Quantitative Finance Collector is a blog on Quantitative finance analysis, methods in mathematical finance focusing on derivative pricing, quantitative trading and quantitative risk management.

Value at Risk is widely used to measure the downside risk, and Copula is a generalized dependence structure instead of linear correlation to model dependence, especially for lower tail dependence, therefore the combination of VaR with Copula is fantastic in terms of accurately capturing the true risk embedded.

I read roughly a working paper Value at Risk – MATLAB Application of Copulas on US and Indian Markets, where the authors calculate the Value at Risk (VaR) using the bivariate Gaussian Copula distribution implemented in MATLAB for the Dow-Jones index and the National Stock Exchange index. It is good to use Gaussian Copula together with some rank correlation (like Spearman's rho or Kendall tau) to model the dependence, however, we must be very careful as basically Gaussian Copula assumes the joint dependence structure normally distributed and as a result, no matter which marginal distribution you choose, the upper and lower tail dependence approach to zero when the significant level limits to one and zero, in other words, in a bivariate case, the probability that X2 exceeds its q-quantile, given that X1 exceeds its q-quantile (upper tail dependence) when q->1, and the probability that X2 is below its q-quantile, given that X1 is below its q-quantile (lower tail dependence) when q->0 are zero.

A collection of codes on Copula estimation and simulation is shown here, where you can find parameter estimation for t-copula, Grouped-t copula, asymmetric copula, etc., another simple recursive routine to estimate by maximum likelihood the correlation matrix and the degrees of freedom for structured t-copula is shared at http://www.mathworks.com/matlabcentral/fileexchange/19751, the authors impose extra structure on the correlation matrix in the estimation process, where the number of variables is large as compared to the number of observations.

The paper "Estimation of Structured t-Copulas" is available at: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1126401, more paper and codes are at the author's homepage: http://www.symmys.com/AttilioMeucci/Home/Home.html

http://www.mathfinance.cn/Grouped-T-copula-simulation-estimation/ shared a sample code for grouped-t copula simulation, further, several copula estimation and simulation package can be found. But, most of the case we talk about an exchangeble copula due to its relatively easier to explain, however, it has limited applications especially in the area of credit risk, or derivative markets where asymmetric dependence plays a crutial role. For example, a desire to maintain the competitiveness of Japanese exports to the United States. with German exports to the United States. would lead the Bank of Japan to intervene to ensure a matching depreciation of the yen against the dollar whenever the Deutsche mark (DM) depreciated against the U.S. dollar. Such rebalancing behavior would also lead to greater dependence during depreciations of the DM and yen against the dollar than during appreciations. It is certainly natural to enquire whether there are extensions that are not rigidly exchangeble.


A scatter plot of the return of S&P 500 index and that of its implied volatility difference series is shown above, clearly the dependence is stronger in left-up corner than right-down corner.

Copula is widely applied to model the dependence of multivariate variable, two popula implicit copulas are Gaussian copula and T copula, however, tail dependence under Gaussian copula is asymptotically equal to zero, which is unrealistic and under-estimate the co-movement of variables, especially in extreme market situation nowdays; T copula, on the other hand, has a global degree of freedom to decide largely the dependence structure, which is over-simple, for instance, risk manager might want to define different degree of freedom for different markets due to their special risk profile. Grouped-T copula was created to overcome this problem, where seperated degree of freedom can be set for each subgroup. sample code is here: http://economia.unipv.it/pagp/pagine_personali/dean/programs/gruped_t_copula_simul_est

Classes (S4) of commonly used copulas including elliptical (normal and t), Archimedean (Clayton, Gumbel, Frank, and Ali-Mikhail-Haq), extreme value (Husler-Reiss and Galambos), and other families (Plackett and Farlie-Gumbel-Morgenstern). Methods for density, distribution, random number generation, bivariate dependence measures, perspective and contour plots. Functions for fitting copula models. Independence tests among random variables and random vectors. Serial independence tests for univariate and multivariate continuous time series. Goodness-of-fit tests for copulas based on multipliers and on the parametric bootstrap.

R package can be downloaded at http://cran.r-project.org/web/packages/copula/index.html