| Preface 1 Brownian Motion 1.1 Origins 1.2 Brownian Motion Specification 1.3 Use of Brownian Motion in Stock Price Dynamics 1.4 Construction of Brownian Motion from a Symmetric Random Walk 1.5 Covariance of Brownian Motion 1.6 Correlated Brownian Motions 1.7 Successive Brownian Motion Increments 1.7.1 Numerical Illustration 1.8 Features of a Brownian Motion Path 1.8.1 Simulation of Brownian Motion Paths 1.8.2 Slope of Path 1.8.3 Non-Differentiability of Brownian Motion Path 1.8.4 Measuring Variability 1.9 Exercises 1.10 Summary 2 Martingales 2.1 Simple Example 2.2 Filtration 2.3 Conditional Expectation 2.3.1 General Properties 2.4 Martingale Description 2.4.1 Martingale Construction by Conditioning 2.5 Martingale Analysis Steps 2.6 Examples of Martingale Analysis 2.6.1 Sum of Independent Trials 2.6.2 Square of Sum of Independent Trials 2.6.3 Product of Independent Identical Trials 2.6.4 Random Process B(t) 2.6.5 Random Process exp[B(t) 2.6.6 Frequently Used Expressions 2.7 Process of Independent Increments 2.8 Exercises 2.9 Summary 3 Ito Stochastic Integral 3.1 How a Stochastic Integral Arises 3.2 Stochastic Integral for Non-Random Step-Functions 3.3 Stochastic Integral for Non-Anticipating Random Step-Functions 3.4 Extension to Non-Anticipating General Random Integrands 3.5 Properties of an It6 Stochastic Integral 3.6 Significance of Integrand Position 3.7 It6 integral of Non-Random Integrand 3.8 Area under a Brownian Motion Path 3.9 Exercises 3.10 Summary 3.11 A Tribute to Kiyosi Ito Acknowledgment 4 Ito Calculus 4.1 Stochastic Differential Notation 4.2 Taylor Expansion in Ordinary Calculus 4.3 Ito's Formula as a Set of Rules 4.4 Illustrations of Itp's Formula 4.4,1 Frequent Expressions for Functions of Two Processes 4.4.2 Function of Brownian Motion fiB(t)] 4.4.3 Function of Time and Brownian Motion f[t, B(t)] …… 5 Stochastic Differential Equations 6 Option Valuation 7 Change of Probability 8 Numeraire A Annex A:Computations with Brownian Motion B aNNEX B:Ordinary Integration C Annex C:Brownian Motion Variability D Annex D:Norms E Annex E:Convergence Concepts Answers to Exercises References Index |
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