity to the futures prices. We show that when far from delivery, electricity futures behave like a basket of futures on fuels.