Applied Quantitative Finance

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Applied Quantitative Finance

Wolfgang Hardle Torsten Kleinow Gerhard Stahl


In cooperation with


Gokhan Aydinli, Oliver Jim Blaskowitz, Song Xi Chen, Matthias Fengler, Jiirgen Franke, Christoph Frisch, Helmut Herwartz, Harriet Holzberger, Steffi Hose, Stefan Huschens, Kim Huynh, Stefan R. Jaschke, Yuze Jiang Pierre Kervella, Rudiger Kiesel, Germar Knochlein,


Sven Knoth, Jens Lussem, Danilo Mercurio,


Marlene Muller, Jorn Rank, Peter Schmidt,


Rainer Schulz, Jurgen 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


1 Approximating Value at Risk in Conditional Gaussian Models    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


1.4    Fourier Inversion …………………….. 16


1.4.1    Error Analysis ………………….. 16


1.4.2    Tail Behavior…………………… 20


1.4.3    Inversion of the cdf minus the Gaussian Approximation 21


1.5    Variance Reduction Techniques in Monte-Carlo Simulation …    24


1.5.1    Monte-Carlo Sampling Method………….. 24


1.5.2    Partial Monte-Carlo with Importance Sampling….. 28


1.5.3    XploRe Examples ………………… 30


2    Applications of Copulas for the Calculation of Value-at-Risk    35


JOrn Rank and Thomas Siegl


2.1    Copulas…………………………. 36


2.1.1    Definition…………………….. 36


2.1.2    Sklar’s Theorem…………………. 37

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