Tools for computational finance:
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| Format: | Electronic eBook |
|---|---|
| Language: | English |
| Published: |
Berlin ; Heidelberg ; New York
Springer
2006
|
| Edition: | 3. ed. |
| Series: | Universitext
|
| Subjects: | |
| Online Access: | BTU01 TUM01 UBA01 UBM01 UBR01 UBT01 UPA01 Volltext Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 283 - 292 |
| Physical Description: | 1 Online-Ressource (XIX, 299 S.) graph. Darst. |
| ISBN: | 9783540279235 9783540279266 |
| DOI: | 10.1007/3-540-27926-1 |
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Record in the Search Index
| _version_ | 1804136432105684992 |
|---|---|
| adam_text | Contents
Prefaces V
Contents XIII
Notation XVII
Chapter 1 Modeling Tools for Financial Options 1
1.1 Options 1
1.2 Model of the Financial Market 8
1.3 Numerical Methods 10
1.4 The Binomial Method 12
1.5 Risk Neutral Valuation 21
1.6 Stochastic Processes 25
1.6.1 Wiener Process 26
1.6.2 Stochastic Integral 28
1.7 Stochastic Differential Equations 31
1.7.1 Ito Process 31
1.7.2 Application to the Stock Market 33
1.7.3 Risk Neutral Valuation 36
1.7.4 Mean Reversion 37
1.7.5 Vector Valued SDEs 39
1.8 Ito Lemma and Implications 40
1.9 Jump Processes 45
Notes and Comments 48
Exercises 52
Chapter 2 Generating Random Numbers with Specified
Distributions 61
2.1 Uniform Deviates 62
2.1.1 Linear Congruential Generators 62
2.1.2 Quality of Generators 63
2.1.3 Random Vectors and Lattice Structure 64
2.1.4 Fibonacci Generators 67
2.2 Transformed Random Variables 69
2.2.1 Inversion 69
2.2.2 Transformations in H1 70
XIV Contents
2.2.3 Transformation in Mn 72
2.3 Normally Distributed Random Variables 72
2.3.1 Method of Box and Muller 72
2.3.2 Variant of Marsaglia 73
2.3.3 Correlated Random Variables 75
2.4 Monte Carlo Integration 77
2.5 Sequences of Numbers with Low Discrepancy 79
2.5.1 Discrepancy 79
2.5.2 Examples of Low Discrepancy Sequences 82
Notes and Comments 85
Exercises 87
Chapter 3 Simulation with Stochastic Differential
Equations 91
3.1 Approximation Error 92
3.2 Stochastic Taylor Expansion 95
3.3 Examples of Numerical Methods 98
3.4 Intermediate Values 102
3.5 Monte Carlo Simulation 102
3.5.1 Integral Representation 103
3.5.2 The Basic Version for European Options 104
3.5.3 Bias 107
3.5.4 Variance Reduction 108
3.5.5 American Options Ill
3.5.6 Further Hints 116
Notes and Comments 117
Exercises 119
Chapter 4 Standard Methods for Standard Options 123
4.1 Preparations 124
4.2 Foundations of Finite Difference Methods 126
4.2.1 Difference Approximation 126
4.2.2 The Grid 127
4.2.3 Explicit Method 128
4.2.4 Stability 130
4.2.5 An Implicit Method 133
4.3 Crank Nicolson Method 135
4.4 Boundary Conditions 138
4.5 American Options as Free Boundary Problems 140
4.5.1 Early Exercise Curve 141
4.5.2 Free Boundary Problems 143
4.5.3 Black Scholes Inequality 146
4.5.4 Obstacle Problems 148
4.5.5 Linear Complementarity for American Put Options . 151
Contents XV
4.6 Computation of American Options 152
4.6.1 Discretization with Finite Differences 152
4.6.2 Iterative Solution 154
4.6.3 An Algorithm for Calculating American Options .... 157
4.7 On the Accuracy 161
4.7.1 Elementary Error Control 162
4.7.2 Extrapolation 165
4.8 Analytic Methods 165
