Mai / Scherer Financial Engineering with Copulas Explained

2014
ISBN: 978-1-137-34631-5
Verlag: Palgrave Macmillan UK
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 167 Seiten

Reihe: Financial Engineering Explained

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.

Autoren/Hrsg.

Mai, J.

Scherer, M.

Inhaltsverzeichnis

1;Cover;1
2;Half-Title;2
3;Title;4
4;Copyright;5
5;Dedication;6
6;Contents;8
7;1 What Are Copulas?;18
7.1;1.1 Two Motivating Examples;18
7.1.1;1.1.1 Example 1: Analyzing Dependence between Asset Movements;18
7.1.2;1.1.2 Example 2: Modeling the Dependence between Default Times;23
7.2;1.2 Copulas and Sklar’s Theorem;23
7.2.1;1.2.1 The Generalized Inverse;26
7.2.2;1.2.2 Sklar’s Theorem for Survival Functions;28
7.2.3;1.2.3 How to Apply Sklar’s Theorem?;29
7.3;1.3 General Copula Properties;31
8;2 Which Rules for Handling Copulas Do I Need?;36
8.1;2.1 The Fréchet–Hoeffding Bounds;36
8.2;2.2 Switching from Distribution to Survival Functions;38
8.3;2.3 Invariance Under Strictly Monotone Transformations;42
8.4;2.4 Computing Probabilities from a Distribution Function;45
8.5;2.5 Copula Derivatives;47
8.6;2.6 Constructing New Copulas from Existing Ones;49
9;3 How to Measure Dependence?;52
9.1;3.1 Pearson’s Correlation Coefficient;52
9.2;3.2 Concordance Measures;56
9.2.1;3.2.1 Using Kendall’s t and Spearman’s ?S;61
9.3;3.3 Tail Dependence;62
10;4 What are Popular Families of Copulas?;66
10.1;4.1 Gaussian Copulas;66
10.1.1;4.1.1 Important Stylized Facts of the (Bivariate) Gaussian Copula;68
10.1.2;4.1.2 Generalization to Elliptical Copulas;72
10.2;4.2 Archimedean Copulas;74
10.2.1;4.2.1 Stylized Facts of Archimedean Copulas;77
10.2.2;4.2.2 Hierarchical Archimedean Copulas;79
10.3;4.3 Extreme-value Copulas;81
10.3.1;4.3.1 Marshall–Olkin Copulas;83
10.3.2;4.3.2 Stylized Facts of Extreme-Value Copulas;87
10.4;4.4 Archimax Copulas;88
11;5 How to SimulateMultivariate Distributions?;91
11.1;5.1 How to Simulate from a Copula?;95
11.1.1;5.1.1 Simulation Based on Analytical Techniques;95
11.1.2;5.1.2 Simulation Along a Stochastic Model;96
11.1.3;5.1.3 Practical Guide for the Implementation;100
12;6 How to Estimate Parameters of a Multivariate Model?;102
12.1;6.1 The Method ofMoments;103
12.1.1;6.1.1 Some Theoretical Background;104
12.2;6.2 Maximum-Likelihood Methods;105
12.2.1;6.2.1 Perfect Information about the Marginal Laws;106
12.2.2;6.2.2 Joint Maximization Over a and ?: Full Maximum-Likelihood;107
12.2.3;6.2.3 Inference Functions forMargins (IFM)Method;107
12.3;6.3 Using A Rank Transformation to Obtain (Pseudo-)Samples;108
12.3.1;6.3.1 Visualization of theMethods;111
12.4;6.4 Estimation of Specific Copula Families;113
12.4.1;6.4.1 Taylor-made Estimation Strategies for Extreme-value Copulas;114
12.5;6.5 A Note on Positive Semi-Definiteness;116
12.6;6.6 Some Remarks Concerning the Implementation;118
13;7 How to Deal with Uncertainty Concerning Dependence?;120
13.1;7.1 Bounds for the VaR of a Portfolio;122
13.2;7.2 What is the Maximal Probability for a Joint Default?;129
13.2.1;7.2.1 Motivation;129
13.2.2;7.2.2 Maximal Coupling;130
14;8 How to Construct a Portfolio-Default Model?;134
14.1;8.1 The Canonical Construction of Default Times;134
14.2;8.2 Classical CopulaModels for Dependent Default Times;136
14.2.1;8.2.1 The Portfolio-loss Distribution;138
14.3;8.3 A FactorModel for CDO Pricing;143
14.3.1;8.3.1 An Excursion to CDO Pricing;143
14.3.2;8.3.2 Calibrating the Two Portfolio-default Models;151
15;References;155
16;Index;166