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