Contents
1 Foundation of Probability Theory and Discrete-Time Martingales 1
1.1 Basic Concepts of Probability Theory 1
1.1.1 Events andProbability 1
1.1.2 Independence, 0-1 Law, and Borel-Cantelli Lemma 3
1.1.3 Integrals, (Mathematical) Expectations of Random
1.1.4 Convergence Theorems 7
1.2 Conditional Mathematical Expectation 9
1.2.1 Definition and Basic Properties 9
1.2.2 Convergence Theorems 14
1.2.3 Two Theorems About Conditional Expectation 15
1.3 Duals of Spaces L∞ (Ω,F) and L∞ (Ω,F, m) 17
1.4 Family of Uniformly Integrable Random Variables 18
1.5 Discrete Time Martingales 22
1.5.1 Basic Definitions 22
1.5.3 Martingale Transforms 27
1.5.4 Snell Envelop 30
1.6 Markov Sequences 31
2 Portfolio Selection Theory in Discrete-Time 33
2.1 Mean-Variance Analysis 34
2.1.1 Mean-Variance Frontier Portfolios Without
2.1.2 Revised Formulations of Mean-Variance Analysis Without Risk-Free Asset 38
2.1.3 Mean-Variance Frontier Portfolios with Risk-Free
2.1.4 Mean-Variance Utility Functions 45
2.2 Capital Asset Pricing Model (CAPM) 47
2.2.1 Market Competitive Equilibrium and Market Portfolio 47
2.2.2 CAPM with Risk-Free Asset 49
2.2.3 CAPM Without Risk-Free Asset 52
2.2.4 Equilibrium Pricing Using CAPM 53
2.3 Arbitrage Pricing Theory (APT) 54
2.4 Mean-Semivariance Model 57
2.5 Multistage Mean-Variance Model 58
2.6 Expected Utility Theory 62
2.6.1 Utility Functions 63
2.6.2 Arrow-Pratt's Risk Aversion Functions 64
2.6.3 Comparison of Risk Aversion Functions 66
2.6.4 Preference Defined by Stochastic Orders 66
2.6.5 Maximization of Expected Utility and Initial Price of Risky Asset 70
2.7 Consumption-Based Asset Pricing Models 72
3 Financial Markets in Discrete Time 75
3.1 Basic Concepts of Financial Markets 75
3.1.2 Pricing and Hedging 76
3.1.3 Put-Call Parity 76
3.1.4 Intrinsic Value and Time Value 77
3.1.6 Efficient Market Hypothesis 78
3.2 Binomial Tree Model 78
3.2.1 The One-Period Case 78
3.2.2 The Multistage Case 79
3.2.3 The Approximately Continuous Trading Case 82
3.3 The General Discrete-Time Model 83
3.3.1 The Basic Framework 83
3.3.2 Arbitrage, Admissible, and Allowable Strategies 85
3.4 Martingale Characterization of No-Arbitrage Markets 86
3.4.1 The Finite Market Case 86
3.4.2 The General Case: Dalang-Morton-Willinger Theorem 87
3.5 Pricing of European Contingent Claims 90
3.6 Maximization of Expected Utility and Option Pricing 92
3.6.1 General Utility Function Case 92
3.6.2 HARA Utility Functions and Their Duality Case 94
3.6.3 Utility Function-Based Pricing 96
3.6.4 Market Equilibrium Pricing 99
3.7 American Contingent Claims Pricing 103
3.7.1 Super-Hedging Strategies in Complete Markets 103
3.7.2 Arbitrage-Free Pricing in Complete Markets 104
3.7.3 Arbitrage-Free Pricing in Non-complete Markets 105
4 Martingale Theory and Ito Stochastic Analysis 107
4.1 Continuous Time Stochastic Processes 107
4.1.1 Basic Concepts of Stochastic Processes 107
4.1.2 Poisson and Compound Poisson Processes 108
4.1.3 Markov Processes 110
4.1.4 Brownian Motion 113
4.1.5 Stopping Times, Martingales, Local Martingales 114
4.1.6 Finite Variation Processes 115
4.1.7 Doob-Meyer Decomposition of Local Submartingales 116
