E-Book, Englisch, 416 Seiten
Cerný / Cerny Mathematical Techniques in Finance
2. Auflage 2009
ISBN: 978-1-4008-3148-7
Verlag: De Gruyter
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)
Tools for Incomplete Markets - Second Edition
E-Book, Englisch, 416 Seiten
ISBN: 978-1-4008-3148-7
Verlag: De Gruyter
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)
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Preface to the Second Edition xiii
From the Preface to the First Edition xix
Chapter 1: The Simplest Model of Financial Markets 1
1.1 One-Period Finite State Model 1
1.2 Securities and Their Payoffs 3
1.3 Securities as Vectors 3
1.4 Operations on Securities 4
1.5 The Matrix as a Collection of Securities 6
1.6 Transposition 6
1.7 Matrix Multiplication and Portfolios 8
1.8 Systems of Equations and Hedging 10
1.9 Linear Independence and Redundant Securities 12
1.10 The Structure of the Marketed Subspace 14
1.11 The Identity Matrix and Arrow-Debreu Securities 16
1.12 Matrix Inverse 17
1.13 Inverse Matrix and Replicating Portfolios 17
1.14 Complete Market Hedging Formula 19
1.15 Summary 20
1.16 Notes 21
1.17 Exercises 22
Chapter 2: Arbitrage and Pricing in the One-Period Model 25
2.1 Hedging with Redundant Securities and Incomplete Market 25
2.2 Finding the Best Approximate Hedge 29
2.3 Minimizing the Expected Squared Replication Error 32
2.4 Numerical Stability of Least Squares 34
2.5 Asset Prices, Returns and Portfolio Units 36
2.6 Arbitrage 38
2.7 No-Arbitrage Pricing 40
2.8 State Prices and the Arbitrage Theorem 41
2.9 State Prices and Asset Returns 44
2.10 Risk-Neutral Probabilities 45
2.11 State Prices and No-Arbitrage Pricing 46
2.12 Asset Pricing Duality 47
2.13 Summary 48
2.14 Notes 49
2.15 Appendix: Least Squares with QR Decomposition 49
2.16 Exercises 52
Chapter 3: Risk and Return in the One-Period Model 55
3.1 Utility Functions 56
3.2 Expected Utility Maximization 59
3.3 The Existence of Optimal Portfolios 61
3.4 Reporting Expected Utility in Terms of Money 62
3.5 Normalized Utility and Investment Potential 63
3.6 Quadratic Utility 67
3.7 The Sharpe Ratio 69
3.8 Arbitrage-Adjusted Sharpe Ratio 71
3.9 The Importance of Arbitrage Adjustment 75
3.10 Portfolio Choice with Near-Arbitrage Opportunities 77
3.11 Summary 79
3.12 Notes 81
3.13 Exercises 82
Chapter 4: Numerical Techniques for Optimal Portfolio Selection in Incomplete Markets 84
4.1 Sensitivity Analysis of Portfolio Decisions with the CRRA Utility 84
4.2 Newton's Algorithm for Optimal Investment with CRRA Utility 88
4.3 Optimal CRRA Investment Using Empirical Return Distribution 90
4.4 HARA Portfolio Optimizer 94
4.5 HARA Portfolio Optimization with Several Risky Assets 96
4.6 Quadratic Utility Maximization with Multiple Assets 99
4.7 Summary 102
4.8 Notes 102
4.9 Exercises 102
Chapter 5: Pricing in Dynamically Complete Markets 104
5.1 Options and Portfolio Insurance 104
5.2 Option Pricing 105
5.3 Dynamic Replicating Trading Strategy 108
5.4 Risk-Neutral Probabilities in a Multi-Period Model 116
5.5 The Law of Iterated Expectations 119
5.6 Summary 121
5.7 Notes 121
5.8 Exercises 121
Chapter 6: Towards Continuous Time 125
6.1 IID Returns, and the Term Structure of Volatility 125
6.2 Towards Brownian Motion 127
6.3 Towards a Poisson Jump Process 136
6.4 Central Limit Theorem and Infinitely Divisible Distributions 142
6.5 Summary 143
