Introduction to Stochastic Calculus for Finance: A New Didactic Approach ” by Di

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Brief Sketch of Lebesgue’s Integral . . . . . . . . . . . . . . . . . . . 3
1.2 Convergence Concepts for Random Variables . . . . . . . . . . 7
1.3 The Lebesgue-Stieltjes Integral . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Introduction to Itˆo-Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Stochastic Calculus vs. Classical Calculus . . . . . . . . . . . . . 15
2.2 Quadratic Variation and 1-dimensional Itˆo-Formula . . . . 18
2.3 Covariation and Multidimensional Itˆo-Formula . . . . . . . . . 26
2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5 First Application to Financial Markets . . . . . . . . . . . . . . . . 33
2.6 Stopping Times and Local Martingales . . . . . . . . . . . . . . . . 36
2.7 Local Martingales and Semimartingales . . . . . . . . . . . . . . . 44
2.8 Itˆo’s Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . . 49
2.9 Application to Option Pricing . . . . . . . . . . . . . . . . . . . . . . . 50
3 The Girsanov Transformation . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1 Heuristic Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 The General Girsanov Transformation . . . . . . . . . . . . . . . . 58
3.3 Application to Brownian Motion . . . . . . . . . . . . . . . . . . . . . 63
4 Application to Financial Economics . . . . . . . . . . . . . . . . . . 67
4.1 The Market Price of Risk and Risk-neutral Valuation . . . 68
4.2 The Fundamental Pricing Rule . . . . . . . . . . . . . . . . . . . . . . . 73
4.3 Connection with the PDE-Approach
(Feynman-Kac Formula) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
X Contents
4.4 Currency Options and Siegel-Paradox . . . . . . . . . . . . . . . . . 78
4.5 Change of Numeraire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.6 Solution of the Siegel-Paradox . . . . . . . . . . . . . . . . . . . . . . . 84
4.7 Admissible Strategies and Arbitrage-free Pricing . . . . . . . 86
4.8 The “Forward Measure” . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.9 Option Pricing Under Stochastic Interest Rates . . . . . . . . 92
5 Term Structure Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.1 Different Descriptions of the Term Structure of Interest
Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.2 Stochastics of the Term Structure . . . . . . . . . . . . . . . . . . . . 99
5.3 The HJM-Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.5 The “LIBOR Market” Model . . . . . . . . . . . . . . . . . . . . . . . . 107
5.6 Caps, Floors and Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6 Why Do We Need Itˆo-Calculus in Finance? . . . . . . . . . . 113
6.1 The Buy-Sell-Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.2 Local Times and Generalized Itˆo Formula . . . . . . . . . . . . . 115
6.3 Solution of the Buy-Sell-Paradox . . . . . . . . . . . . . . . . . . . . . 120
6.4 Arrow-Debreu Prices in Finance. . . . . . . . . . . . . . . . . . . . . . 121
6.5 The Time Value of an Option as Expected Local Time . . 123
7 Appendix: Itˆo Calculus Without Probabilities . . . . . . . . 125
 
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关键词：introduction Stochastic troduction Calculus Stochast Finance introduction Calculus Approach Dieter