Bounds for Functions of Dependent Risks

The problem of finding the best-possible lower bound on the distribution of a non-decreasing function of n dependent risks is solved when n=2 and a lower bound on the copula of the portfolio is provided. The problem gets much more complicated in arbitrary dimensions. When no information on the structure of dependence of the random vector is available, we provide a bound on the distribution function of the sum of risks which we prove to be better than the one generally used in the literature.     

Measuring financial risks with copulas

This paper is concerned with the statistical modeling of the dependence structure of multivariate financial data using the concept of copulas. We select some special copulas and identify the type of dependency captured by each one. We fit copulas to daily returns and simulate from the fitted models. We compare the effect of the choice of copula on risk measures and assess the variability of one-step-ahead predictions of portfolio losses. We analyze extreme scenarios and fit extreme value copulas to the block maxima and minima from daily returns. The stress scenarios constructed are compared to those obtained using models from the extreme value theory. We illustrate the usefulness of the copula approach using two stock market indexes.     

Can investors of Chinese energy stocks benefit from diversification into commodity futures?

We attempt to evaluate the diversification potential of commodity futures for energy stocks in China. With a variety of copula functions and three risk-based dynamic measures, our results show that even though commodity futures are not helpful in improving the risk-adjusted returns of energy stocks, they can significantly reduce the volatilities and expected-shortfalls of the diversified portfolios. Such diversification benefits are much larger during large market downturns than during normal times. In particular, gold (copper) futures are the most (least) attractive in diversifying risks of energy stocks in most cases. The results also highlight that the non-linear dependence cannot be ignored when estimating the diversification benefits, and more various risk hedging strategies are expected for investors holding energy stocks, especially coal company stocks.     

Impact of copula choice on the modeling of crop yield basis risk

A number of problems in agricultural economics involve modeling joint distributions for which the assumption of multivariate normality may not be warranted. Yet, very little work has been conducted evaluating competing methods for modeling joint dependence. We develop a simulation framework to evaluate the bias and efficiency impacts of copula choice in the context of evaluating county‐to‐farm basis risk. The results suggest significant differences in performance across various copulas and approaches. The findings have important implications for risk analysis, insurance, and policy modeling problems in agriculture regarding the selection of method to model dependence among random variables.     

Criteria for the Stochastic Ordering of Random Sums,with Actuarial Applications

It is shown that vectors ( S M 1 , … , S Mn ) and ( S' M'1 , …, S' M'n ) of random sums of positive random variables are stochastically ordered by upper orthant dependence, lower orthant dependence, concordance or by the supermodular ordering whenever their corresponding random numbers of terms ( M 1 , … , M n ) and ( M' 1 , … , M' n ) are themselves ordered in this fashion. Actuarial applications of these results are given to different dependence structures for the collective risk model with several classes of business.     

Bivariate income distributions for assessing inequality and poverty under dependent samples

As indicators of social welfare, the incidence of inequality and poverty is of ongoing concern to policy makers and researchers alike. Of particular interest are the changes in inequality and poverty over time, which are typically assessed through the estimation of income distributions. From this, income inequality and poverty measures, along with their differences and standard errors, can be derived and compared. With panel data becoming more frequently used to make such comparisons, traditional methods which treat income distributions from different years independently and estimate them on a univariate basis, fail to capture the dependence inherent in a sample taken from a panel study. Consequently, parameter estimates are likely to be less efficient, and the standard errors for between-year differences in various inequality and poverty measures will be incorrect. This paper addresses the issue of sample dependence by suggesting a number of bivariate distributions, with Singh–Maddala or Dagum marginals, for a partially dependent sample of household income for two years. Specifically, the distributions considered are the bivariate Singh–Maddala distribution, proposed by Takahasi (1965), and bivariate distributions belonging to the copula class of multivariate distributions, which are an increasingly popular approach to modelling joint distributions. Each bivariate income distribution is estimated via full information maximum likelihood using data from the Household, Income and Labour Dynamics in Australia (HILDA) Survey for 2001 and 2005. Parameter estimates for each bivariate income distribution are used to obtain values for mean income and modal income, the Gini inequality coefficient and the headcount ratio poverty measure, along with their differences, enabling the assessment of changes in such measures over time. In addition, the standard errors of each summary measure and their differences, which are of particular interest in this analysis, are calculated using the delta method.     

Correlation smile matching for collateralized debt obligation tranches with α-stable distributions and fitted Archimedean copula models

As an extension of the standard Gaussian copula model to price collateralized debt obligation (CDO) tranche swaps we present a generalization of a one-factor copula model based on stable distributions. For special parameter values these distributions coincide with Gaussian or Cauchy distributions, but changing the parameters allows a continuous deformation away from the Gaussian copula. All these factor copulas are embedded in a framework of stochastic correlations. We furthermore generalize the linear dependence in the usual factor approach to a more general Archimedean copula dependence between the individual trigger variable and the common latent factor. Our analysis is carried out on a non-homogeneous correlation structure of the underlying portfolio. CDO tranche market premia, even throughout the correlation crisis in May 2005, can be reproduced by certain models. From a numerical perspective, all these models are simple, since calculations can be reduced to one-dimensional numerical integrals.     

Oil price and Gulf Corporation Council stock indices: New evidence from time-varying copula models

Using a time-varying GJR copula approach, we determine the conditional dependence of the GCC stock indices on oil price between 2007 and 2016. We show how to improve the forecasting accuracy of the co-movement of energy and stock prices in an equally weighted portfolio. Contrary to prior findings, we demonstrate that due to the different co-movements across the GCC stock indices, portfolios of oil assets and several GCC stocks are less likely to be affected by systemic risk. The different co-movements across several stock indices over time provide different entry and exit points for stock investors. This approach is in line with the ‘buy low/sell high’ adage.     

WTI crude oil Futures in portfolio diversification: The time-to-maturity effect

The aim of the paper is to analyze the diversification effect brought by crude oil Futures contracts, the most liquid commodity Futures, into a portfolio of stocks. The studies that have documented the very low- and essentially negative-correlations between commodities and equities typically rely on normally distributed returns, which is not the case for crude oil Futures and stocks indexes. Moreover, the particular time-to-maturity chosen for the Future contract used as an investment vehicle is an important matter that needs to be addressed, in presence of forward curves switching between backwardation and contango shapes.Our goal in this paper is twofold: (a) we introduce copula functions to have a better representation of the dependence structure of oil Futures with equity indexes; (b) using this copula representation, we are able to analyze in a precise manner the “maturity effect” in the choice of crude oil Future contract with respect to its diversification benefits. Our finding is that, in the case of distant maturities Futures, e.g., 18 months, the negative correlation effect is more pronounced whether stock prices increase or decrease. This property has the merit to avoid the hurdles of a frequent roll over while being quite desirable in the current trendless equity markets. Empirical evidence is exhibited on a database comprising the NYMEX WTI crude oil Futures and S&P 500 index over a 15 year-time period.     

On an efficient multiple time step Monte Carlo simulation of the SABR model

In this paper, we will present a multiple time step Monte Carlo simulation technique for pricing options under the Stochastic Alpha Beta Rho model. The proposed method is an extension of the one time step Monte Carlo method that we proposed in an accompanying paper Leitao et al. [Appl. Math. Comput. 2017, 293, 461–479], for pricing European options in the context of the model calibration. A highly efficient method results, with many very interesting and nontrivial components, like Fourier inversion for the sum of log-normals, stochastic collocation, Gumbel copula, correlation approximation, that are not yet seen in combination within a Monte Carlo simulation. The present multiple time step Monte Carlo method is especially useful for long-term options and for exotic options.