Dynamic Copula Methods in Finance

About the authorsUMBERTO 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 RiskManagers. 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.

• Preface ix1 Correlation Risk in Finance 11.1 Correlation Risk in Pricing and Risk Management 11.2 Implied vs Realized Correlation 31.3 Bottom-up vs Top-down Models 41.4 Copula Functions 41.5 Spatial and Temporal Dependence 51.6 Long-range Dependence 51.7 Multivariate GARCH Models 71.8 Copulas and Convolution 82 Copula Functions: The State of the Art 112.1 Copula Functions: The Basic Recipe 112.2 Market Co-movements 142.3 Delta Hedging Multivariate Digital Products 162.4 Linear Correlation 192.5 Rank Correlation 202.6 Multivariate Spearman’s Rho 222.7 Survival Copulas and Radial Symmetry 232.8 Copula Volume and Survival Copulas 242.9 Tail Dependence 272.10 Long/Short Correlation 272.11 Families of Copulas 292.11.1 Elliptical Copulas 292.11.2 Archimedean Copulas 312.12 Kendall Function 332.13 Exchangeability 342.14 Hierarchical Copulas 352.15 Conditional Probability and Factor Copulas 392.16 Copula Density and Vine Copulas 422.17 Dynamic Copulas 452.17.1 Conditional Copulas 452.17.2 Pseudo-copulas 463 Copula Functions and Asset Price Dynamics 493.1 The Dynamics of Speculative Prices 493.2 Copulas and Markov Processes: The DNO approach 513.2.1 The * and _ Product Operators 523.2.2 Product Operators and Markov Processes 553.2.3 Self-similar Copulas 583.2.4 Simulating Markov Chains with Copulas 623.3 Time-changed Brownian Copulas 633.3.1 CEV Clock Brownian Copulas 643.3.2 VG Clock Brownian Copulas 653.4 Copulas and Martingale Processes 663.4.1 C-Convolution 673.4.2 Markov Processes with Independent Increments 753.4.3 Markov Processes with Dependent Increments 783.4.4 Extracting Dependent Increments in Markov Processes 813.4.5 Martingale Processes 833.5 Multivariate Processes 863.5.1 Multivariate Markov Processes 863.5.2 Granger Causality and the Martingale Condition 884 Copula-based Econometrics of Dynamic Processes 914.1 Dynamic Copula Quantile Regressions 914.2 Copula-based Markov Processes: Non-linear Quantile Autoregression 934.3 Copula-based Markov Processes: Semi-parametric Estimation 994.4 Copula-based Markov Processes: Non-parametric Estimation 1084.5 Copula-based Markov Processes: Mixing Properties 1104.6 Persistence and Long Memory 1134.7 C-convolution-based Markov Processes: The Likelihood Function 1165 Multivariate Equity Products 1215.1 Multivariate Equity Products 1215.1.1 European Multivariate Equity Derivatives 1225.1.2 Path-dependent Equity Derivatives 1255.2 Recursions of Running Maxima and Minima 1265.3 The Memory Feature 1305.4 Risk-neutral Pricing Restrictions 1325.5 Time-changed Brownian Copulas 1335.6 Variance Swaps 1355.7 Semi-parametric Pricing of Path-dependent Derivatives 1365.8 The Multivariate Pricing Setting 1375.9 H-Condition and Granger Causality 1375.10 Multivariate Pricing Recursion 1385.11 Hedging Multivariate Equity Derivatives 1415.12 Correlation Swaps 1445.13 The Term Structure of Multivariate Equity Derivatives 1475.13.1 Altiplanos 1485.13.2 Everest 1505.13.3 Spread Options 1506 Multivariate Credit Products 1536.1 Credit Transfer Finance 1536.1.1 Univariate Credit Transfer Products 1546.1.2 Multivariate Credit Transfer Products 1556.2 Credit Information: Equity vs CDS 1586.3 Structural Models 1606.3.1 Univariate Model: Credit Risk as a Put Option 1606.3.2 Multivariate Model: Gaussian Copula 1616.3.3 Large Portfolio Model: Vasicek Formula 1636.4 Intensity-based Models 1646.4.1 Univariate Model: Poisson and Cox Processes 1656.4.2 Multivariate Model: Marshall–Olkin Copula 1656.4.3 Homogeneous Model: Cuadras Aug´e Copula 1676.5 Frailty Models 1706.5.1 Multivariate Model: Archimedean Copulas 1706.5.2 Large Portfolio Model: Sch¨onbucher Formula 1716.6 Granularity Adjustment 1716.7 Credit Portfolio Analysis 1726.7.1 Semi-unsupervised Cluster Analysis: K-means 1726.7.2 Unsupervised Cluster Analysis: Kohonen Self-organizing Maps 1746.7.3 (Semi-)unsupervised Cluster Analysis: Hierarchical Correlation  Model 1756.8 Dynamic Analysis of Credit Risk Portfolios 1767 Risk Capital Management 1817.1 A Review of Value-at-Risk and Other Measures 1817.2 Capital Aggregation and Allocation 1857.2.1 Aggregation: C-Convolution 1877.2.2 Allocation: Level Curves 1897.2.3 Allocation with Constraints 1917.3 Risk Measurement of Managed Portfolios 1937.3.1 Henriksson–Merton Model 1957.3.2 Semi-parametric Analysis of Managed Funds 2007.3.3 Market-neutral Investments 2017.4 Temporal Aggregation of Risk Measures 2027.4.1 The Square-root Formula 2037.4.2 Temporal Aggregation by C-convolution 2038 Frontier Issues 2078.1 Levy Copulas 2078.2 Pareto Copulas 2108.3 Semi-martingale Copulas 212A Elements of Probability 215A.1 Elements of Measure Theory 215A.2 Integration 216A.2.1 Expected Values and Moments 217A.3 The Moment-generating Function or Laplace Transform 218A.4 The Characteristic Function 219A.5 Relevant Probability Distributions 219A.6 Random Vectors and Multivariate Distributions 224A.6.1 The Multivariate Normal Distribution 225A.7 Infinite Divisibility 226A.8 Convergence of Sequences of Random Variables 228A.81 The Strong Law of Large Numbers 229A.9 The Radon–Nikodym Derivative 229A.10 Conditional Expectation 229B Elements of Stochastic Processes Theory 231B.1 Stochastic Processes 231B.1.1 Filtrations 231B.1.2 Stopping Times 232B.2 Martingales 233B.3 Markov Processes 234B.4 L´evy Processes 237B.4.1 Subordinators 240B.5 Semi-martingales 240References 245Extra Reading 251Index 259