4.8.1 Approximation Based on Interpolation 167
4.8.2 Quadratic Approximation 169
4.8.3 Analytic Method of Lines 172
4.8.4 Methods Evaluating Probabilities 173
Notes and Comments 174
Exercises 178
Chapter 5 Finite Element Methods 183
5.1 Weighted Residuals 184
5.1.1 The Principle of Weighted Residuals 184
5.1.2 Examples of Weighting Functions 186
5.1.3 Examples of Basis Functions 187
5.2 Galerkin Approach with Hat Functions 188
5.2.1 Hat Functions 189
5.2.2 Assembling 191
5.2.3 A Simple Application 192
5.3 Application to Standard Options 194
5.4 Error Estimates 198
5.4.1 Classical and Weak Solutions 199
5.4.2 Approximation on Finite Dimensional Subspaces . . . 201
5.4.3 Cea s Lemma 202
Notes and Comments 205
Exercises 206
Chapter 6 Pricing of Exotic Options 209
6.1 Exotic Options 210
6.2 Options Depending on Several Assets 211
6.3 Asian Options 214
6.3.1 The Payoff 214
6.3.2 Modeling in the Black Scholes Framework 215
6.3.3 Reduction to a One Dimensional Equation 216
6.3.4 Discrete Monitoring 220
6.4 Numerical Aspects 222
6.4.1 Convection Diffusion Problems 222
6.4.2 Von Neumann Stability Analysis 225
6.5 Upwind Schemes and Other Methods 226
XVI Contents
6.5.1 Upwind Scheme 226
6.5.2 Dispersion 230
6.6 High Resolution Methods 231
6.6.1 The Lax Wendroff Method 231
6.6.2 Total Variation Diminishing 232
6.6.3 Numerical Dissipation 233
Notes and Comments 235
Exercises 237
Appendices 239
A Financial Derivatives 239
Al Investment and Risk 239
A2 Financial Derivatives 240
A3 Forwards and the No Arbitrage Principle 243
A4 The Black Scholes Equation 244
A5 Early Exercise Curve 249
B Stochastic Tools 253
Bl Essentials of Stochastics 253
B2 Advanced Topics 257
B3 State Price Process 260
C Numerical Methods 265
Cl Basic Numerical Tools 265
C2 Iterative Methods for Ax = b 270
C3 Function Spaces 272
D Complementary Material 277
Dl Bounds for Options 277
D2 Approximation Formula 279
D3 Software 281
References 283
Index 293
|
| adam_txt |
Contents
Prefaces V
Contents XIII
Notation XVII
Chapter 1 Modeling Tools for Financial Options 1
1.1 Options 1
1.2 Model of the Financial Market 8
1.3 Numerical Methods 10
1.4 The Binomial Method 12
1.5 Risk Neutral Valuation 21
1.6 Stochastic Processes 25
1.6.1 Wiener Process 26
1.6.2 Stochastic Integral 28
1.7 Stochastic Differential Equations 31
1.7.1 Ito Process 31
1.7.2 Application to the Stock Market 33
1.7.3 Risk Neutral Valuation 36
1.7.4 Mean Reversion 37
1.7.5 Vector Valued SDEs 39
1.8 Ito Lemma and Implications 40
1.9 Jump Processes 45
Notes and Comments 48
Exercises 52
Chapter 2 Generating Random Numbers with Specified
Distributions 61
2.1 Uniform Deviates 62
2.1.1 Linear Congruential Generators 62
2.1.2 Quality of Generators 63
2.1.3 Random Vectors and Lattice Structure 64
2.1.4 Fibonacci Generators 67
2.2 Transformed Random Variables 69
2.2.1 Inversion 69
2.2.2 Transformations in H1 70
XIV Contents
2.2.3 Transformation in Mn 72
2.3 Normally Distributed Random Variables 72
2.3.1 Method of Box and Muller 72
2.3.2 Variant of Marsaglia 73
2.3.3 Correlated Random Variables 75
2.4 Monte Carlo Integration 77
2.5 Sequences of Numbers with Low Discrepancy 79
2.5.1 Discrepancy 79
2.5.2 Examples of Low Discrepancy Sequences 82
Notes and Comments 85
Exercises 87
Chapter 3 Simulation with Stochastic Differential
