Financial engineering with copulas explained

The modeling of dependence structures (or copulas) is undoubtedly one of the key challenges for modern financial engineering. First applied to credit-risk modeling, 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. However, by their very nature, copulas are complex and their applications are often misunderstood. Incorrectly applied, copulas can be hugely detrimental to a model or algorithm. Financial Engineering with Copulas Explained is a reader-friendly, yet rigorous introduction to the state-of-the-art regarding the theory of copulas, their simulation and estimation, and their use in financial applications. Starting with an introduction to the basic notions, such as required definitions and dependence measures, the book looks at statistical issues comprising parameter estimation and stochastic simulation. The book will show, from a financial engineering perspective, how copula theory can be applied in the context of portfolio credit-risk modeling, and how it can help to derive model-free bounds for relevant risk measures. The book will cover numerous different market applications of copulas, and enable readers to construct stable, high-dimensional models for asset pricing and risk modeling. Written to appeal to quantitatively minded practitioners across the trading floors and in risk management, academics and students, Financial Engineering with Copulas Explained will be a valuable, accessible and practical guide to this complex topic.

[1]  David X. Li.  On Default Correlation: A Copula Function Approach , 1999 .

[2]  J. Segers.  Non-Parametric Inference for Bivariate Extreme-Value Copulas , 2004 .

[3]  M. Fréchet.  Sur les tableaux de correlation dont les marges sont donnees , 1951 .

[4]  Harry Joe,et al.  Parametric families of multivariate distributions with given margins , 1993 .

[5]  Dorota Kurowicka,et al.  Dependence Modeling: Vine Copula Handbook , 2010 .

[6]  Bruno Rémillard,et al.  Copula-Based Semiparametric Models for Multivariate Time Series , 2011, J. Multivar. Anal..

[7]  Uwe Schmock,et al.  On the Asymptotic Behaviour of the Estimator of Kendall's Tau , 2010 .

[8]  H. Thorisson.  Coupling, stationarity, and regeneration , 2000 .

[9]  Jingping Yang,et al.  Saddlepoint approximation method for pricing CDOs , 2006 .

[10]  Agostino Capponi,et al.  Arbitrage‐Free Bilateral Counterparty Risk Valuation Under Collateralization and Application to Credit Default Swaps , 2014 .

[11]  S. Kotz,et al.  Symmetric Multivariate and Related Distributions , 1989 .

[12]  Arthur Charpentier,et al.  Tails of multivariate Archimedean copulas , 2009, J. Multivar. Anal..

[13]  Eckhard Liebscher,et al.  Construction of asymmetric multivariate copulas , 2008 .

[14]  T. Bielecki,et al.  Credit Risk: Modeling, Valuation And Hedging , 2004 .

[15]  Christian Genest,et al.  A nonparametric estimation procedure for bivariate extreme value copulas , 1997 .

[16]  Cornelis W. Oosterlee,et al.  Higher order saddlepoint approximations in the Vasicek portfolio credit loss model , 2006 .

[17]  P. Carr,et al.  Stock Options and Credit Default Swaps: A Joint Framework for Valuation and Estimation , 2006 .

[18]  W. Kruskal.  Ordinal Measures of Association , 1958 .

[19]  R. Williamson.  Multiply monotone functions and their Laplace transforms , 1956 .

[20]  P. Embrechts,et al.  Risk Management: Correlation and Dependence in Risk Management: Properties and Pitfalls , 2002 .

[21]  A. Sklar,et al.  Random variables, distribution functions, and copulas---a personal look backward and forward , 1996 .

[22]  Damiano Brigo,et al.  Counterparty Credit Risk, Collateral and Funding: With Pricing Cases For All Asset Classes , 2013 .

[23]  Liang Peng,et al.  Nonparametric estimation of the dependence function for a multivariate extreme value distribution , 2008 .

[24]  L. Devroye.  Non-Uniform Random Variate Generation , 1986 .

[25]  V. Linetsky.  PRICING EQUITY DERIVATIVES SUBJECT TO BANKRUPTCY , 2006 .

[26]  H. Joe.  Multivariate models and dependence concepts , 1998 .

[27]  Paul Embrechts,et al.  The AEP algorithm for the fast computation of the distribution of the sum of dependent random variables , 2011, 1106.2920.

[28]  Andrew Ang,et al.  Asymmetric Correlations of Equity Portfolios , 2001 .

[29]  Friedrich Schmid,et al.  Copula-Based Measures of Multivariate Association , 2010 .

[30]  Improved estimation for a model arising in reliability and competing risks , 1991 .

[31]  Wlodzimierz Ogryczak,et al.  Dual Stochastic Dominance and Related Mean-Risk Models , 2002, SIAM J. Optim..

