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Applied Quantitative Finance Wolfgang H¨ ardle Torsten Kleinow Gerhard Stahl In cooperation with G¨ okhan Aydınlı, Oliver Jim Blaskowitz, Song Xi Chen, Matthias Fengler, J¨ urgen Franke, Christoph Frisch, Helmut Herwartz, Harriet Holzberger, Steffi H¨ ose, Stefan Huschens, Kim Huynh, Stefan R. Jaschke, Yuze Jiang Pierre Kervella, R¨ udiger Kiesel, Germar Kn¨ ochlein, Sven Knoth, Jens L¨ ussem, Danilo Mercurio, Marlene M¨ uller, J¨ orn Rank, Peter Schmidt, Rainer Schulz, J¨ urgen Schumacher, Thomas Siegl, Robert Wania, Axel Werwatz, Jun Zheng June 20, 2002 Contents Preface xv Contributors xix Frequently Used Notation xxi I Value at Risk 1 Approximating Value at Risk in Conditional Gaussian Models 1 3 Stefan R. Jaschke and Yuze Jiang 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 The Practical Need . . . . . . . . . . . . . . . . . . . . . 3 1.1.2 Statistical Modeling for VaR . . . . . . . . . . . . . . . 4 1.1.3 VaR Approximations . . . . . . . . . . . . . . . . . . . . 6 1.1.4 Pros and Cons of Delta-Gamma Approximations . . . . 7 1.2 General Properties of Delta-Gamma-Normal Models . . . . . . 8 1.3 Cornish-Fisher Approximations . . . . . . . . . . . . . . . . . . 12 1.3.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Fourier Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4 iv Contents 1.5 1.4.1 Error Analysis . . . . . . . . . . . . . .