1 Foundation of Probability Theory and Discrete-Time Martingales 1.1 Basic Concepts of Probability Theory 1.1.1 Events and Probability 1.1.2 Independence, 0-1 Law, and Borel-Cantelli Lemma 1.1.3 Integrals, (Mathematical) Expectations of Random Variables 1.1.4 Convergence Theorems 1.2 Conditional Mathematical Expectation 1.2.1 Definition and Basic Properties 1.2.2 Convergence Theorems 1.2.3 Two Theorems About Conditional Expectation 1.3 Duals of Spaces L∞(Ω, F) and L∞(Ω, F, m) 1.4 Family of Uniformly Integrable Random Variables 1.5 Discrete Time Martingales 1.5.1 Basic Definitions 1.5.2 Basic Theorems 1.5.3 Martingale Transforms 1.5.4 Snell Envelop 1.6 Markoy Sequences 2 Portfolio Selection Theory in Discrete.Time 2.1 Mean-Variance Analysis 2.1.1 Mean-Variance Frontier Portfolios Without Risk-Free Asset 2.1.2 Revised Formulations of Mean-Variance Analysis Without Risk-Free Asset 2.1.3 Mean-Variance Frontier Portfolios with Risk-Free Asset 2.1.4 Mean-Variance Utility Functions 2.2 Capital Asset Pricing Model (CAPM) 2.2.1 Market Competitive Equilibrium and Market Portfolio., 2.2.2 CAPM with Risk-Free Asset 2.2.3 CAPM Without Risk-Free Asset 2.2.4 Equilibrium Pricing Using CAPM 2.3 Arbitrage Pricing Theory (APT) 2.4 Mean-Sernivariance Model 2.5 Multistage Mean-Variance Model 2.6 Expected Utility Theory 2.6.1 Utility Functions 2.6.2 Arrow-Pratt's Risk Aversion Functions 2.6.3 Comparison of Risk Aversion Functions 2.6.4 Preference Defined by Stochastic Orders 2.6.5 Maximization of Expected Utility and Initial Price of Risky Asset 2.7 Consumption-Based Asset Pricing Models 3 Financial Markets in Discrete Time 3.1 Basic Concepts of Financial Markets 3.1.1 Numeraire 3.1.2 Pricing and Hedging 3.1.3 Put-Call Parity 3.1.4 Intrinsic Value and Time Value 3.1.5 Bid-Ask Spread 3.1.6 Efficient Market Hypothesis 3.2 Binomial Tree Model 3.2.1 The One-Period Case 3.2.2 The Multistage Case 3.2.3 The Approximately Continuous Trading Case 3.3 The General Discrete-Time Model 3.3.1 The Basic Framework 3.3.2 Arbitrage, Admissible, and Allowable Strategies 3.4 Martingale Characterization of No-Arbitrage Markets 3.4.1 The Finite Market Case 3.4.2 The General Case: Dalang-Morton-Willinger Theorem.. 3.5 Pricing of European Contingent Claims 3.6 Maximization of Expected Utility and Option Pricing 3.6.1 General Utility Function Case 3.6.2 HARA Utility Functions and Their Duality Case 3.6.3 Utility Function-Based Pricing 3.6.4 Market Equilibrium Pricing 3.7 American Contingent Claims Pricing 3.7.1 Super-Hedging Strategies in Complete Markets 3.7.2 Arbitrage-Free Pricing in Complete Markets 3.7.3 Arbitrage-Free Pricing in Non-complete Markets 4 Martingale Theory and It8 Stochastic Analysis 4.1 Continuous Time Stochastic Processes 4.1.1 Basic Concepts of Stochastic Processes 4.1.2 Poisson and Compound Poisson Processes 4.1.3 Markov Processes 4.1.4 Brownian Motion 4.1.5 Stopping Times, Martingales, Local Martingales 4.1.6 Finite Variation Processes 4.1.7 Doob-Meyer Decomposition of Local Submartingales 4.1.8 Quadratic Variation Processes of Semimartingales 4.2 Stochastic Integrals w.t.t. Brownian Motion 4.2.1 Wiener Integrals 4.2.2 Ito Stochastic Integrals 4.3 Itr's Formula and Girsanov's Theorem 4.3.1 Itr's Formula 4.3.2 Lrvy's Martingale Characterization of Brownian Motion 4.3.3 Reflection Principle of Brownian Motion 4.3.4 Stochastic Exponentials and Novikov Theorem 4.3.5 Girsanov's Theorem 4.4 Martingale Representation Theorem 4.5 Ito Stochastic Differential Equations 4.5.1 Existence and Uniqueness of Solutions 4.5.2 Examples 4.6 Ito Diffusion Processes 4.7 Feynman-Kac Formula 4.8 Snell Envelop (Continuous Time Case) 5 The Black-Scholes Model and Its Modifications 5.1 Martingale Method for Option Pricing and Hedging 5.1.1 The Black-Scholes Model 5.1.2 Equivalent Martingale Measures 5.1.3 Pricing and Hedging of European Contingent Claims 5.1.4 Pricing of American Contingent Claims 5.2 Some Examples of Option Pricing 5.2.1 Options on a Stock with Proportional Dividends 5.2.2 Foreign Currency Option 5.2.3 Compound Option 5.2.4 Chooser Option 5.3 Practical Uses of the Black-Scholes Formulas 5.3.1 Historical and Implied Volatilities 5.3.2 Delta Hedging and Analyses of Option Price Sensitivities 5.4 Capturing Biases in Black-Scholes Formulas 5.4.1 CEV Model and Level-Dependent Volatility Model 5.4.2 Stochastic Volatility Model 5.4.3 SABR Model 5.4.4 Variance-Gamma (VG) Model 5.4.5 GARCH Model 6 Pricing