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 3.2.1 Symmetric Random Walk 3.2.2 Increments of the Symmetric Random Walk 3.2.3 Martingale Property for the Symmetric Random Walk 3.2.4 Quadratic Variation of the Symmetric Random Walk 3.2.5 Scaled Symmetric Random Walk 3.2.6 Limiting Distribution of the Scaled Random Walk 3.2.7 Log-Normal Distribution as the Limit of the 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 Reflection 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 Ito's Integral for Simple Integrands 4.2.1 Construction of the Integral 4.2.2 Properties of the Integral 4.3 Ito's Integral for General Integrands 4.4 Ito-Doeblin Formula 4.4.1 Formula for Brownian Motion 4.4.2 Formula for Ito 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 The Greeks 4.5.6 Put-Call Parity 4.6 Multivariable Stochastic Calculus 4.6.1 Multiple Brownian Motions 4.6.2 Ito-Doeblin Formula for Multiple Processes 4.6.3 Recognizing a Brownian Motion 4.7 Brownian Bridge 4.7.1 Gaussian Processes 4.7.2 Brownian Bridge as a Gaussian Process 4.7.3 Brownian Bridge as a Scaled Stochastic Integral 4.7.4 Multidimensional Distribution of the Brownian Bridge 4.7.5 Brownian Bridge as a Conditioned Brownian Motion 4.8 Summary 4.9 Notes 4.10 Exercises 5 Risk-Neutral Pricing 5.1 Introduction 5.2 Risk-Neutral Measure 5.2.1 Girsanov's Theorem for a Single Brownian Motion 5.2.2 Stock Under the Risk-Neutral Measure 5.2.3 Value of Portfolio Process Under the Risk-Neutral Measure 5.2.4 Pricing Under the Risk-Neutral Measure 5.2.5 Deriving the Black-Scholes-Merton Formula 5.3 Martingale Representation Theorem 5.3.1 Martingale Representation with One Brownian Motion 5.3.2 Hedging with One Stock 5.4 Fundamental Theorems of Asset Pricing 5.4.1 Girsanov and Martingale Representation Theorems 5.4.2 Multidimensional Market Model 5.4.3 Existence of the Risk-Neutral Measure 5.4.4 Uniqueness of the Risk-Neutral Measure 5.5 Dividend-Paying Stocks 5.5.1 Continuously Paying Dividend 5.5.2 Continuously Paying Dividend with Constant Coefficients 5.5.3 Lump Payments of Dividends 5.5.4 Lump Payments of Dividends with Constant Coefficients 5.6 Forwards and Futures 5.6.1 Forward Contracts 5.6.2 Futures Contracts 5.6.3 Forward-Futures Spread 5.7 Summary 5.8 Notes 5.9 Exercises 6 Connections with Partial Differential Equations 6.1 Introduction 6.2 Stochastic Differential Equations 6.3 The Markov Property 6.4 Partial Differential Equations 6.5 Interest Rate Models 6.6 Multidimensional Feynman-Kac Theorems 6.7 Summary 6.8 Notes 6.9 Exercises 7 Exotic Options 7.1 Introduction 7.2 Maximum of Brownian Motion with Drift 7.3 Knock-out Barrier Options 7.3.1 Up-and-Out Call 7.3.2 Black-Scholes-Merton Equation 7.3.3 Computation of the Price of the Up-and-Out Call 7.4 Lookback Options 7.4.1 Floating Strike Lookback Option 7.4.2 Black-Scholes-Merton Equation 7.4.3 Reduction of Dimension 7.4.4 Computation of the Price of the Lookback Option 7.5 Asian Options 7.5.1 Fixed-Strike Asian Call 7.5.2 Augmentation of the State 7.5.3 Change of Numeraire 7.6 Summary 7.7 Notes 7.8 Exercises 8 American Derivative Securities 8.1 Introduction 8.2 Stopping Times 8.3 Perpetual American Put 8.3.1 Price Under Arbitrary Exercise 8.3.2 Price Under Optimal Exercise 8.3.3 Analytical Characterization of the Put Price 8.3.4 Probabilistic Characterization of the Put Price 8.4 Finite-Expiration American Put 8.4.1 Analytical Characterization of the Put Price 8.4.2 Probabilistic Characterization of the Put Price 8.5 American Call 8.5.1 Underlying Asset Pays No Dividends 8.5.2 Underlying Asset Pays Dividends 8.6 Summary 8.7 Notes 8.8 Exercises 9 Change of Numeraire 9.1 Introduction 9.2 Numeraire 9.3 Foreign and Domestic Risk-Neutral Measures 9.3.1 The Basic Processes 9.3.2 Domestic Risk-Neutral Measure 9.3.3 Foreign Risk-Neutral Measure 9.3.4 Siegel's Exchange Rate Paradox 9.3.5 Forward Exchange Rates 9.3.6 Garman-Kohlhagen Formula 9.3.7 Exchange Rate Put-Call Duality 9.4 Forward Measures 9.4.1 Forward Price 9.4.2 Zero-Coupon Bond as Numeraire 9.4.3 Option Pricing with a Random Interest Rate 9.5 Summary 9.6 Notes 9.7 Exercises 10 Term-Structure Models 10.1 Introduction 10.2 Affine-Yield Models 10.2.1 Two-Factor Vasicek Model 10.2.2 Two-Factor CIR Model 10.2.3 Mixed Model 10.3 Heath-Jarrow-Morton Model 10.3.1 Forward Rates 10.3.2 Dynamics of Forward Rates and Bond Prices 10.3.3 No-Arbitrage Condition 10.3.4 HJM Under Risk-Neutral Measure 10.3.5 Relation to Afine-Yield Models 10.3.6 Implementation of HJM 10.4 Forward LIBOR Model 10.4.1 The Problem with Forward Rates 10.4.2 LIBOR and Forward LIBOR 10.4.3 Pricing a Backset LIBOR Contract 10.4.4 Black Caplet Formula 10, .4.5 Forward LIBOR and Zero-Coupon Bond Volatilities 10.4.6 A Forward LIBOR Term-Structure Model 10.5 Summary 10.6 Notes 10.7 Exercises 11 Introduction to Jump Processes 11.1 Introduction 11.2 Poisson Process 11.2.1 Exponential Random Variables 11.2.2 Construction of a Poisson Process 11.2.3 Distribution of Poisson Process Increments 11.2.4 Mean and Variance of Poisson Increments 11.2.5 Martingale Property 11.3 Compound Poisson Process 11.3.1 Construction of a Compound Poisson Process 11.3.2 Moment-Generating Function 11.4 Jump Processes and Their Integrals 11.4.1 Jump Processes 11.4.2 Quadratic Variation 11.5 Stochastic Calculus for Jump Processes 11.5.1 It6-Doeblin Formula for One Jump Process 11.5.2 Ito-Doeblin Formula for Multiple Jump Processes 11.6 Change of Measure 11.6.1 Change of Measure for a Poisson Process 11.6.2 Change of Measure for a Compound Poisson Process 11.6.3 Change of Measure for a Compound Poisson Process and a Brownian Motion 11.7 Pricing a European Call in a Jump Model 11.7.1 Asset Driven by a Poisson Process 11.7.2 Asset Driven by a Brownian Motion and a Compound Poisson Process 11.8 Summary 11.9 Notes 11.10 Exercises A Advanced Topics in Probability Theory A.1 Countable Additivity A.2 Generating o-algebras A.3 Random Variable with Neither Density nor Probability Mass Function B Existence of Conditional Expectations C Completion of the Proof of the Second Fundamental Theorem of Asset Pricing References Index