4.1.8 Quadratic Variation Processes of Semimartingales 119
4.2 Stochastic Integrals w.t.t Brownian Motion 124
4.2.1 Wiener Integrals 124
4.2.2 Ito Stochastic Integrals 125
4.3 Ito's Formula and Girsanov's Theorem 130
4.3.1 Ito's Formula 131
4.3.2 Levy's Martingale Characterization of Brownian
4.3.3 Reflection Principle of Brownian Motion 134
4.3.4 Stochastic Exponentials and Novikov Theorem 134
4.3.5 Girsanov's Theorem 136
4.4 Martingale Representation Theorem 137
4.5 Ito Stochastic Differential Equations 140
4.5.1 Existence and Uniqueness of Solutions 140
4.6 Ito Diffusion Processes 147
4.7 Feynman-Kac Formula 148
4.8 Snell Envelop (Continuous Time Case) 149
5 The Black-Scholes Model and Its Modifications 153
5.1 Martingale Method for Option Pricing and Hedging 154
5.1.1 The Black-Scholes Model 154
5.1.2 Equivalent Martingale Measures 155
5.1.3 Pricing and Hedging of European Contingent Claims 157
5.1.4 Pricing of American Contingent Claims 160
5.2 Some Examples of Option Pricing 162
5.2.1 Options on a Stock with Proportional Dividends 162
5.2.2 Foreign Currency Option 163
5.2.3 Compound Option 164
5.3 Practical Uses of the Black-Scholes Formulas 166
5.3.1 Historical and Implied Volatilities 166
5.3.2 Delta Hedging and Analyses of Option Price Sensitivities 166
5.4 Capturing Biases in Black-Scholes Formulas 168
5.4.1 CEV Model and Level-Dependent Volatility Model 168
5.4.2 Stochastic Volatility Model 170
5.4.3 SABR Model 172
5.4.4 Variance-Gamma (VG) Model 172
6 Pricing and Hedging of Exotic Options 175
6.1 Running Extremum of Brownian Motion with Drift 175
6.2.1 Single-Barrier Options 179
6.2.2 Double-Barrier Options 180
6.3 Asian Options 180
6.3.1 Geometric Average Asian Options 181
6.3.2 Arithmetic Average Asian Options 183
6.4 Lookback Options 189
6.4.1 Lookback Strike Options 190
6.4.2 Lookback Rate Options 192
6.5 Reset Options 193
7 Ito Process and Diffusion Models 195
7.1 Ito Process Models 195
7.1.1 Self-Financing Trading Strategies 195
7.1.2 Equivalent Martingale Measures and No Arbitrage 197
7.1.3 Pricing and Hedging of European Contingent Claims 201
7.1.4 Change of Numeraire 203
7.1.5 Arbitrage Pricing Systems 205
7.2 PDE Approach to Option Pricing 208
7.3 Probabilistic Methods for Option Pricing 209
7.3.1 Time and Scale Changes 209
7.3.2 Option Pricing in Merton's Model 210
7.3.3 General Nonlinear Reduction Method 211
7.3.4 Option Pricing Under the CEV Model 212
7.4 Pricing American Contingent Claims 214
8 Term Structure Models for Interest Rates 217
8.1 The Bond Market 218
8.1.1 Basic Concepts 218
8.1.2 Bond Price Process 219
8.2 Short Rate Models 220
8.2.1 0ne-Factor Models and Affine Term Structures 221
8.2.2 Functional Approach to One-Factor Models 225
8.2.3 Multifactor Short Rate Models 229
8.2.4 Forward Rate Models: The HJM Model 231
8.3 Forward Price and Futures Price 234
8.3.1 Forward Price 234
8.3.2 Futures Price 235
8.4 Pricing Interest Rate Derivatives 236
8.4.1 PDE Method 236
8.4.2 Forward Measure Method 239
8.4.3 Changing Numeraire Method 239
8.5 The Flesaker-Hughston Model 242
8.6 BGM Models 244
9 Optimal Investment-Consumption Strategies in Diffusion Models 247
9.1 Market Models and Investment-Consumption Strategies 247
