| |
| |
| |
General Probability Theory | |
| |
| |
| |
In.nite Probability Spaces | |
| |
| |
| |
Random Variables and Distributions | |
| |
| |
| |
Expectations | |
| |
| |
| |
Convergence of Integrals | |
| |
| |
| |
Computation of Expectations | |
| |
| |
| |
Change of Measure | |
| |
| |
| |
Summary | |
| |
| |
| |
Notes | |
| |
| |
| |
Exercises | |
| |
| |
| |
Information and Conditioning | |
| |
| |
| |
Information and s-algebras | |
| |
| |
| |
Independence | |
| |
| |
| |
General Conditional Expectations | |
| |
| |
| |
Summary | |
| |
| |
| |
Notes | |
| |
| |
| |
Exercises | |
| |
| |
| |
Brownian Motion | |
| |
| |
| |
Introduction | |
| |
| |
| |
Scaled Random Walks | |
| |
| |
| |
Symmetric Random Walk | |
| |
| |
| |
Increments of Symmetric Random Walk | |
| |
| |
| |
Martingale Property for Symmetric Random Walk | |
| |
| |
| |
Quadratic Variation of Symmetric Random Walk | |
| |
| |
| |
Scaled Symmetric Random Walk | |
| |
| |
| |
Limiting Distribution of Scaled Random Walk | |
| |
| |
| |
Log-Normal Distribution as Limit of Binomial Model | |
| |
| |
| |
Brownian Motion | |
| |
| |
| |
Definition of Brownian Motion | |
| |
| |
| |
Distribution of Brownian Motion | |
| |
| |
| |
Filtration for Brownian Motion | |
| |
| |
| |
Martingale Property for Brownian Motion | |
| |
| |
| |
Quadratic Variation | |
| |
| |
| |
First-Order Variation | |
| |
| |
| |
Quadratic Variation | |
| |
| |
| |
Volatility of Geometric Brownian Motion | |
| |
| |
| |
Markov Property | |
| |
| |
| |
First Passage Time Distribution | |
| |
| |
| |
Re.ection Principle | |
| |
| |
| |
Reflection Equality | |
| |
| |
| |
First Passage Time Distribution | |
| |
| |
| |
Distribution of Brownian Motion and Its Maximum | |
| |
| |
| |
Summary | |
| |
| |
| |
Notes | |
| |
| |
| |
Exercises | |
| |
| |
| |
Stochastic Calculus | |
| |
| |
| |
Introduction | |
| |
| |
| |
Ito's Integral for Simple Integrands | |
| |
| |
| |
Construction of the Integral | |
| |
| |
| |
Properties of the Integral | |
| |
| |
| |
Ito's Integral for General Integrands | |
| |
| |
| |
Ito-Doeblin Formula | |
| |
| |
| |
Formula for Brownian Motion | |
| |
| |
| |
Formula for Ito Processes | |
| |
| |
| |
Examples | |
| |
| |
| |
Black-Scholes-Merton Equation | |
| |
| |
| |
Evolution of Portfolio Value | |
| |
| |
| |
Evolution of Option Value | |
| |
| |
| |
Equating the Evolutions | |
| |
| |
| |
Solution to the Black-Scholes-Merton Equation | |
| |
| |
| |
The Greeks | |
| |
| |
| |
Put-Call Parity | |
| |
| |
| |
Multivariable Stochastic Calculus | |
| |
| |
| |
Multiple Brownian Motions | |
| |
| |
| |
Ito-Doeblin Formula for Multiple Processes | |
| |
| |
| |
Recognizing a Brownian Motion | |
| |
| |
| |
Brownian Bridge | |
| |
| |
| |
Gaussian Processes | |
| |
| |
| |
Brownian Bridge as a Gaussian Process | |
| |
| |
| |
Brownian Bridge as a Scaled Stochastic Integral | |
| |
| |
| |
Multidimensional Distribution of Brownian Bridge | |
| |
| |
| |
Brownian Bridge as Conditioned Brownian Motion | |
| |
| |
| |
Summary | |
| |
| |
| |
Notes | |
| |
| |
| |
Exercises | |
| |
| |
| |
Risk-Neutral Pricing | |
| |
| |
| |
Introduction | |
| |
| |
| |
Risk-Neutral Measure | |
| |
| |
| |
Girsanov's Theorem for a Single Brownian Motion | |
| |
| |
| |
Stock Under the Risk-Neutral Measure | |
| |
| |
| |
Value of Portfolio Process Under the Risk-Neutral Measure | |
| |
| |
| |
Pricing Under the Risk-Neutral Measure | |
| |
| |
| |
Deriving the Black-Scholes-Merton Formula | |
| |
| |
| |
Martingale Representation Theorem | |
| |
| |
| |
Martingale Representation with One Brownian Motion | |
| |
| |
| |
Hedging with One Stock | |
| |
| |
| |
Fundamental Theorems of Asset Pricing | |
| |
| |
| |
Girsanov and Martingale Representation Theorems | |
| |
| |
| |
Multidimensional Market Model | |
| |
| |
| |
Existence of Risk-Neutral Measure | |
| |
| |
| |
Uniqueness of the Risk-Neutral Measure | |
| |
| |
| |
Dividend-Paying Stocks | |
| |
| |
| |
Continuously Paying Dividend | |
| |
| |
| |
Continuously Paying Dividend with Constant Coeffcients | |
| |
| |
| |
Lump Payments of Dividends | |
| |
| |
| |
Lump Payments of Dividends with Constant Coeffcients | |
| |
| |
| |
Forwards and Futures | |
| |
| |
| |
Forward Contracts | |
| |
| |
| |
Futures Contracts | |
| |
| |
| |
Forward-Futures Spread | |
| |
| |
| |
Summary | |
| |
| |
| |
Notes | |
| |
| |
| |
Exercises | |
| |
| |
| |
Connections with Partial Differential Equations | |
| |
| |
| |
Introduction | |
| |
| |
| |
Stochastic Differential Equations | |
| |
| |
| |
The Markov Property | |
| |
| |
| |
Partial Differential Equations | |
| |
| |
| |
Interest Rate Models | |
| |
| |
| |
Multidimensional Feynman-Kac Theorems | |
| |
| |
| |
Summary | |
| |
| |
| |
Notes | |
| |
| |
| |
Exercises | |
| |
| |
| |
Exotic Options | |
| |
| |
| |
Introduction | |
| |
| |
| |
Maximum of Brownian Motion with Drift | |
| |
| |
| |
Knock-Out Barrier Options | |
| |
| |
| |
Up-and-Out Call | |
| |
| |
| |
Black-Scholes-Merton Equation | |
| |
| |
| |
Computation of the Price of the Up-and-Out Call | |
| |
| |
| |
Lookback Options | |
| |
| |
| |
Floating Strike Lookback Option | |
| |
| |
| |
Black-Scholes-Merton Equation | |
| |
| |
| |
Reduction of Dimension | |
| |
| |
| |
Computation of the Price of the Lookback Option | |
| |
| |
| |
Asian Options | |
| |
| |
| |
Fixed-Strike Asian Call | |
| |
| |
| |
Augmentation of the State | |
| |
| |
| |
Change of Num'eraire | |
| |
| |
| |
Summary | |
| |
| |
| |
Notes | |
| |
| |
| |
Exercises | |
| |
| |
| |