Dynamic Copula Methods in Finance

by Cherubini , Umberto; Mulinacci, Sabrina; Gobbi, Fabio; Romagnoli, Silvia

• ISBN13: 9780470683071
• ISBN10: 0470683074
• Edition: 1st
• Format: Hardcover
• Copyright: 2011-11-21
• Publisher: Wiley

Summary

The latest tools and techniques for pricing and risk management This book introduces readers to the use of copula functions to represent the dynamics of financial assets and risk factors, integrated temporal and cross-section applications. The first part of the book will briefly introduce the standard the theory of copula functions, before examining the link between copulas and Markov processes. It will then introduce new techniques to design Markov processes that are suited to represent the dynamics of market risk factors and their co-movement, providing techniques to both estimate and simulate such dynamics. The second part of the book will show readers how to apply these methods to the evaluation of pricing of multivariate derivative contracts in the equity and credit markets. It will then move on to explore the applications of joint temporal and cross-section aggregation to the problem of risk integration.

Author Biography

UMBERTO CHERUBINI is Associate Professor of Financial Mathematics at the University of Bologna, where he heads the Graduate Degree in Quantitative Finance. He is a fellow of the Financial Econometrics Research Center (FERC), a member of the Scientific Committees of Abiformazione – the professional education arm of the Italian banking association, and AIFIRM – the Italian Association of Financial Risk Managers. He has been consulting and teaching in the field of finance and risk management for more than ten years. Before joining academia he worked as an economist at the Economic Research Department of BCI Milan. He has published papers in finance and economics in international journals, and is co-author of six books on topics of risk management and financial mathematics, including Fourier Transform Methods in Finance, John Wiley & Sons, Ltd, 2009; and Copula Methods in Finance, John Wiley & Sons, Ltd, 2004.

FABIO GOBBI is a post-doctoral researcher at the University of Bologna. He has a PhD in Statistics from the University of Florence and his areas of research focus on probability and financial econometrics. This is his first book.

SABRINA MULINACCI is Associate Professor of Mathematical Methods for Economics and Finance at the University of Bologna, Italy. Prior to this Sabrina was Associate Professor of Mathematical Methods for Economics and Actuarial Sciences at the Catholic University of Milan. She has a PhD in Mathematics from the University of Pisa and has published a number of research papers in international journals on probability and mathematical finance. She is co-author of Fourier Transform Methods in Finance, John Wiley & Sons, Ltd, 2009.

SILVIA ROMAGNOLI is Assistant Professor of Mathematical Models for Economics and Actuarial and Financial Sciences at the University of Bologna. Her scientific research is mainly addressed to the applications of stochastic models to finance and insurance. She has published several research papers in international journals on mathematical finance.