[32]  L. Adler.  A Modification of Kendall's Tau for the Case of Arbitrary Ties in Both Rankings , 1957 .

[33]  Michael B. Gordy.  Saddlepoint approximation of CreditRisk , 2002 .

[34]  Stanislav Volgushev,et al.  Empirical and sequential empirical copula processes under serial dependence , 2011, J. Multivar. Anal..

[35]  Ludger Rüschendorf,et al.  Solution of a statistical optimization problem by rearrangement methods , 1983 .

[36]  S. Bernstein,et al.  Sur les fonctions absolument monotones , 1929 .

[37]  C. Genest,et al.  ESTIMATORS BASED ON KENDALL'S TAU IN MULTIVARIATE COPULA MODELS , 2011 .

[38]  Christian Genest,et al.  Multivariate Archimax copulas , 2014, J. Multivar. Anal..

[39]  G. O'Brien.  The Comparison Method for Stochastic Processes , 1975 .

[40]  Holger Dette,et al.  Minimum distance estimators of the Pickands dependence function and related tests of multivariate extreme-value dependence , 2013 .

[41]  Jan-Frederik Mai,et al.  A Tractable Multivariate Default Model Based on a Stochastic Time-Change , 2009 .

[42]  Haijun Li,et al.  Orthant tail dependence of multivariate extreme value distributions , 2009, J. Multivar. Anal..

[43]  Jan-Frederik Mai,et al.  Simulating from the Copula that Generates the Maximal Probability for a Joint Default Under Given (Inhomogeneous) Marginals , 2014 .

[44]  R. Nelsen.  An Introduction to Copulas , 1998 .

[45]  Xiaohong Chen,et al.  Estimation and model selection of semiparametric copula-based multivariate dynamic models under copula misspecification , 2006 .

[46]  C. Genest,et al.  Bivariate Distributions with Given Extreme Value Attractor , 2000 .

[47]  Holger Dette,et al.  A test for Archimedeanity in bivariate copula models , 2011, J. Multivar. Anal..

[48]  C. Genest,et al.  A semiparametric estimation procedure of dependence parameters in multivariate families of distributions , 1995 .

[49]  Paul Ressel.  Homogeneous distributions - And a spectral representation of classical mean values and stable tail dependence functions , 2013, J. Multivar. Anal..

[50]  Paul Deheuvels,et al.  On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions , 1991 .

[51]  D. J. G. Farlie,et al.  The performance of some correlation coefficients for a general bivariate distribution , 1960 .

[52]  Jan-Frederik Mai,et al.  Bivariate extreme-value copulas with discrete Pickands dependence measure , 2011 .

[53]  Jan-Frederik Mai,et al.  Constructing hierarchical Archimedean copulas with Lévy subordinators , 2010, J. Multivar. Anal..

[54]  Ludger Rüschendorf,et al.  Computation of sharp bounds on the distribution of a function of dependent risks , 2012, J. Comput. Appl. Math..

[55]  Ludger Rüschendorf,et al.  Mathematical Risk Analysis: Dependence, Risk Bounds, Optimal Allocations and Portfolios , 2013 .

[56]  J. Segers,et al.  RANK-BASED INFERENCE FOR BIVARIATE EXTREME-VALUE COPULAS , 2007, 0707.4098.

[57]  Jan-Frederik Mai,et al.  Default models based on scale mixtures of Marshall-Olkin copulas: properties and applications , 2013 .

[58]  C. Genest,et al.  Statistical Inference Procedures for Bivariate Archimedean Copulas , 1993 .

[59]  P. J. Schonbucher.  Credit Derivatives Pricing Models , 2003 .

[60]  M. Sklar.  Fonctions de repartition a n dimensions et leurs marges , 1959 .

[61]  Paul Embrechts,et al.  Quantitative Risk Management , 2011, International Encyclopedia of Statistical Science.

[62]  Kay Giesecke,et al.  Correlated Default with Incomplete Information , 2002 .

[63]  Paul Embrechts,et al.  A note on generalized inverses , 2013, Math. Methods Oper. Res..

[64]  Ruodu Wang,et al.  Bounds for the sum of dependent risks and worst Value-at-Risk with monotone marginal densities , 2013, Finance Stochastics.

[65]  W. Hoeffding.  The strong law of large numbers for u-statistics. , 1961 .

[66]  D. Brigo,et al.  Interest Rate Models , 2001 .

[67]  N. Higham.  Computing the nearest correlation matrix—a problem from finance , 2002 .

[68]  Dilip B. Madan,et al.  Local Volatility Enhanced by a Jump to Default , 2010, SIAM J. Financial Math..