Equations 91
3.1 Approximation Error 92
3.2 Stochastic Taylor Expansion 95
3.3 Examples of Numerical Methods 98
3.4 Intermediate Values 102
3.5 Monte Carlo Simulation 102
3.5.1 Integral Representation 103
3.5.2 The Basic Version for European Options 104
3.5.3 Bias 107
3.5.4 Variance Reduction 108
3.5.5 American Options Ill
3.5.6 Further Hints 116
Notes and Comments 117
Exercises 119
Chapter 4 Standard Methods for Standard Options 123
4.1 Preparations 124
4.2 Foundations of Finite Difference Methods 126
4.2.1 Difference Approximation 126
4.2.2 The Grid 127
4.2.3 Explicit Method 128
4.2.4 Stability 130
4.2.5 An Implicit Method 133
4.3 Crank Nicolson Method 135
4.4 Boundary Conditions 138
4.5 American Options as Free Boundary Problems 140
4.5.1 Early Exercise Curve 141
4.5.2 Free Boundary Problems 143
4.5.3 Black Scholes Inequality 146
4.5.4 Obstacle Problems 148
4.5.5 Linear Complementarity for American Put Options . 151
Contents XV
4.6 Computation of American Options 152
4.6.1 Discretization with Finite Differences 152
4.6.2 Iterative Solution 154
4.6.3 An Algorithm for Calculating American Options . 157
4.7 On the Accuracy 161
4.7.1 Elementary Error Control 162
4.7.2 Extrapolation 165
4.8 Analytic Methods 165
4.8.1 Approximation Based on Interpolation 167
4.8.2 Quadratic Approximation 169
4.8.3 Analytic Method of Lines 172
4.8.4 Methods Evaluating Probabilities 173
Notes and Comments 174
Exercises 178
Chapter 5 Finite Element Methods 183
5.1 Weighted Residuals 184
5.1.1 The Principle of Weighted Residuals 184
5.1.2 Examples of Weighting Functions 186
5.1.3 Examples of Basis Functions 187
5.2 Galerkin Approach with Hat Functions 188
5.2.1 Hat Functions 189
5.2.2 Assembling 191
5.2.3 A Simple Application 192
5.3 Application to Standard Options 194
5.4 Error Estimates 198
5.4.1 Classical and Weak Solutions 199
5.4.2 Approximation on Finite Dimensional Subspaces . . . 201
5.4.3 Cea's Lemma 202
Notes and Comments 205
Exercises 206
Chapter 6 Pricing of Exotic Options 209
6.1 Exotic Options 210
6.2 Options Depending on Several Assets 211
6.3 Asian Options 214
6.3.1 The Payoff 214
6.3.2 Modeling in the Black Scholes Framework 215
6.3.3 Reduction to a One Dimensional Equation 216
6.3.4 Discrete Monitoring 220
6.4 Numerical Aspects 222
6.4.1 Convection Diffusion Problems 222
6.4.2 Von Neumann Stability Analysis 225
6.5 Upwind Schemes and Other Methods 226
XVI Contents
6.5.1 Upwind Scheme 226
6.5.2 Dispersion 230
6.6 High Resolution Methods 231
6.6.1 The Lax Wendroff Method 231
6.6.2 Total Variation Diminishing 232
6.6.3 Numerical Dissipation 233
Notes and Comments 235
Exercises 237
Appendices 239
A Financial Derivatives 239
Al Investment and Risk 239
A2 Financial Derivatives 240
A3 Forwards and the No Arbitrage Principle 243
A4 The Black Scholes Equation 244
A5 Early Exercise Curve 249
B Stochastic Tools 253
Bl Essentials of Stochastics 253
B2 Advanced Topics 257
B3 State Price Process 260
C Numerical Methods 265
Cl Basic Numerical Tools 265
C2 Iterative Methods for Ax = b 270
C3 Function Spaces 272
D Complementary Material 277
Dl Bounds for Options 277
D2 Approximation Formula 279
D3 Software 281
References 283
Index 293 |
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| discipline_str_mv | Informatik Mathematik Wirtschaftswissenschaften |
