Introduction to Stochastic Calculus for Finance (eBook)

Preface 6
Contents 8
Introduction 10
1 Preliminaries 12
1.1 Brief Sketch of Lebesgue’s Integral 12
1.2 Convergence Concepts for Random Variables 16
1.3 The Lebesgue-Stieltjes Integral 19
1.4 Exercises 22
2 Introduction to Ito-Calculus 24
2.1 Stochastic Calculus vs. Classical Calculus 24
2.2 Quadratic Variation and 1-dimensional Itˆ o-Formula 27
2.3 Covariation and Multidimensional Itˆ o-Formula 35
2.4 Examples 40
2.5 First Application to Financial Markets 42
2.6 Stopping Times and Local Martingales 45
2.7 Local Martingales and Semimartingales 53
2.8 Ito’s Representation Theorem 58
2.9 Application to Option Pricing 59
3 The Girsanov Transformation 63
3.1 Heuristic Introduction 63
3.2 The General Girsanov Transformation 66
3.3 Application to Brownian Motion 71
4 Application to Financial Economics 74
4.1 The Market Price of Risk and Risk-neutral Valuation 75
4.2 The Fundamental Pricing Rule 80
4.3 Connection with the PDE-Approach ( Feynman- Kac Formula) 83
4.4 Currency Options and Siegel-Paradox 85
4.5 Change of Numeraire 86
4.6 Solution of the Siegel-Paradox 91
4.7 Admissible Strategies and Arbitrage-free Pricing 93
4.8 The “Forward Measure” 96
4.9 Option Pricing Under Stochastic Interest Rates 99
5 Term Structure Models 102
5.1 Different Descriptions of the Term Structure of Interest Rates 103
5.2 Stochastics of the Term Structure 106
5.3 The HJM-Model 109
5.4 Examples 112
5.5 The “LIBOR Market” Model 114
5.6 Caps, Floors and Swaps 118
6 Why Do We Need Ito-Calculus in Finance? 120
6.1 The Buy-Sell-Paradox 121
6.2 Local Times and Generalized Ito Formula 122
6.3 Solution of the Buy-Sell-Paradox 127
6.4 Arrow-Debreu Prices in Finance 128
6.5 The Time Value of an Option as Expected Local Time 130
7 Appendix: Ito Calculus Without Probabilities 132
References 141