Stochastic Calculus of Variations in Mathematical Finance

1. Auflage. Softcover version of original hardcover Auflage 2006
ISBN: 978-3-642-07783-8
Verlag: Springer

Buch, Englisch, 142 Seiten, Paperback, Format (B × H): 155 mm x 235 mm, Gewicht: 248 g

Reihe: Springer Finance

Malliavin calculus provides an infinite-dimensional differential calculus in the context of continuous paths stochastic processes. The calculus includes formulae of integration by parts and Sobolev spaces of differentiable functions defined on a probability space. This new book, demonstrating the relevance of Malliavin calculus for Mathematical Finance, starts with an exposition from scratch of this theory. Greeks (price sensitivities) are reinterpreted in terms of Malliavin calculus. Integration by parts formulae provide stable Monte Carlo schemes for numerical valuation of digital options. Finite-dimensional projections of infinite-dimensional Sobolev spaces lead to Monte Carlo computations of conditional expectations useful for computing American options. The discretization error of the Euler scheme for a stochastic differential equation is expressed as a generalized Watanabe distribution on the Wiener space. Insider information is expressed as an infinite-dimensional drift. The last chapter gives an introduction to the same objects in the context of jump processes where incomplete markets appear.

Zielgruppe

Research

Autoren/Hrsg.

Thalmaier, Anton

Malliavin, Paul

Fachgebiete

• Wirtschaftswissenschaften Volkswirtschaftslehre Volkswirtschaftslehre Allgemein Ökonometrie
• Wirtschaftswissenschaften Betriebswirtschaft Wirtschaftsmathematik und -statistik
• Wirtschaftswissenschaften Finanzsektor & Finanzdienstleistungen Bankwirtschaft
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle

Inhaltsverzeichnis

Gaussian Stochastic Calculus of Variations.- Computation of Greeks and Integration by Parts Formulae.- Market Equilibrium and Price-Volatility Feedback Rate.- Multivariate Conditioning and Regularity of Law.- Non-Elliptic Markets and Instability in HJM Models.- Insider Trading.- Asymptotic Expansion and Weak Convergence.- Stochastic Calculus of Variations for Markets with Jumps.