Algorithms for Finding Copulas Minimizing Convex Functions of Sums

In this paper, we develop improved rearrangement algorithms to find the dependence structure that minimizes a convex function of the sum of dependent variables with given margins. We propose a new multivariate dependence measure, which can assess the convergence of the rearrangement algorithms and can be used as a stopping rule. We show how to apply these algorithms for example to finding the dependence among variables for which the marginal distributions and the distribution of the sum or the difference are known. As an example, we can find the dependence between two uniformly distributed variables that makes the distribution of the sum of two uniform variables indistinguishable from a normal distribution. Using MCMC techniques, we design an algorithm that converges to the global optimum.

[1]  Ruodu Wang,et al.  Extremal Dependence Concepts , 2015, 1512.03232.

[2]  Carol Alexander 3,et al.  Bivariate normal mixture spread option valuation , 2003 .

[3]  P. Embrechts,et al.  An Academic Response to Basel 3.5 , 2014 .

[4]  Giovanni Puccetti,et al.  Bounds on total economic capital: the DNB case study , 2014 .

[5]  Ludger Rüschendorf,et al.  Value-at-Risk Bounds with Variance Constraints , 2015 .

[6]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[7]  Éva Krauczi.  A study of the quantile correlation test for normality , 2009 .

[8]  Bin Wang,et al.  The complete mixability and convex minimization problems with monotone marginal densities , 2011, J. Multivar. Anal..

[9]  Oleg Bondarenko.  Estimation of Risk-Neutral Densities Using Positive Convolution Approximation , 2002 .

[10]  Ruodu Wang,et al.  Risk aggregation with dependence uncertainty , 2014 .

[11]  Douglas T. Breeden,et al.  Prices of State-Contingent Claims Implicit in Option Prices , 1978 .

[12]  A. Lo,et al.  Nonparametric Risk Management and Implied Risk Aversion , 2000 .

[13]  On the multidimensional extension of countermonotonicity and its applications , 2014 .

[14]  Erhan Çinlar,et al.  Introduction to stochastic processes , 1974 .

[15]  Ludger Rüschendorf,et al.  Computation of sharp bounds on the distribution of a function of dependent risks , 2012, J. Comput. Appl. Math..

[16]  Ruodu Wang,et al.  Detecting Complete and Joint Mixability , 2015, J. Comput. Appl. Math..

[17]  S. Vanduffel,et al.  A New Approach to Assessing Model Risk in High Dimensions , 2015 .

[18]  René Carmona,et al.  Pricing and Hedging Spread Options , 2003, SIAM Rev..

[19]  J. Hammersley,et al.  Monte Carlo Methods , 1965 .

[20]  J. A. Cuesta-Albertos,et al.  Tests of goodness of fit based on the $L_2$-Wasserstein distance , 1999 .

[21]  Mihalis Yannakakis,et al.  Permuting Elements Within Columns of a Matrix in Order to Minimize Maximum Row Sum , 1984, Math. Oper. Res..

[22]  Wen-Lian Hsu,et al.  Approximation Algorithms for the Assembly Line Crew Scheduling Problem , 1984, Math. Oper. Res..

[23]  Utz-Uwe Haus,et al.  Bounding stochastic dependence, joint mixability of matrices, and multidimensional bottleneck assignment problems , 2014, Oper. Res. Lett..

[24]  P. Embrechts,et al.  Model Uncertainty and VaR Aggregation , 2013 .

[25]  Hal S. Stern,et al.  Models for Distributions on Permutations , 1990 .