Stochastic Calculus for Finance II

1 General Probability Theory
1.1 In.nite 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 s-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
3.2.1 Symmetric Random Walk
3.2.2 Increments of Symmetric Random Walk
3.2.3 Martingale Property for Symmetric Random Walk
3.2.4 Quadratic Variation of Symmetric Random Walk
3.2.5 Scaled Symmetric Random Walk
3.2.6 Limiting Distribution of Scaled Random Walk
3.2.7 Log-Normal Distribution as Limit of Binomial Model
3.3 Brownian Motion
3.3.1 Definition of Brownian Motion
3.3.2 Distribution of Brownian Motion
3.3.3 Filtration for Brownian Motion
3.3.4 Martingale Property for Brownian Motion
3.4 Quadratic Variation
3.4.1 First-Order Variation
3.4.2 Quadratic Variation
3.4.3 Volatility of Geometric Brownian Motion
3.5 Markov Property
3.6 First Passage Time Distribution
3.7 Re.ection Principle
3.7.1 Reflection Equality
3.7.2 First Passage Time Distribution
3.7.3 Distribution of Brownian Motion and Its Maximum
3.8 Summary
3.9 Notes
3.10 Exercises 4 Stochastic Calculus
4.1 Introduction
4.2 Itˆo's Integral for Simple Integrands
4.2.1 Construction of the Integral
4.2.2 Properties of the Integral
4.3 Itˆo's Integral for General Integrands
4.4 Itˆo-Doeblin Formula
4.4.1 Formula for Brownian Motion
4.4.2 Formula for Itˆo Processes
4.4.3 Examples
4.5 Black-Scholes-Merton Equation
4.5.1 Evolution of Portfolio Value
4.5.2 Evolution of Option Value
4.5.3 Equating the Evolutions
4.5.4 Solution to the Black-Scholes-Merton Equation
4.5.5 TheGreeks
4.5.6 Put-Call Parity
4.6 Multivariable Stochastic Calculus