9.2 Expected Utility Maximization 250
9.3 Mean-Risk Portfolio Selection 258
9.3.1 General Framework for Mean-Risk Models 258
9.3.2 Weighted Mean-Variance Model 259
10 Static Risk Measures 263
10.1 Coherent Risk Measures 263
10.1.1 Monetary Risk Measures and Coherent Risk Measures 264
10.1.2 Representation of Coherent Risk Measures 266
10.2 Co-monotonic Subadditive Risk Measures 268
10.2.1 Representation: The Model-Free Case 269
10.2.2 Representation: The Model-Dependent Case 272
10.3 Convex Risk Measures 274
10.3.1 Representation: The Model-Free Case 274
10.3.2 Representation: The Model-Dependent Case 275
10.4 Co-monotonic Convex Risk Measures 276
10.4.1 The Model-Free Case 276
10.4.2 The Model-Dependent Case 278
10.5 Law-Invariant Risk Measures 280
10.5.1 Law-Invariant Coherent Risk Measures 280
10.5.2 Law-Invariant Convex Risk Measures 285
10.5.3 Some Results About Stochastic Orders and Quantiles 286
10.5.4 Law-Invariant Co-monotonic Subadditive Risk
10.5.5 Law-Invariant Co-monotonic Convex Risk Measures 298
11 Stochastic Calculus and Semimartingale Model 307
11.1 Semimartingales and Stochastic Calculus 308
11.1.1 Doob-Meyer's Decomposition of Supermartingales 308
11.1.2 Local Martingales and Semimartingales 310
11.1.3 Stochastic Integrals wrt Local Martingales 311
11.1.4 Stochasticlntegrals wrt Semimartingales 313
11.1.5 Ito's Formula and Doleans Exponential Formula 314
11.2 Semimartingale Model 315
11.2.1 Basic Concepts and Notations 316
11.2.2 Vector Stochastic Integrals wrt Semimartingales 318
11.2.3 Optional Decomposition Theorem 319
11.3 Superhedging 321
11.4 Fair Prices and Attainable Claims 322
12 Optimal Investment in Incomplete Markets 327
12.1 Convex Duality on Utility Maximization 328
12.1.1 The Problem 328
12.1.2 Complete Market Case 329
12.1.3 Incomplete Market Case 330
12.1.4 Results of Kramkov and Schachermayer 332
12.2 A Numeraire-Free Framework 334
12.2.1 Martingale Deflators and Superhedging 335
12.2.2 Reformulation of Theorem 12.1 337
12.3 Utility-Based Approaches to Option Pricing 338
12.3.1 Minimax Martingale Deflator Approach 338
12.3.2 Marginal Utility-Based Approach 340
13 Martingale Method for Utility Maximization 343
13.1 Expected Utility Maximization and Valuation 344
13.1.1 Expected Utility Maximization 344
13.1.2 Utility-Based Valuation 346
13.2 Minimum Relative Entropy and Maximum Hellingerlntegral 348
13.2.1 HARA Utility Functions 348
13.2.2 Another Type of Utility Function 350
13.2.3 Utility Function W0(x)=-e-x 351
13.3 Market Driven by a Levy Process 352
13.3.1 The Market Model 352
13.3.2 Results for HARA Utility Functions 354
13.3.3 Results for Utility Functions of the Form Wy(y<0) 359
13.3.4 Results for Utility Function W0(x)=-e-x 359
14 Optimal Growth Portfolios and Option Pricing 365
14.1 OptimalGrowthPortfolio 365
14.1.1 OptimalGrowth Strategy 366
14.1.2 A Geometric Levy Process Model 367
14.1.3 A Jump-Diffusion-Like Process Model 373
14.2 Pricing in a Geometric Levy Process Model 377
14.3 Other Approaches to Option Pricing 383
14.3.1 The Follmer-Schwarzer Approach 383
14.3.2 The Davis' Approach 383
14.3.3 Esscher Transform Approach 384
References 387
Index 397