Table of Contents

Preface	p. ix
Correlation Risk in Finance	p. 1
Correlation Risk in Pricing and Risk Management	p. 1
Implied vs Realized Correlation	p. 3
Bottom-up vs Top-down Models	p. 4
Copula Functions	p. 4
Spatial and Temporal Dependence	p. 5
Long-range Dependence	p. 5
Multivariate GARCH Models	p. 7
Copulas and Convolution	p. 8
Copula Functions: The State of the Art	p. 11
Copula Functions: The Basic Recipe	p. 11
Market Co-movements	p. 14
Delta Hedging Multivariate Digital Products	p. 16
Linear Correlation	p. 19
Rank Correlation	p. 20
Multivariate Spearman's Rho	p. 22
Survival Copulas and Radial Symmetry	p. 23
Copula Volume and Survival Copulas	p. 24
Tail Dependence	p. 27
Long/Short Correlation	p. 27
Families of Copulas	p. 29
Elliptical Copulas	p. 29
Archimedean Copulas	p. 31
Kendall Function	p. 33
Exchangeability	p. 34
Hierarchical Copulas	p. 35
Conditional Probability and Factor Copulas	p. 39
Copula Density and Vine Copulas	p. 42
Dynamic Copulas	p. 45
Conditional Copulas	p. 45
Pseudo-copulas	p. 46
Copula Functions and Asset Price Dynamics	p. 49
The Dynamics of Speculative Prices	p. 49
Copulas and Markov Processes: The DNO approach	p. 51
The * and * Product Operators	p. 52
Product Operators and Markov Processes	p. 55
Self-similar Copulas	p. 58
Simulating Markov Chains with Copulas	p. 62
Time-changed Brownian Copulas	p. 63
CEV Clock Brownian Copulas	p. 64
VG Clock Brownian Copulas	p. 65
Copulas and Martingale Processes	p. 66
C-Convolution	p. 67
Markov Processes with Independent Increments	p. 75
Markov Processes with Dependent Increments	p. 78
Extracting Dependent Increments in Markov Processes	p. 81
Martingale Processes	p. 83
Multivariate Processes	p. 86
Multivariate Markov Processes	p. 86
Granger Causality and the Martingale Condition	p. 88
Copula-based Econometrics of Dynamic Processes	p. 91
Dynamic Copula Quantile Regressions	p. 91
Copula-based Markov Processes: Non-linear Quantile Autoregression	p. 93
Copula-based Markov Processes: Semi-parametric Estimation	p. 99
Copula-based Markov Processes: Non-parametric Estimation	p. 108
Copula-based Markov Processes: Mixing Properties	p. 110
Persistence and Long Memory	p. 113
C-convolution-based Markov Processes: The Likelihood Function	p. 116
Multivariate Equity Products	p. 121
Multivariate Equity Products	p. 121
European Multivariate Equity Derivatives	p. 122
Path-dependent Equity Derivatives	p. 125
Recursions of Running Maxima and Minima	p. 126
The Memory Feature	p. 130
Risk-neutral Pricing Restrictions	p. 132
Time-changed Brownian Copulas	p. 133
Variance Swaps	p. 135
Semi-parametric Pricing of Path-dependent Derivatives	p. 136
The Multivariate Pricing Setting	p. 137
H-Condition and Granger Causality	p. 137
Multivariate Pricing Recursion	p. 138
Hedging Multivariate Equity Derivatives	p. 141
Correlation Swaps	p. 144
The Term Structure of Multivariate Equity Derivatives	p. 147
Altiplanos	p. 148
Everest	p. 150
Spread Options	p. 150
Multivariate Credit Products	p. 153
Credit Transfer Finance	p. 153
Univariate Credit Transfer Products	p. 154
Multivariate Credit Transfer Products	p. 155
Credit Information: Equity vs CDS	p. 158
Structural Models	p. 160
Univariate Model: Credit Risk as a Put Option	p. 160
Multivariate Model: Gaussian Copula	p. 161
Large Portfolio Model: Vasicek Formula	p. 163
Intensity-based Models	p. 164
Univariate Model: Poisson and Cox Processes	p. 165
Multivariate Model: Marshall-Olkin Copula	p. 165
Homogeneous Model: Cuadras Augé Copula	p. 167
Frailty Models	p. 170
Multivariate Model: Archimedean Copulas	p. 170
Large Portfolio Model: Schönbucher Formula	p. 171
Granularity Adjustment	p. 171
Credit Portfolio Analysis	p. 172
Semi-unsupervised Cluster Analysis: K-means	p. 172
Unsupervised Cluster Analysis: Kohonen Self-organizing Maps	p. 174
(Semi-)unsupervised Cluster Analysis: Hierarchical Correlation Model	p. 175
Dynamic Analysis of Credit Risk Portfolios	p. 176
Risk Capital Management	p. 181
A Review of Value-at-Risk and Other Measures	p. 181
Capital Aggregation and Allocation	p. 185
Aggregation: C-Convolution	p. 187
Allocation: Level Curves	p. 189
Allocation with Constraints	p. 191
Risk Measurement of Managed Portfolios	p. 193
Henriksson-Merton Model	p. 195
Semi-parametric Analysis of Managed Funds	p. 200
Market-neutral Investments	p. 201
Temporal Aggregation of Risk Measures	p. 202
The Square-root Formula	p. 203
Temporal Aggregation by C-convolution	p. 203
Frontier Issues	p. 207
L'evy Copulas	p. 207
Pareto Copulas	p. 210
Semi-martingale Copulas	p. 212
A Elements of Probability	p. 215
Elements of Measure Theory	p. 215
Integration	p. 216
Expected Values and Moments	p. 217
The Moment-generating Function or Laplace Transform	p. 218
The Characteristic Function	p. 219
Relevant Probability Distributions	p. 219
Random Vectors and Multivariate Distributions	p. 224
The Multivariate Normal Distribution	p. 225
Infinite Divisibility	p. 226
Convergence of Sequences of Random Variables	p. 228
The Strong Law of Large Numbers	p. 229
The Radon-Nikodym Derivative	p. 229
Conditional Expectation	p. 229
Elements of Stochastic Processes Theory	p. 231
Stochastic Processes	p. 231
Filtrations	p. 231
Stopping Times	p. 232
Martingales	p. 233
Markov Processes	p. 234
Lévy Processes	p. 237
Subordinators	p. 240
Semi-martingales	p. 240
References	p. 245
Extra Reading	p. 251
Index	p. 259
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