6.6 Notes 145
6.7 Exercises 145
Chapter 7: Fast Fourier Transform 147
7.1 Introduction to Complex Numbers and the Fourier Transform 147
7.2 Discrete Fourier Transform (DFT) 152
7.3 Fourier Transforms in Finance 153
7.4 Fast Pricing via the Fast Fourier Transform (FFT) 158
7.5 Further Applications of FFTs in Finance 162
7.6 Notes 166
7.7 Appendix 167
7.8 Exercises 169
Chapter 8: Information Management 170
8.1 Information: Too Much of a Good Thing? 170
8.2 Model-Independent Properties of Conditional Expectation 174
8.3 Summary 178
8.4 Notes 179
8.5 Appendix: Probability Space 179
8.6 Exercises 183
Chapter 9: Martingales and Change of Measure in Finance 187
9.1 Discounted Asset Prices Are Martingales 187
9.2 Dynamic Arbitrage Theorem 192
9.3 Change of Measure 193
9.4 Dynamic Optimal Portfolio Selection in a Complete Market 198
9.5 Summary 206
9.6 Notes 208
9.7 Exercises 208
Chapter 10: Brownian Motion and Itˆo Formulae 213
10.1 Continuous-Time Brownian Motion 213
10.2 Stochastic Integration and Itˆo Processes 218
10.3 Important Itˆo Processes 220
10.4 Function of a Stochastic Process: the Itˆo Formula 222
10.5 Applications of the Itˆo Formula 223
10.6 Multivariate Itˆo Formula 225
10.7 Itˆo Processes as Martingales 228
10.8 Appendix: Proof of the Itˆo Formula 229
10.9 Summary 229
10.10 Notes 230
10.11 Exercises 231
Chapter 11: Continuous-Time Finance 233
11.1 Summary of Useful Results 233
11.2 Risk-Neutral Pricing 234
11.3 The Girsanov Theorem 237
11.4 Risk-Neutral Pricing and Absence of Arbitrage 241
11.5 Automatic Generation of PDEs and the Feynman-Kac Formula 246
11.6 Overview of Numerical Methods 250
11.7 Summary 251
11.8 Notes 252
11.9 Appendix: Decomposition of Asset Returns into Uncorrelated Components 252
11.10 Exercises 255
Chapter 12: Finite-Difference Methods 261
12.1 Interpretation of PDEs 261
12.2 The Explicit Method 263
12.3 Instability 264
12.4 Markov Chains and Local Consistency 266
12.5 Improving Convergence by Richardson's Extrapolation 268
12.6 Oscillatory Convergence Due to Grid Positioning 269
12.7 Fully Implicit Scheme 270
12.8 Crank-Nicolson Scheme 273
12.9 Summary 274
12.10 Notes 276
12.11 Appendix: Efficient Gaussian Elimination for Tridiagonal Matrices 276
12.12 Appendix: Richardson's Extrapolation 277
12.13 Exercises 277
Chapter 13: Dynamic Option Hedging and Pricing in Incomplete Markets 280
13.1 The Risk in Option Hedging Strategies 280
13.2 Incomplete Market Option Price Bounds 299
13.3 Towards Continuous Time 304
13.4 Derivation of Optimal Hedging Strategy 309
13.5 Summary 318
13.6 Notes 319
13.7 Appendix: Expected Squared Hedging Error in the Black-Scholes Model 320
13.8 Exercises 322
Appendix A Calculus 326
A.1 Notation 326
A.2 Differentiation 329
A.3 Real Function of Several Real Variables 332
A.4 Power Series Approximations 334
A.5 Optimization 336
A.6 Integration 338
A.7 Exercises 344
Appendix B Probability 348
B.1 Probability Space 348
B.2 Conditional Probability 348
B.3 Marginal and Joint Distribution 351
B.4 Stochastic Independence 352
B.5 Expectation Operator 354
B.6 Properties of Expectation 355
B.7 Mean and Variance 356
B.8 Covariance and Correlation 357
B.9 Continuous Random Variables 360
B.10 Normal Distribution 364
B.11 Quantiles 370
B.12 Relationships among Standard Statistical Distributions 371
B.13 Notes 372
B.14 Exercises 372
References 381
Index 385
