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Introduction to Stochastic Calculus for Finance A New Didactic Approach Contents 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 Ito-Calculus ..................................................................... 2.1 Stochastic Calculus vs. Classical Calculus .......................... 2.2 Quadratic Variation and 1-dimensional Ito-Formula ........... 2.3 Covariation and Multidimensional Ito-Formula.................... 2.4 Examples ............................................................................. 2.5 First Application to Financial Markets.................................. 2.6 Stopping Times and Local Martingales.................................. 2.7 Local Martingales and Semimartingales ............................... 2.8 Itô's Representation Theorem................................................ 2.9 Application to Option Pricing ................................................ 3 The Girsanov Transformation.................................................................... 3.1 Heuristic Introduction ......................................................... 3.2 The General Girsanov Transformation .................................. 3.3 Application to Brownian Motion ............................................ 4 Application to Financial Economics ....................................................... 4.1 The Market Price of Risk and Risk-neutral Valuation ........... 4.2 The Fundamental Pricing Rule.............................................. 4.3 Connection with the PDE-Approach (Feynrnan-Kac Formula) ..................................................... 15 15 18 26 31 33 36 44 49 50 55 55 58 63 67 68 73 76 4.4 Currency Options and Siegel-Paradox........................................ 4.5 Change of Numeraire ....................................................................... 4.6 Solution of the Siegel-Paradox ...................................................... 4.7 Admissible Strategies and Arbitrage-free Pricing ................... 4.8 The "Forward Measure" ................................................................... 4.9 Option Pricing Under Stochastic Interest Rates ..................... 78 79 84 86 89 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 Ito-Calculus in Finance? .................................................................. 113 6.1 The Buy-Sell-Paradox .................................................................... 114 6.2 Local Times and Generalized Ito 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: Ito Calculus Without Probabilities.................................. 125 References........................................................................................................ 135