Simulating Copulas: Stochastic Models, Sampling Algorithms, and Applications PDF
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SIMULATING COPULAS Stochastic Models, Sampling Algorithms and Applications p842.9781848168749-tp.indd 1 11/21/11 10:01 AM Series in Quantitative Finance ISSN: 1756-1604 Series Editor: Ralf Korn (University of Kaiserslautern, Germany) Editorial Members: Tang Shanjian (Fudan University, China) Kwok Yue Kuen (Hong Kong University of Science and Technology, China) Published Vol. 1 An Introduction to Computational Finance by Ömür U—ur Vol. 2 Advanced Asset Pricing Theory by Chenghu Ma Vol. 3 Option Pricing in Incomplete Markets: Modeling Based on Geometric Lévy Processes and Minimal Entropy Martingale Measures by Yoshio Miyahara Vol. 4 Simulating Copulas: Stochastic Models, Sampling Algorithms, and Applications by Jan-Frederik Mai and Matthias Scherer Catherine - Simulating Copulas.pmd 1 6/4/2012, 7:43 PM Series in Quantitative Finance – Vol. 4 SIMULATING COPULAS Stochastic Models, Sampling Algorithms and Applications Jan-Frederik Mai Assenagon Credit Management GmbH, Germany Matthias Scherer Techn ische Universität München, Germany Imperial College Press ICP p842.9781848168749-tp.indd 2 11/21/11 10:01 AM Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Series in Quantitative Finance — Vol. 4 SIMULATING COPULAS Stochastic Models, Sampling Algorithms, and Applications Copyright © 2012 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-1-84816-874-9 ISBN-10 1-84816-874-8 Printed in Singapore. Catherine - Simulating Copulas.pmd 2 6/4/2012, 7:43 PM March5,2012 14:59 WorldScientificBook-9inx6in Main˙MaiScherer˙SimulatingCopulas˙resub3 To our amazing wives, Jasna and Leni. And to Paul Jonah, who has just seen the light of day. v TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk March5,2012 14:59 WorldScientificBook-9inx6in Main˙MaiScherer˙SimulatingCopulas˙resub3 Preface The joint treatment of d 2 random variables requires vector-valued ≥ stochastic models. In the financial industry, multivariate models are ap- pliedto,e.g.,assetallocationproblems(portfoliooptimization),thepricing of basket options, risk management, and the modeling of credit portfolios. In particular, the development during the past years highlighted that the financial industry is in urgent need of realistic and viable models in large dimensions. Other fields of application for multivariate stochastic mod- els include geostatistics, hydrology, insurance mathematics, medicine, and reliability theory. Besides specifying the univariate marginals, for multivariate distribu- tions it is additionally required to appropriately define the dependence structureamongthemodeledobjects. Inmostapplications,aportfolioper- spective is significantly more demanding compared to modeling univariate marginals. One consequence is that analytical solutions for the aforemen- tioned applications can typically be derived under restrictive assumptions only. An increasingly popular alternative to accepting unrealistic simplifi- cations is to solve the model in question by Monte Carlo simulation. This allows for very general models but requires efficient simulation schemes for multivariate distributions. This book aims at providing a toolbox for the simulation of random vectors with a considerable spectrum of dependence structures. vii March5,2012 14:59 WorldScientificBook-9inx6in Main˙MaiScherer˙SimulatingCopulas˙resub3 viii Simulating Copulas: Stochastic Models, Sampling Algorithms, and Applications Why Sampling Copulas? Thisbookfocusesonsamplingcopulas, i.e.distributionfunctionson[0,1]d with uniform univariate marginals. On the first view, this standardiza- tion to univariate margins seems to be a rather artificial assumption. The justification for considering copulas instead of more general multivariate distribution functions is provided by Sklar’s seminal decomposition (see Sklar (1959) and Section 1.1.2). Heuristically speaking, Sklar’s theorem allows us to decompose any d-dimensional multivariate distribution func- tion F into its univariate marginsF ,...,F and the dependence structure 1 d among them. The latter is described by the copula behind the model, de- noted C. More precisely, we have F(x ,...,x ) = C(F (x ),...,F (x )) 1 d 1 1 d d for(x ,...,x ) Rd. Theconverseimplicationalsoholds,i.e.couplinguni- 1 d ∈ variate margins with some copula yields a multivariate distribution. This observation is especially convenient for the specification of a multivariate model, since a separate treatment of the dependence structure and uni- variate margins is usually easier compared to specifying the multivariate distribution in one step. Sklar’s decomposition also applies to sampling applications. Assume that we want to simulate from a multivariate distribution function F with univariate marginal distribution functions F ,...,F and copula C. Given 1 d a sampling scheme for the copula C, the following algorithm generates a sample from the distribution F by applying the generalized inverses F1−1,...,Fd−1 (see Lemma 1.4) to the sample of the copula. Algorithm 0.1 (Sampling Multivariate Distributions) Let F(x ,...,x )=C(F (x ),...,F (x )) be a d-dimensional distribution 1 d 1 1 d d function. Let sample C() be a function that returns a sample from C. Sampling F is then possible via the following scheme: FUNCTION sample F() Set (U ,...,U ):=sample C() 1 d RETURN F1−1(U1),...,Fd−1(Ud) (cid:0) (cid:1) April19,2012 12:15 WorldScientificBook-9inx6in 01-Main Preface ix Why Another Book on Copulas? Our main motivation for writing this book was to summarize the fast- growing literature on simulation algorithms for copulas. Several results on new sampling techniques for classical copulas, e.g. the Archimedean and Marshall–Olkin families, have lately been published. Moreover, new fam- ilies and construction principles have been discovered; an example is the pair-copula construction. At the same time, the financial industry has be- come aware that copula models (beyond a Gaussian dependence structure) are required to realistically model various aspects of quantitative finance. This book takes account of this fact by providing a comprehensive toolbox for financial engineering, and, of course, for other applications as well. All algorithms are described in pseudo-code. Thus, they can easily be imple- mentedintheuser’spreferredprogramminglanguage. Moreover,weaimat being comprehensive with respect to sampling schemes for univariate ran- dom variables as well as with respect to the use of Monte Carlo sampling engines in general. We purposely included sampling schemes for very basic copulas, even though this might not be required for an expert in the field. Anotherintentionistoprovideanelementaryintroductiontocopulasfrom the perspective of probabilistic representations. Hence, an experienced re- searcher might skip some parts of the book. But someone who is new to the field of copulas can use the book as a stand-alone textbook. The book, however, does not treat statistical estimation of dependence models. Especially for sampling applications, the dimension of the copula plays a crucial role. To give an example, the original probabilistic model behind the so-called d-dimensional Marshall–Olkin copula is based on 2d 1 ran- − dom variables, i.e. the dimension d enters exponentially. Hence, this book explicitly focuses on the d-dimensional case and discusses the efficiency of the provided algorithms with respect to their dimension. Especially in the fieldofportfoliocreditriskmodeling,therearesomeapplicationsrequiring high-dimensional models with d=125 or even more. Copulascanbeinvestigatedfromtwo(notnecessarilydisjoint)perspec- tives: (1)analytically,i.e.viewingthemasd-dimensionalfunctions,and(2) probabilistically,i.e.viewingthemasthedependencestructurebehindsome random vector. Both perspectives have their distinct advantages. (1) The analytical perspective aims at deriving statements about cop- ulas from their functional form. This is especially successful for analytically tractable families. In this case, it is often possible
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