Financial Engineering with Copulas Explained (Financial Engineering Explained)

## List of Figures 1.1: Left: daily stock returns of the BMW stock, plotted against daily stock returns of the Daimler stock. Their empirical correlation is around 79. 6 %. Middle: daily stock returns of the Daimler stock, plotted against daily returns of a Gold Index. Right: daily stock returns of the BMW stock, plotted against daily returns of a Gold Index. Both stock returns have an empirical correlation to Gold Index returns of about 4. 4 %. The time series comprise business days between April 2008 and May 2013. The R command used to create this figure is plot(). 1.2: Left: transformed daily stock returns of the BMW stock, plotted against transformed daily stock returns of the Daimler stock. Middle: transformed daily stock returns of the Daimler stock, plotted against transformed daily returns of a Gold Index. Right: transformed daily stock returns of the BMW stock, plotted against transformed daily returns of a Gold Index. The time series comprise business days between April 2008 and May 2013. The rank transformation was done using the command pobs() from the R-package copula. 1.3: Plot of a distribution function F, its generalized inverse F -1 , and their concatenations F -1 • F and F • F -1 in the case where F is not continuous. 1.4: Left to right: Frank copula (see Section 4.2), Maresias copula (see [Mai, Scherer (2012), p. 45]), and BC 2 copula (see Section 4.3). 2.1: Left: bivariate scatter plot of samples drawn from the independence copula 2 . Note that there are no systematic clusterings observable. Middle: scatter plot of the comonotonicity copula M 2 , where in each sample both components take precisely the same value, so all samples fall on the diagonal. Right: countermonotonicity copula W 2 , where all samples fall on the counterdiagonal. 2.2: Left: histogram of n = 5 000 iid realizations X (1) , ..., X (n) from the exponential law with mean 10. Right: histogram of the transformed variates U (1) := 1exp ( -0. 1 X (1) ), ..., U (n) := 1exp ( -0. 1 X (n) ), which are U[0, 1]-distributed. This plot is created using the R command hist(). 2 2.3: Computation of cube probabilities in low dimensions (d = 2 and d = 3). 2.4: Bivariate Gaussian copula density c ρ for increasing parameter ρ, visualized using persp() from the R-package graphics. 3 10.1057/9781137346315 -Financial Engineering with Copulas Explained, Jan- This is a succinct guide to the application and modelling of dependence models or copulas in the financial markets. First applied to credit risk modelling, copulas are now widely used across a range of derivatives transactions, asset pricing techniques and risk models and are a core part of the financial engineer's toolkit. The modeling of dependence structures (or copulas) is undoubtedly one of the key challenges for modern financial engineering. First applied to credit-risk modeling, copulas are now widely used across a range of derivatives transactions, asset pricing techniques, and risk models, and are a core part of the financial engineer's toolkit. However, by their very nature, copulas are complex and their applications are often misunderstood. Incorrectly applied, copulas can be hugely detrimental to a model or algorithm. Financial Engineering with Copulas Explained is a reader-friendly, yet rigorous introduction to the state-of-the-art regarding the theory of copulas, their simulation and estimation, and their use in financial applications. Starting with an introduction to the basic notions, such as required definitions and dependence measures, the book looks at statistical issues comprising parameter estimation and stochastic simulation. The book will show, from a financial engineering perspective, how copula theory can be applied in the context of portfolio credit-risk modeling, and how it can help to derive model-free bounds for relevant risk measures. The book will cover numerous different market applications of copulas, and enable readers to construct stable, high-dimensional models for asset pricing and risk modeling. Written to appeal to quantitatively minded practitioners across the trading floors and in risk management, academics and students, Financial Engineering with Copulas Explained will be a valuable, accessible and practical guide to this complex topic "Financial Engineering with Copulas Explained is a reader-friendly, yet rigorous introduction to the state-of-the-art regarding the theory of copulas, their simulation and estimation, and their use in financial applications. Starting with an introduction to the basic notions, such as required definitions and dependence measures, the book looks at statistical issues comprising parameter estimation and stochastic simulation. The book will show, from a financial engineering perspective, how copula theory can be applied in the context of portfolio credit-risk modeling, and how it can help to derive model-free bounds for relevant risk measures. The book will cover numerous different market applications of copulas, and enable readers to construct stable, high-dimensional models for asset pricing and risk modeling."-- Portion of summary from book