E-Book, Englisch, Band 579, 138 Seiten
Sondermann Introduction to Stochastic Calculus for Finance
2006
ISBN: 978-3-540-34837-5
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark
A New Didactic Approach
E-Book, Englisch, Band 579, 138 Seiten
Reihe: Lecture Notes in Economics and Mathematical Systems
ISBN: 978-3-540-34837-5
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark
Although there are many textbooks on stochastic calculus applied to finance, this volume earns its place with a pedagogical approach. The text presents a quick (but by no means 'dirty') road to the tools required for advanced finance in continuous time, including option pricing by martingale methods, term structure models in a HJM-framework and the Libor market model. The reader should be familiar with elementary real analysis and basic probability theory.
Autoren/Hrsg.
Weitere Infos & Material
1;Preface;6
2;Contents;8
3;Introduction;10
4;1 Preliminaries;12
4.1;1.1 Brief Sketch of Lebesgue’s Integral;12
4.2;1.2 Convergence Concepts for Random Variables;16
4.3;1.3 The Lebesgue-Stieltjes Integral;19
4.4;1.4 Exercises;22
5;2 Introduction to Ito-Calculus;24
5.1;2.1 Stochastic Calculus vs. Classical Calculus;24
5.2;2.2 Quadratic Variation and 1-dimensional Itˆ o-Formula;27
5.3;2.3 Covariation and Multidimensional Itˆ o-Formula;35
5.4;2.4 Examples;40
5.5;2.5 First Application to Financial Markets;42
5.6;2.6 Stopping Times and Local Martingales;45
5.7;2.7 Local Martingales and Semimartingales;53
5.8;2.8 Ito’s Representation Theorem;58
5.9;2.9 Application to Option Pricing;59
6;3 The Girsanov Transformation;63
6.1;3.1 Heuristic Introduction;63
6.2;3.2 The General Girsanov Transformation;66
6.3;3.3 Application to Brownian Motion;71
7;4 Application to Financial Economics;74
7.1;4.1 The Market Price of Risk and Risk-neutral Valuation;75
7.2;4.2 The Fundamental Pricing Rule;80
7.3;4.3 Connection with the PDE-Approach ( Feynman- Kac Formula);83
7.4;4.4 Currency Options and Siegel-Paradox;85
7.5;4.5 Change of Numeraire;86
7.6;4.6 Solution of the Siegel-Paradox;91
7.7;4.7 Admissible Strategies and Arbitrage-free Pricing;93
7.8;4.8 The “Forward Measure”;96
7.9;4.9 Option Pricing Under Stochastic Interest Rates;99
8;5 Term Structure Models;102
8.1;5.1 Different Descriptions of the Term Structure of Interest Rates;103
8.2;5.2 Stochastics of the Term Structure;106
8.3;5.3 The HJM-Model;109
8.4;5.4 Examples;112
8.5;5.5 The “LIBOR Market” Model;114
8.6;5.6 Caps, Floors and Swaps;118
9;6 Why Do We Need Ito-Calculus in Finance?;120
9.1;6.1 The Buy-Sell-Paradox;121
9.2;6.2 Local Times and Generalized Ito Formula;122
9.3;6.3 Solution of the Buy-Sell-Paradox;127
9.4;6.4 Arrow-Debreu Prices in Finance;128
9.5;6.5 The Time Value of an Option as Expected Local Time;130
10;7 Appendix: Ito Calculus Without Probabilities;132
11;References;141
