This research project explores some properties of quasi-copulas and their applications in various fields, including finance and actuarial science. In probability theory and statistics, a copula is a bivariate distribution function, which describes the dependence between random variables. Every cumulative distribution of a random vector can be written in terms of marginal distribution of each component of the random vector and the copula describes the dependence structure between the components.
The first section of the paper develops sufficient conditions for both lower and upper bounds of a set of quasi-copulas to be copulas. Because the sufficient conditions are very simple to verify, this main theorem has a broad application and it is the cornerstone of the whole paper. It extends two papers published in the Journal of Applied Probability by Tankov in 2011 and by my research supervisor and her co-authors in 2012.
The second section gives specific examples to show how the main theorem can be applied to quantitative risk management in the financial industry. These examples are based on real-world situations, for which only partial information is available. For instance, the second example studies a bivariate derivative linked to two stocks. The prospective losses of two underlying stocks may look independent most of the time. However, when the market is stressed, both companies can be hit by common shocks, so that two companies become dependent. With the information collected in a normal period, it is possible to calculate the worst scenario, which may happen in a crisis (using our theorem).
The last section proves another theorem about bounds on copula, which restricts the constraints to be exactly two points. The theorem derives explicit expressions for the smallest copula above the lower bound and the larges copula below the upper bound for the infinite norm and extends earlier work by Nelson.