十年的书，算是经典了。
CONTENTS
1 Introduction 1
1.1 Problem Formulation 1
2 The Binomial Model 5
2.1 The One Period Model 5
2.1.1 Model Description 5
2.1.2 Portfolios and Arbitrage 6
2.1.3 Contingent Claims 9
2.1.4 Risk Neutral Valuation 11
2.2 The Multiperiod Model 15
2.2.1 Portfolios and Arbitrage 15
2.2.2 Contingent Claims 17
2.3 Exercises 25
2.4 Notes 25
3 A More General One Period Model 26
3.1 The Model 26
3.2 Absence of Arbitrage 27
3.3 Martingale Measures 32
3.4 Martingale Pricing 34
3.5 Completeness 35
3.6 Stochastic Discount Factors 38
3.7 Exercises 39
4 Stochastic Integrals 40
4.1 Introduction 40
4.2 Information 42
4.3 Stochastic Integrals 44
4.4 Martingales 46
4.5 Stochastic Calculus and the Itˆo Formula 49
4.6 Examples 54
4.7 The Multidimensional Itˆo Formula 57
4.8 Correlated Wiener Processes 59
4.9 Exercises 63
4.10 Notes 65
5 Differential Equations 66
5.1 Stochastic Differential Equations 66
5.2 Geometric Brownian Motion 67
5.3 The Linear SDE 70
5.4 The Infinitesimal Operator 71
5.5 Partial Differential Equations 72
5.6 The Kolmogorov Equations 76
5.7 Exercises 79
5.8 Notes 83
6 Portfolio Dynamics 84
6.1 Introduction 84
6.2 Self-financing Portfolios 87
6.3 Dividends 89
6.4 Exercises 91
7 Arbitrage Pricing 92
7.1 Introduction 92
7.2 Contingent Claims and Arbitrage 93
7.3 The Black–Scholes Equation 98
7.4 Risk Neutral Valuation 102
7.5 The Black–Scholes Formula 104
7.6 Options on Futures 106
7.6.1 Forward Contracts 106
7.6.2 Futures Contracts and the Black Formula 107
7.7 Volatility 108
7.7.1 Historic Volatility 109
7.7.2 Implied Volatility 110
7.8 American Options 110
7.9 Exercises 112
7.10 Notes 114
8 Completeness and Hedging 115
8.1 Introduction 115
8.2 Completeness in the Black–Scholes Model 116
8.3 Completeness—Absence of Arbitrage 121
8.4 Exercises 122
8.5 Notes 124
9 Parity Relations and Delta Hedging 125
9.1 Parity Relations 125
9.2 The Greeks 127
9.3 Delta and Gamma Hedging 130
9.4 Exercises 134
10 The Martingale Approach to Arbitrage Theory* 137
10.1 The Case with Zero Interest Rate 137
10.2 Absence of Arbitrage 140
10.2.1 A Rough Sketch of the Proof 141
10.2.2 Precise Results 144
10.3 The General Case 146
10.4 Completeness 149
10.5 Martingale Pricing 151
10.6 Stochastic Discount Factors 153
10.7 Summary for the Working Economist 154
10.8 Notes 156
11 The Mathematics of the Martingale Approach* 158
11.1 Stochastic Integral Representations 158
11.2 The Girsanov Theorem: Heuristics 162
11.3 The Girsanov Theorem 164
11.4 The Converse of the Girsanov Theorem 168
11.5 Girsanov Transformations and Stochastic Differentials 168
11.6 Maximum Likelihood Estimation 169
11.7 Exercises 171
11.8 Notes 172
12 Black–Scholes from a Martingale Point of View* 173
12.1 Absence of Arbitrage 173
12.2 Pricing 175
12.3 Completeness 176
13 Multidimensional Models: Classical Approach 179
13.1 Introduction 179
13.2 Pricing 181
13.3 Risk Neutral Valuation 187
13.4 Reducing the State Space 188
13.5 Hedging 192
13.6 Exercises 195
14 Multidimensional Models: Martingale Approach* 196
14.1 Absence of Arbitrage 197
14.2 Completeness 199
14.3 Hedging 200
14.4 Pricing 202
14.5 Markovian Models and PDEs 203
14.6 Market Prices of Risk 204
14.7 Stochastic Discount Factors 205
14.8 The Hansen–Jagannathan Bounds 205
14.9 Exercises 208
14.10 Notes 208
15 Incomplete Markets 209
15.1 Introduction 209
15.2 A Scalar Nonpriced Underlying Asset 209
15.3 The Multidimensional Case 218
15.4 A Stochastic Short Rate 222
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