Mathematical techniques in finance: tools for incomplete markets
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Princeton, NJ [u.a.]
Princeton Univ. Press
2009
|
| Edition: | 2. ed. |
| Subjects: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017680902&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Physical Description: | XX, 390 S. graph. Darst. |
| ISBN: | 9780691141213 0691141215 |
Staff View
MARC
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| 100 | 1 | |a Černý, Aleš |d 1971- |e Verfasser |0 (DE-588)130444898 |4 aut | |
| 245 | 1 | 0 | |a Mathematical techniques in finance |b tools for incomplete markets |c Aleš Černý |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Princeton, NJ [u.a.] |b Princeton Univ. Press |c 2009 | |
| 300 | |a XX, 390 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 650 | 4 | |a Mathematik | |
| 650 | 4 | |a Mathematisches Modell | |
| 650 | 4 | |a Derivative securities |x Mathematics | |
| 650 | 4 | |a Finance |x Mathematical models | |
| 650 | 4 | |a Pricing |x Mathematical models | |
| 650 | 4 | |a Risk management |x Mathematical models | |
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Record in the Search Index
| DE-BY-TUM_call_number | 0048 WIR 160f 2010 A 9166(2) |
|---|---|
| DE-BY-TUM_katkey | 1745881 |
| DE-BY-TUM_location | LSB |
| DE-BY-TUM_media_number | 040010200921 |
| _version_ | 1821934883457466368 |
| adam_text | Contents
Preface
to the Second Edition
xiii
From the Preface to the First Edition
xix
lhe
í
1.1
simplest Model or Financial Markets
One-Period Finite State Model
1
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
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-Neutrai 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
x
Contents
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
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
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
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
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
156
7.7
Appendix j67
7.8
Exercises J69
Contents xi
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
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
10
Brownian Motion and
Ito
Formulae
213
10.1
Continuous-Time Brownian Motion
213
10.2
Stochastic Integration and
Ito
Processes
218
10.3
Important
Ito
Processes
220
10.4
Function of a Stochastic Process: the
Ito
Formula
222
10.5
Applications of the
Ito
Formula
223
10.6
Multivariate
Ito
Formula
225
10.7
Ito
Processes as Martingales
228
10.8
Appendix: Proof of the
Ito
Formula
229
10.9
Summary
229
10.10
Notes
230
10.11
Exercises
231
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
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-Nieolson Scheme
273
12.9
Summary
274
12.10
Notes
276
12.11
Appendix: Efficient Gaussian Elimination for Tridiagonal Matrices
276
xii Contents
12.12 Appendix:
Richardson s
Extrapolation 277
12.13
Exercises
277
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
В
Probability
348
B.I 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
335
|
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| discipline | Mathematik Wirtschaftswissenschaften |
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| id | DE-604.BV035625852 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T13:39:12Z |
| institution | BVB |
| isbn | 9780691141213 0691141215 |
| language | English |
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| owner_facet | DE-355 DE-BY-UBR DE-N2 DE-384 DE-M382 DE-11 DE-91G DE-BY-TUM DE-29T |
| physical | XX, 390 S. graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Princeton Univ. Press |
| record_format | marc |
| spellingShingle | Černý, Aleš 1971- Mathematical techniques in finance tools for incomplete markets Mathematik Mathematisches Modell Derivative securities Mathematics Finance Mathematical models Pricing Mathematical models Risk management Mathematical models Finanzmathematik (DE-588)4017195-4 gnd |
| subject_GND | (DE-588)4017195-4 |
| title | Mathematical techniques in finance tools for incomplete markets |
| title_auth | Mathematical techniques in finance tools for incomplete markets |
| title_exact_search | Mathematical techniques in finance tools for incomplete markets |
| title_full | Mathematical techniques in finance tools for incomplete markets Aleš Černý |
| title_fullStr | Mathematical techniques in finance tools for incomplete markets Aleš Černý |
| title_full_unstemmed | Mathematical techniques in finance tools for incomplete markets Aleš Černý |
| title_short | Mathematical techniques in finance |
| title_sort | mathematical techniques in finance tools for incomplete markets |
| title_sub | tools for incomplete markets |
| topic | Mathematik Mathematisches Modell Derivative securities Mathematics Finance Mathematical models Pricing Mathematical models Risk management Mathematical models Finanzmathematik (DE-588)4017195-4 gnd |
| topic_facet | Mathematik Mathematisches Modell Derivative securities Mathematics Finance Mathematical models Pricing Mathematical models Risk management Mathematical models Finanzmathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017680902&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT cernyales mathematicaltechniquesinfinancetoolsforincompletemarkets |
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