and Hedging of Exotic Options 6.1 Running Extremum of Brownian Motion with Drift 6.2 Barrier Options 6.2.1 Single-Barrier Options 6.2.2 Double-Barrier Options 6.3 Asian Options 6.3.1 Geometric Average Asian Options 6.3.2 Arithmetic Average Asian Options 6.4 Lookback Options 6.4.1 Lookback Strike Options 6.4.2 Lookback Rate Options 6.5 Reset Options 7 Ito Process and Diffusion Models 7.1 Ito Process Models 7.1.1 Self-Financing Trading Strategies 7.1.2 Equivalent Martingale Measures and No Arbitrage 7.1.3 Pricing and Hedging of European Contingent Claims 7.1.4 Change of Numeraire 7.1.5 Arbitrage Pricing Systems 7.2 PDE Approach to Option Pricing 7.3 Probabilistic Methods for Option Pricing 7.3.1 Time and Scale Changes 7.3.2 Option Pricing in Merton's Model 7.3.3 General Nonlinear Reduction Method 7.3.4 Option Pricing Under the CEV Model 7.4 Pricing American Contingent Claims 8 Term Structure Models for Interest Rates 8.1 The Bond Market 8.1.1 Basic Concepts 8.1.2 Bond Price Process 8.2 Short Rate Models 8.2.1 One-Factor Models and Affine Term Structures 8.2.2 Functional Approach to One-Factor Models 8.2.3 Multifactor Short Rate Models 8.2.4 Forward Rate Models: The HJM Model 8.3 Forward Price and Futures Price 8.3.1 Forward Price 8.3.2 Futures Price 8.4 Pricing Interest Rate Derivatives 8.4.1 PDE Method 8.4.2 Forward Measure Method 8.4.3 Changing Numeraire Method 8.5 The Flesaker-Hughston Model 8.6 BGM Models 9 Optimal Investment-Consumption Strategies in Diffusion Models 9.1 Market Models and Investment-Consumption Strategies 9.2 Expected Utility Maximization 9.3 Mean-Risk Portfolio Selection 9.3.1 General Framework for Mean-Risk Models 9.3.2 Weighted Mean-Variance Model 10 Static Risk Measures 10.1 Coherent Risk Measures 10.1.1 Monetary Risk Measures and Coherent Risk Measures 10.1.2 Representation of Coherent Risk Measures 10.2 Co-monotonic Subadditive Risk Measures 10.2.1 Representation: The Model-Free Case 10.2.2 Representation: The Model-Dependent Case 10.3 Convex Risk Measures 10.3.1 Representation: The Model-Free Case 10.3.2 Representation: The Model-Dependent Case 10.4 Co-monotonic Convex Risk Measures 10.4.1 The Model-Free Case 10.4.2 The Model-Dependent Case 10.5 Law-Invariant Risk Measures 10.5.1 Law-Invariant Coherent Risk Measures 10.5.2 Law-Invariant Convex Risk Measures 10.5.3 Some Results About Stochastic Orders and Quantiles 10.5.4 Law-Invariant Co-monotonic Subadditive Risk Measures 10.5.5 Law-Invariant Co-monotonic Convex Risk Measures 11 Stochastic Calculus and Semimartingale Model 11.1 Semimartingales and Stochastic Calculus 11.1.1 Doob-Meyer's Decomposition of Supermartingales 11.1.2 Local Martingales and Semimartingales 11.1.3 Stochastic Integrals w.r.t. Local Martingales 11.1.4 Stochastic Integrals w.r.t. Semimartingales 11.1.5 Ito's Formula and Dol6ans Exponential Formula 11.2 Semimartingale Model 11.2.1 Basic Concepts and Notations 11.2.2 Vector Stochastic Integrals w.r.t. Semimartingales 11.2.3 Optional Decomposition Theorem 11.3 Superhedging 11.4 Fair Prices and Attainable Claims 12 Optimal Investment in Incomplete Markets 12.1 Convex Duality on Utility Maximization 12.1.1 The Problem 12.1.2 Complete Market Case 12.1.3 Incomplete Market Case 12.1.4 Results of Kramkov and Schachermayer 12.2 A Numeraire-Free Framework 12.2.1 Martingale Deflators and Superhedging 12.2.2 Reformulation of Theorem 12.1 12.3 Utility-Based Approaches to Option Pricing 12.3.1 Minimax Martingale Deflator Approach 12.3.2 Marginal Utility-Based Approach 13 Martingale Method for Utility Maximization 13.1 Expected Utility Maximization and Valuation 13.1.1 Expected Utility Maximization 13.1.2 Utility-B ased Valuation 13.2 Minimum Relative Entropy and Maximum Hellinger Integral 13.2.1 HARA Utility Functions 13,2.2 Another Type of Utility Function 13.2.3 Utility Function Wo(x) = -e-x 13.3 Market Driven by a Levy Process 13.3.1 The Market Model 13.3.2 Results for HARA Utility Functions 13.3.3 Results for Utility Functions of the Form Wy (γ< 0) 13.3.4 Results for Utility Function Wo(x) = -e-x 14 Optimal Growth Portfolios and Option Pricing 14.1 Optimal Growth Portfolio 14.1.1 Optimal Growth Strategy 14.1.2 A Geometric L6vy Process Model 14.1.3 A Jump-Diffusion-Like Process Model 14.2 Pricing in a Geometric Levy Process Model 14.3 Other Approaches to Option Pricing 14.3.1 The Follmer-Schwarzer Approach 14.3.2 The Davis' Approach 14.3.3 Esscher Transform Approach References Index
