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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 . . . . . . . . . . . . . .