Ubbo Wiersema

Brownian Motion Calculus (eBook, PDF)

• Format: PDF

Produktbeschreibung

BROWNIAN MOTION CALCULUS Brownian Motion Calculus presents the basics of Stochastic Calculus with a focus on the valuation of financial derivatives. It is intended as an accessible introduction to the technical literature. The sequence of chapters starts with a description of Brownian motion, the random process which serves as the basic driver of the irregular behaviour of financial quantities. That exposition is based on the easily understood discrete random walk. Thereafter the gains from trading in a random environment are formulated in a discrete-time setting. The continuous-time equivalent requires a new concept, the Ito stochastic integral. Its construction is explained step by step, using the so-called norm of a random process (its magnitude), of which a motivated exposition is given in an Annex. The next topic is Ito's formula for evaluating stochastic integrals; it is the random process counter part of the well known Taylor formula for functions in ordinary calculus. Many examples are given. These ingredients are then used to formulate some well established models for the evolution of stock prices and interest rates, so-called stochastic differential equations, together with their solution methods. Once all that is in place, two methodologies for option valuation are presented. One uses the concept of a change of probability and the Girsanov transformation, which is at the core of financial mathematics. As this technique is often perceived as a magic trick, particular care has been taken to make the explanation elementary and to show numerous applications. The final chapter discusses how computations can be made more convenient by a suitable choice of the so-called numeraire. A clear distinction has been made between the mathematics that is convenient for a first introduction, and the more rigorous underpinnings which are best studied from the selected technical references. The inclusion of fully worked out exercises makes the book attractive for self study. Standard probability theory and ordinary calculus are the prerequisites. Summary slides for revision and teaching can be found on the book website www.wiley.com/go/brownianmotioncalculus.

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Produktdetails

• Verlag: John Wiley & Sons
• Seitenzahl: 330
• Erscheinungstermin: 14. September 2008
• Englisch
• ISBN-13: 9780470021712
• Artikelnr.: 37289843

Autorenporträt

UBBO WIERSEMA was educated in Applied Mathematics at Delft, in Operations Research at Berkeley, and in Financial Economics and Financial Mathematics at the London School of Economics. He joined The Business School for Financial Markets (the ICMA Centre) at the University of Reading, UK, in 1997, to develop and teach its curriculum in Quantitative Finance. Prior to that, he was a derivatives mathematician at the merchant bank Robert Fleming in the City of London. His earlier career was in Operations Research in the US and Europe.

Inhaltsangabe

1. Brownian Motion Origins Notion of a Random Process Brownian Motion Stock
Price Dynamics Construction of Brownian Motion from a Discrete Symmetric
Random Walk Features of Brownian Motion Paths Computations with Brownian
motion References Examples 2. Martingales Introduction Filtration
Conditional Expectation Martingale Martingale examples References Examples
3. Ito Stochastic Integration How a Stochastic integral arises in stock
trading Construction of Ito Stochastic integral for random step functions
Extension to general random integrands Summary of properties of an Ito
stochastic integral Reference Examples 4. Ito Calculus Stochastic
differential notation Taylor's expansion in ordinary calculus Ito's formula
as a set of rules Illustrations of Ito's formula Justification of Ito's
formula References Examples 5. Stochastic differential equations Structure
of a stochastic differental equation Stocastic differntail equations
arising in finance Finding a closed form solution Checking the solution of
an sde General method for solving sde's References Examples 6. Risk-neutral
probability Risk-neutral valuation - the basic concept Risk-neutral
probability construction in discrete one period binomial framework
Risk-neutral probability construction in the continuous framework
Girsanov's theorem Radon-Nikodym derivative Numerical Illustration
Motivation for Girsanov's theorem Summary References 7. Feynman-Kac
Representation Stochastic Representation Derivation of simple Feynman-Kac
formula Application to Black Scholes pde Generalisations Solution by
Simulation References Annexes Computations with Brownian motion Riemann
Integration Brownian Motion Variability Norms Einstrin's Model of Brownian
Motion