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Tables of Contents for Mathematical Techniques in Finance
Chapter/Section Title
Page #
Page Count
Preface
xiii
1 The Simplest Model of Financial Markets
1
24
1.1 One-Period Finite State Model
1
2
1.2 Securities and Their Pay-Offs
3
1
1.3 Securities as Vectors
3
1
1.4 Operations on Securities
4
2
1.5 The Matrix as a Collection of Securities
6
1
1.6 Transposition
6
2
1.7 Matrix Multiplication and Portfolios
8
2
1.8 Systems of Equations and Hedging
10
2
1.9 Linear Independence and Redundant Securities
12
2
1.10 The Structure of the Marketed Subspace
14
2
1.11 The Identity Matrix and Arrow-Debreu Securities
16
1
1.12 Matrix Inverse
17
1
1.13 Inverse Matrix and Replicating Portfolios
17
2
1.14 Complete Market Hedging Formula
19
1
1.15 Summary
20
1
1.16 Notes
21
1
1.17 Exercises
21
4
2 Arbitrage and Pricing in the One-Period Model
25
30
2.1 Hedging with Redundant Securities and Incomplete Market
25
4
2.2 Finding the Best Approximate Hedge
29
3
2.3 Minimizing the Expected Squared Replication Error
32
2
2.4 Numerical Stability of Least Squares
34
3
2.5 Asset Prices, Returns and Portfolio Units
37
1
2.6 Arbitrage
38
2
2.7 No-Arbitrage Pricing
40
2
2.8 State Prices and the Arbitrage Theorem
42
3
2.9 State Prices and Asset Returns
45
1
2.10 Risk-Neutral Probabilities
45
1
2.11 State Prices and No-Arbitrage Pricing
46
2
2.12 Summary
48
1
2.13 Notes
49
1
2.14 Appendix: Least Squares with QR Decomposition
49
3
2.15 Exercises
52
3
3 Risk and Return in the One-Period Model
55
32
3.1 Utility Functions
55
4
3.2 Expected Utility Maximization
59
1
3.3 Reporting Expected Utility in Terms of Money
60
2
3.4 Scale-Free Formulation of the Optimal Investment Problem with the HARA Utility
62
4
3.5 Quadratic Utility
66
4
3.6 Reporting Investment Potential in Terms of Sharpe Ratios
70
7
3.7 The Importance of Arbitrage Adjustment
77
1
3.8 Portfolio Choice with Near-Arbitrage Opportunities
78
4
3.9 Generalization of the Sharpe Ratio
82
1
3.10 Summary
83
1
3.11 Notes
84
1
3.12 Exercises
85
2
4 Numerical Techniques for Optimal Portfolio Selection in Incomplete Markets
87
22
4.1 Sensitivity Analysis of Portfolio Decisions with the CRRA Utility
87
4
4.2 Newton's Algorithm for Optimal Investment with CRRA Utility
91
2
4.3 Optimal CRRA Investment Using Empirical Return Distribution
93
6
4.4 HARA Portfolio Optimizer
99
1
4.5 HARA Portfolio Optimization with Several Risky Assets
100
4
4.6 Quadratic Utility Maximization with Multiple Assets
104
2
4.7 Summary
106
1
4.8 Notes
107
1
4.9 Exercises
107
2
5 Pricing in Dynamically Complete Markets
109
22
5.1 Options and Portfolio Insurance
109
1
5.2 Option Pricing
110
3
5.3 Dynamic Replicating Trading Strategy
113
8
5.4 Risk-Neutral Probabilities in a Multi-Period Model
121
3
5.5 The Law of Iterated Expectations
124
2
5.6 Summary
126
1
5.7 Notes
126
1
5.8 Exercises
126
5
6 Towards Continuous Time
131
22
6.1 IID Returns, and the Term Structure of Volatility
131
2
6.2 Towards Brownian Motion
133
9
6.3 Towards a Poisson Jump Process
