Mathematical Techniques in Finance: Tools for Incomplete Markets (Second Edition)

Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Dedication......Page 6
Contents......Page 10
Preface to the Second Edition......Page 14
From the Preface to the First Edition......Page 20
1.1 One-Period Finite State Model......Page 24
1.3 Securities as Vectors......Page 26
1.4 Operations on Securities......Page 27
1.6 Transposition......Page 29
1.7 Matrix Multiplication and Portfolios......Page 31
1.8 Systems of Equations and Hedging......Page 33
1.9 Linear Independence and Redundant Securities......Page 35
1.10 The Structure of the Marketed Subspace......Page 37
1.11 The Identity Matrix and Arrow-Debreu Securities......Page 39
1.13 Inverse Matrix and Replicating Portfolios......Page 40
1.14 Complete Market Hedging Formula......Page 42
1.15 Summary......Page 43
1.16 Notes......Page 44
1.17 Exercises......Page 45
2.1 Hedging with Redundant Securities and Incomplete Market......Page 48
2.2 Finding the Best Approximate Hedge......Page 52
2.3 Minimizing the Expected Squared Replication Error......Page 55
2.4 Numerical Stability of Least Squares......Page 57
2.5 Asset Prices, Returns and Portfolio Units......Page 59
2.6 Arbitrage......Page 61
2.7 No-Arbitrage Pricing......Page 63
2.8 State Prices and the Arbitrage Theorem......Page 64
2.9 State Prices and Asset Returns......Page 67
2.10 Risk-Neutral Probabilities......Page 68
2.11 State Prices and No-Arbitrage Pricing......Page 69
2.12 Asset Pricing Duality......Page 70
2.13 Summary......Page 71
2.15 Appendix: Least Squares with QR Decomposition......Page 72
2.16 Exercises......Page 75
3 Risk and Return in the One-Period Model......Page 78
3.1 Utility Functions......Page 79
3.2 Expected Utility Maximization......Page 82
3.3 The Existence of Optimal Portfolios......Page 84
3.4 Reporting Expected Utility in Terms of Money......Page 85
3.5 Normalized Utility and Investment Potential......Page 86
3.6 Quadratic Utility......Page 90
3.7 The Sharpe Ratio......Page 92
3.8 Arbitrage-Adjusted Sharpe Ratio......Page 94
3.9 The Importance of Arbitrage Adjustment......Page 98
3.10 Portfolio Choice with Near-Arbitrage Opportunities......Page 100
3.11 Summary......Page 102
3.12 Notes......Page 104
3.13 Exercises......Page 105
4.1 Sensitivity Analysis of Portfolio Decisions with the CRRA Utility......Page 107
4.2 Newton's Algorithm for Optimal Investment with CRRA Utility......Page 111
4.3 Optimal CRRA Investment Using Empirical Return Distribution......Page 113
4.4 HARA Portfolio Optimizer......Page 117
4.5 HARA Portfolio Optimization with Several Risky Assets......Page 119
4.6 Quadratic Utility Maximization with Multiple Assets......Page 122
4.9 Exercises......Page 125
5.1 Options and Portfolio Insurance......Page 127
5.2 Option Pricing......Page 128
5.3 Dynamic Replicating Trading Strategy......Page 131
5.4 Risk-Neutral Probabilities in a Multi-Period Model......Page 139
5.5 The Law of Iterated Expectations......Page 142
5.8 Exercises......Page 144
6.1 IID Returns, and the Term Structure of Volatility......Page 148
6.2 Towards Brownian Motion......Page 150
6.3 Towards a Poisson Jump Process......Page 159
6.4 Central Limit Theorem and Infinitely Divisible Distributions......Page 165
6.5 Summary......Page 166
6.7 Exercises......Page 168
7.1 Introduction to Complex Numbers and the Fourier Transform......Page 170
7.2 Discrete Fourier Transform (DFT)......Page 175
7.3 Fourier Transforms in Finance......Page 176
7.4 Fast Pricing via the Fast Fourier Transform (FFT)......Page 181
7.5 Further Applications of FFTs in Finance......Page 185
7.6 Notes......Page 189
7.7 Appendix......Page 190
7.8 Exercises......Page 192
8.1 Information: Too Much of a Good Thing?......Page 193
8.2 Model-Independent Properties of Conditional Expectation......Page 197
8.3 Summary......Page 201
8.5 Appendix: Probability Space......Page 202
8.6 Exercises......Page 206
9.1 Discounted Asset Prices Are Martingales......Page 210
9.2 Dynamic Arbitrage Theorem......Page 215
9.3 Change of Measure......Page 216
9.4 Dynamic Optimal Portfolio Selection in a Complete Market......Page 221
9.5 Summary......Page 229
9.7 Exercises......Page 231
10.1 Continuous-Time Brownian Motion......Page 236
10.2 Stochastic Integration and Ito Processes......Page 241
10.3 Important Ito Processes......Page 243
10.4 Function of a Stochastic Process: the Ito Formula......Page 245
10.5 Applications of the Ito Formula......Page 246
10.6 Multivariate Ito Formula......Page 248
10.7 Ito Processes as Martingales......Page 251
10.9 Summary......Page 252
10.10 Notes......Page 253
10.11 Exercises......Page 254
11.1 Summary of Useful Results......Page 256
11.2 Risk-Neutral Pricing......Page 257
11.3 The Girsanov Theorem......Page 260
11.4 Risk-Neutral Pricing and Absence of Arbitrage......Page 264
11.5 Automatic Generation of PDEs and the Feynman-Kac Formula......Page 269
11.6 Overview of Numerical Methods......Page 273
11.7 Summary......Page 274
11.9 Appendix: Decomposition of Asset Returns into Uncorrelated Components......Page 275
11.10 Exercises......Page 278
12.1 Interpretation of PDEs......Page 284
12.2 The Explicit Method......Page 286
12.3 Instability......Page 287
12.4 Markov Chains and Local Consistency......Page 289
12.5 Improving Convergence by Richardson's Extrapolation......Page 291
12.6 Oscillatory Convergence Due to Grid Positioning......Page 292
12.7 Fully Implicit Scheme......Page 293
12.8 Crank-Nicolson Scheme......Page 296
12.9 Summary......Page 297
12.11 Appendix: Efficient Gaussian Elimination for Tridiagonal Matrices......Page 299
12.13 Exercises......Page 300
13.1 The Risk in Option Hedging Strategies......Page 303
13.2 Incomplete Market Option Price Bounds......Page 322
13.3 Towards Continuous Time......Page 327
13.4 Derivation of Optimal Hedging Strategy......Page 332
13.5 Summary......Page 341
13.6 Notes......Page 342
13.7 Appendix: Expected Squared Hedging Error in the Black-Scholes Model......Page 343
13.8 Exercises......Page 345
A.1 Notation......Page 349
A.2 Differentiation......Page 352
A.3 Real Function of Several Real Variables......Page 355
A.4 Power Series Approximations......Page 357
A.5 Optimization......Page 359
A.6 Integration......Page 361
A.7 Exercises......Page 367
B.2 Conditional Probability......Page 371
B.3 Marginal and Joint Distribution......Page 374
B.4 Stochastic Independence......Page 375
B.5 Expectation Operator......Page 377
B.6 Properties of Expectation......Page 378
B.7 Mean and Variance......Page 379
B.8 Covariance and Correlation......Page 380
B.9 Continuous Random Variables......Page 383
B.10 Normal Distribution......Page 387
B.11 Quantiles......Page 393
B.12 Relationships among Standard Statistical Distributions......Page 394
B.14 Exercises......Page 395
References......Page 404
Index......Page 408