Financial Engineering with Copulas Explained

By J. Mai and M. Scherer

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.

• Language: English
• Publisher: Palgrave Macmillan
• Release date: Oct 2, 2014
• ISBN: 9781137346315

Book preview

Financial Engineering with Copulas Explained

Jan-Frederik Mai

XAIA Investment, Munich, Germany

and

Matthias Scherer

Technische Universität München, Germany

Jan-Frederik Mai and Matthias Scherer 2014

All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission.

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The authors have asserted their rights to be identified as the authors of this work in accordance with the Copyright, Designs and Patents Act 1988.

First published 2014

by PALGRAVE MACMILLAN

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ISBN: 978–1–137–34630–8

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To the new generation: Paul, Fabian, and Lara.

Contents

List of Figures

List of Tables

Preface and Acknowledgments

1 What Are Copulas?

1.1 Two Motivating Examples

1.1.1 Example 1: Analyzing Dependence between Asset Movements

1.1.2 Example 2: Modeling the Dependence between Default Times

1.2 Copulas and Sklar’s Theorem

1.2.1 The Generalized Inverse

1.2.2 Sklar’s Theorem for Survival Functions

1.2.3 How to Apply Sklar’s Theorem?

1.3 General Copula Properties

2 Which Rules for Handling Copulas Do I Need?

2.1 The Fréchet–Hoeffding Bounds

2.2 Switching from Distribution to Survival Functions

2.3 Invariance Under Strictly Monotone Transformations

2.4 Computing Probabilities from a Distribution Function

2.5 Copula Derivatives

2.6 Constructing New Copulas from Existing Ones

3 How to Measure Dependence?

3.1 Pearson’s Correlation Coefficient

3.2 Concordance Measures

3.2.1 Using Kendall’s τ and Spearman’s ρ S

3.3 Tail Dependence

4 What are Popular Families of Copulas?

4.1 Gaussian Copulas

4.1.1 Important Stylized Facts of the (Bivariate) Gaussian Copula

4.1.2 Generalization to Elliptical Copulas

4.2 Archimedean Copulas

4.2.1 Stylized Facts of Archimedean Copulas

4.2.2 Hierarchical Archimedean Copulas

4.3 Extreme-value Copulas

4.3.1 Marshall–Olkin Copulas

4.3.2 Stylized Facts of Extreme-Value Copulas

4.4 Archimax Copulas

5 How to Simulate Multivariate Distributions?

5.1 How to Simulate from a Copula?

5.1.1 Simulation Based on Analytical Techniques

5.1.2 Simulation Along a Stochastic Model

5.1.3 Practical Guide for the Implementation

6 How to Estimate Parameters of a Multivariate Model?

6.1 The Method of Moments

6.1.1 Some Theoretical Background

6.2 Maximum-Likelihood Methods

6.2.1 Perfect Information about the Marginal Laws

6.2.2 Joint Maximization Over α and θ : Full Maximum-Likelihood

6.2.3 Inference Functions for Margins (IFM) Method

6.3 Using A Rank Transformation to Obtain (Pseudo-)Samples

6.3.1 Visualization of the Methods

6.4 Estimation of Specific Copula Families

6.4.1 Taylor-made Estimation Strategies for Extreme-value Copulas

6.5 A Note on Positive Semi-Definiteness

6.6 Some Remarks Concerning the Implementation

7 How to Deal with Uncertainty Concerning Dependence?

7.1 Bounds for the VaR of a Portfolio

7.2 What is the Maximal Probability for a Joint Default?

7.2.1 Motivation

7.2.2 Maximal Coupling

8 How to Construct a Portfolio-Default Model?

8.1 The Canonical Construction of Default Times

8.2 Classical Copula Models for Dependent Default Times

8.2.1 The Portfolio-loss Distribution

8.3 A Factor Model for CDO Pricing

8.3.1 An Excursion to CDO Pricing

8.3.2 Calibrating the Two Portfolio-default Models

References

Index

List of Figures

List of Tables

Preface and Acknowledgments

What can you expect from this book? We aim to provide you with an easy-to-read introduction to current problems (and solutions, of course) in the field of dependence modeling as it is required in today’s financial and insurance industry. If you enjoy reading a chapter of the book after a long day at work or during a continental flight, and understand the essence of the exposition, then we have succeeded. Clearly, ‘easy-to-read’ strongly depends on your mathematical training. We take as granted familiarity with probability calculus and elementary statistics.¹ Aimed at readers from the financial industry, we try to illustrate the theory with real world examples. We are always mathematically precise, but we do not aim at being complete with respect to proofs and the latest generalizations of the presented results. Instead, we visualize the results and explain how they can be applied. Finally, we direct the interested reader to deeper literature, where proofs and further results are given. The field of dependence modeling has grown impressively in recent years.² Having said this, it is clear that this introduction has to prioritize certain aspects, but we provide many references for those readers with an appetite for further material.

The Importance of Dependence

Both authors strongly believe that the role of dependence has been underestimated for a long time (and still is) in the financial and insurance industry. A bank or insurance company hardly runs into substantial trouble if individual options or insurance contracts are mis-priced by a few basis points. Events where many things go wrong at the same time are those that create true financial distress. Think of an event that causes multiple stocks to fall jointly or multiple insurance claims to trigger simultaneously. Or, even more severe, consider events that affect prices across different markets (stocks, credit, FX, ...) at the same time. Many financial institutions have a very high level of sophistication inside each of their specialized trading desks, but might not have the simplest model available for how these different markets and trading desks interact. So the question of risk aggregation within such an institution is difficult to answer – if it is possible at all. As a different example, the reader might collect all univariate