| 1 General Probability Theory 1.1 Infinite Probability Spaces 1.2 Random Variables and Distributions 1.3 Expectations 1.4 Convergence of Integrals 1.5 Computation of Expectations 1.6 Change of Measure 1.7 Summary 1.8 Notes 1.9 Exercises 2 Information and Conditioning 2.1 Information and o-algebras 2.2 Independence 2.3 General Conditional Expectations 2.4 Summary 2.5 Notes 2.6 Exercises 3 Brownian Motion 3.1 Introduction 3.2 Scaled Random Walks 3.3 Brownian Motion 3.4 Quadratic Variation 3.5 Markov Property 3.6 First Passage Time Distribution 3.7 Reflection Principle 3.8 Summary 3.9 Notes 3.10 Exercises 4 Stochastic Calculus 4.1 Introduction 4.2 Ito's Integral for Simple Integrands 4.3 Ito's Integral for General Integrands 4.4 Ito-Doeblin Formula 4.5 Black-Scholes-Merton Equation 4.6 Multivariable Stochastic Calculus 4.7 Brownian Bridge …… 5 Risk-Neutral Pricing 6 Connections with Partial Differential Equations 7 Exotic Options 8 American Derivative Securities 9 Change of Numeraire 10 Term-Structure Models 11 Introduction to Jump Processes A Advanced Topics in Probability Theory B Existence of Conditional Expectations C Completion of the Proof of the Second Fundamental Theorem of Asset Pricing References Index |
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