142
6
6.4 Central Limit Theorem and Infinitely Divisible Distributions
148
1
6.5 Summary
149
2
6.6 Notes
151
1
6.7 Exercises
151
2
7 Fast Fourier Transform
153
7.1 Introduction to Complex Numbers and the Fourier Transform
153
5
7.2 Discrete Fourier Transform (DFT)
158
1
7.3 Fourier Transforms in Finance
159
5
7.4 Fast Pricing via the Fast Fourier Transform (FFT)
164
3
7.5 Further Applications of FFTs in Finance
167
4
7.6 Notes
171
1
7.7 Appendix
172
2
7.8 Exercises
174
8 Information Management
115
78
8.1 Information: Too Much of a Good Thing?
175
4
8.2 Model-Independent Properties of Conditional Expectation
179
4
8.3 Summary
183
1
8.4 Notes
184
1
8.5 Appendix: Probability Space
184
4
8.6 Exercises
188
5
9 Martingales and Change of Measure in Finance
193
26
9.1 Discounted Asset Prices Are Martingales
193
5
9.2 Dynamic Arbitrage Theorem
198
1
9.3 Change of Measure
199
5
9.4 Dynamic Optimal Portfolio Selection in a Complete Market
204
8
9.5 Summary
212
2
9.6 Notes
214
1
9.7 Exercises
214
5
10 Brownian Motion and Itô Formulae
219
20
10.1 Continuous-Time Brownian Motion
219
5
10.2 Stochastic Integration and Itô Processes
224
2
10.3 Important Itô Processes
226
2
10.4 Function of a Stochastic Process: the Itô Formula
228
1
10.5 Applications of the Itô Formula
229
2
10.6 Multivariate Itô Formula
231
3
10.7 Itô Processes as Martingales
234
1
10.8 Appendix: Proof of the Itô Formula
235
1
10.9 Summary
235
1
10.10 Notes
236
1
10.11 Exercises
237
2
11 Continuous-Time Finance
239
28
11.1 Summary of Useful Results
239
1
11.2 Risk-Neutral Pricing
240
3
11.3 The Girsanov Theorem
243
4
11.4 Risk-Neutral Pricing and Absence of Arbitrage
247
5
11.5 Automatic Generation of PDEs and the Feynman-Kac Formula
252
4
11.6 Overview of Numerical Methods
256
1
11.7 Summary
257
1
11.8 Notes
258
1
11.9 Appendix: Decomposition of Asset Returns into Uncorrelated Components
258
3
11.10 Exercises
261
6
12 Dynamic Option Hedging and Pricing in Incomplete Markets
267
46
12.1 The Risk in Option Hedging Strategies
267
16
12.2 Incomplete Market Option Price Bounds
283
8
12.3 Towards Continuous Time
291
6
12.4 Derivation of Optimal Hedging Strategy
297
9
12.5 Summary
306
1
12.6 Notes
307
1
12.7 Appendix: Expected Squared Hedging Error in the Black-Scholes Model
307
2
12.8 Exercises
309
4
Appendix A Calculus
313
24
A.1 Notation
313
3
A.2 Differentiation
316
3
A.3 Real Function of Several Real Variables
319
2
A.4 Power Series Approximations
321
3
A.5 Optimization
324
2
A.6 Integration
326
6
A.7 Exercises
332
5
Appendix B Probability
337
32
B.1 Probability Space
337
1
B.2 Conditional Probability
337
3
B.3 Marginal and Joint Distribution
340
1
B.4 Stochastic Independence
341
2
B.5 Expectation Operator
343
1
B.6 Properties of Expectation
344
1
B.7 Mean and Variance
345
1
B.8 Covariance and Correlation
346
3
B.9 Continuous Random Variables
349
5
B.10 Normal Distribution
354
5
B.11 Quantiles
359
1
B.12 Relationships among Standard Statistical Distributions
360
1
B.13 Notes
361
1
B.14 Exercises
361
8
References
369
4
Index
373