Dynamic Copula Methods in Finance

Transcription

1 Dynamic Copula Methods in Finance Umberto Cherubini Fabio Gofobi Sabriea Mulinacci Silvia Romageoli A John Wiley & Sons, Ltd., Publication

2 Contents Preface ix 1 Correlation Risk in Finance Correlation Risk in Pricing and Risk Management Implied vs Realized Correlation Bottom-up vs Top-down Models Copula Functions Spatial and Temporal Dependence Long-range Dependence Multivariate GARCH Models CJopulas and Convolution 8 2 Copula Functions: The State of the Art Copula Functions: The Basic Recipe Market Co-movements Delta Hedging Multivariate Digital Products Linear Correlation Rank Correlation Multivariate Spearman's Rho Survival Copulas and Radial Symmetry Copula Volume and Survival Copulas Tail Dependence Long/Short Correlation Families of Copulas Elliptical Copulas Archimedean Copulas Kendall Function Exchangeability Hierarchical Copulas Conditional Probability and Factor Copulas Copula Density and Vine Copulas Dynamic Copulas Conditional Copulas Pseudo-copulas 46

3 vi Contents Copula Functions and Asset Price Dynamics The Dynamics of Speculative Prices Copulas and Markov Processes: The DNO approach The * and Product Operators Product Operators and Markov Processes Self-similar Copulas Simulating Markov Chains with Copulas Time-changed Brownian Copulas CEV Clock Brownian Copulas VG Clock Brownian Copulas Copulas and Martingale Processes C-Convolution Markov Processes with Independent Increments Markov Processes with Dependent Increments Extracting Dependent Increments in Markov Processes Martingale Processes Multivariate Processes Multivariate Markov Processes Granger Causality and the Martingale Condition 88 Copula-based Econometrics of Dynamic Processes Dynamic Copula Quantile Regressions Copula-based Markov Processes: Non-linear Quantile Autoregression Copula-based Markov Processes: Semi-parametric Estimation 99 ^4.4 Copula-based Markov Processes: Non-parametric Estimation Copula-based Markov Processes: Mixing Properties Persistence and Long Memory C-convolution-based Markov Processes: The Likelihood Function 116 Multivariate Equity Products Multivariate Equity Products European Multivariate Equity Derivatives Path-dependent Equity Derivatives Recursions of Running Maxima and Minima The Memory Feature Risk-neutral Pricing Restrictions Time-changed Brownian Copulas Variance Swaps Semi-parametric Pricing of Path-dependent Derivatives The Multivariate Pricing Setting H-Condition and Granger Causality Multivariate Pricing Recursion Hedging Multivariate Equity Derivatives Correlation Swaps The Term Structure of Multivariate Equity Derivatives Altiplanos Everest Spread Options 150

4 Contents 6 Multivariate Credit Products Credit Transfer Finance Univariate Credit Transfer Products Multivariate Credit Transfer Products Credit Information: Equity vs CDS Structural Models Univariate Model: Credit Risk as a Put Option Multivariate Model: Gaussian Copula Large Portfolio Model: Vasicek Formula Intensity-based Models _ Univariate Model: Poisson and Cox Processes Multivariate Model: Marshall-Olkin Copula Homogeneous Model: Cuadras Auge Copula Frailty Models Multivariate Model: Archimedean Copulas Large Portfolio Model: Schonbucher Formula Granularity Adjustment Credit Portfolio Analysis Semi-unsupervised Cluster Analysis: K-means Unsupervised Cluster Analysis: Kohonen Self-organizing Maps (Semi-)unsupervised Cluster Analysis: Hierarchical Correlation Model Dynamic Analysis of Credit Risk Portfolios Risk Capital Management A Review of Value-at-Risk and Other Measures Capital Aggregation and Allocation Aggregation: C-Convolution Allocation: Level Curves Allocation with Constraints Risk Measurement of Managed Portfolios Henriksson-Merton Model Semi-parametric Analysis of Managed Funds Market-neutral Investments Temporal Aggregation of Risk Measures The Square-root Formula Temporal Aggregation by C-convolution Frontier Issues Levy Copulas Pareto Copulas Semi-martingale Copulas 212 A Elements of Probability 215 A. 1 Elements of Measure Theory 215 A.2 Integration 216 A.2.1 Expected Values and Moments 217 A.3 The Moment-generating Function or Laplace Transform 218

5 viii Contents A.4 The Characteristic Function 219 A.5 Relevant Probability Distributions 219 A.6 Random Vectors and Multivariate Distributions 224 A.6.1 The Multivariate Normal Distribution 225 A.7 Infinite Divisibility 226 A.8 Convergence of Sequences of Random Variables 228 A.8.1 The Strong Law of Large Numbers 229 A.9 The Radon-Nikodym Derivative 229 A. 10 Conditional Expectation 229 B Elements of Stochastic Processes Theory 231 B.I Stochastic Processes 231 B.I.I Filtrations 231 B.1.2 Stopping Times 232 B.2 Martingales 233 B.3 Markov Processes 234 B.4 Levy Processes 237 B.4.1 Subordinators 240 B.5 Semi-martingales 240 References 245 Extra Reading 251 Index 259