| doi_str_mv | 10.1007/3-540-27926-1 |
| edition | 3. ed. |
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| id | DE-604.BV022377133 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T17:09:43Z |
| indexdate | 2024-07-09T20:56:18Z |
| institution | BVB |
| isbn | 9783540279235 9783540279266 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015586187 |
| oclc_num | 850842987 |
| open_access_boolean | |
| owner | DE-739 DE-355 DE-BY-UBR DE-634 DE-91 DE-BY-TUM DE-384 DE-703 DE-83 DE-19 DE-BY-UBM |
| owner_facet | DE-739 DE-355 DE-BY-UBR DE-634 DE-91 DE-BY-TUM DE-384 DE-703 DE-83 DE-19 DE-BY-UBM |
| physical | 1 Online-Ressource (XIX, 299 S.) graph. Darst. |
| psigel | ZDB-2-SMA |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Springer |
| record_format | marc |
| series2 | Universitext |
| spelling | Tools for computational finance Rüdiger U. Seydel 3. ed. Berlin ; Heidelberg ; New York Springer 2006 1 Online-Ressource (XIX, 299 S.) graph. Darst. txt rdacontent c rdamedia cr rdacarrier Universitext Literaturverz. S. 283 - 292 Black-Scholes-Modell (DE-588)4206283-4 gnd rswk-swf Wertpapieranalyse (DE-588)4124458-8 gnd rswk-swf Stochastisches Modell (DE-588)4057633-4 gnd rswk-swf Finanzmathematik (DE-588)4017195-4 gnd rswk-swf Derivat Wertpapier (DE-588)4381572-8 gnd rswk-swf Optionspreistheorie (DE-588)4135346-8 gnd rswk-swf Black-Scholes-Modell (DE-588)4206283-4 s Optionspreistheorie (DE-588)4135346-8 s DE-604 Wertpapieranalyse (DE-588)4124458-8 s Stochastisches Modell (DE-588)4057633-4 s 1\p DE-604 Finanzmathematik (DE-588)4017195-4 s Derivat Wertpapier (DE-588)4381572-8 s 2\p DE-604 Seydel, Rüdiger 1947- Sonstige (DE-588)13662782X oth https://doi.org/10.1007/3-540-27926-1 Verlag Volltext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015586187&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Tools for computational finance Black-Scholes-Modell (DE-588)4206283-4 gnd Wertpapieranalyse (DE-588)4124458-8 gnd Stochastisches Modell (DE-588)4057633-4 gnd Finanzmathematik (DE-588)4017195-4 gnd Derivat Wertpapier (DE-588)4381572-8 gnd Optionspreistheorie (DE-588)4135346-8 gnd |
| subject_GND | (DE-588)4206283-4 (DE-588)4124458-8 (DE-588)4057633-4 (DE-588)4017195-4 (DE-588)4381572-8 (DE-588)4135346-8 |
| title | Tools for computational finance |
| title_auth | Tools for computational finance |
| title_exact_search | Tools for computational finance |
| title_exact_search_txtP | Tools for computational finance |
| title_full | Tools for computational finance Rüdiger U. Seydel |
| title_fullStr | Tools for computational finance Rüdiger U. Seydel |
| title_full_unstemmed | Tools for computational finance Rüdiger U. Seydel |
| title_short | Tools for computational finance |
| title_sort | tools for computational finance |
| topic | Black-Scholes-Modell (DE-588)4206283-4 gnd Wertpapieranalyse (DE-588)4124458-8 gnd Stochastisches Modell (DE-588)4057633-4 gnd Finanzmathematik (DE-588)4017195-4 gnd Derivat Wertpapier (DE-588)4381572-8 gnd Optionspreistheorie (DE-588)4135346-8 gnd |
| topic_facet | Black-Scholes-Modell Wertpapieranalyse Stochastisches Modell Finanzmathematik Derivat Wertpapier Optionspreistheorie |
| url | https://doi.org/10.1007/3-540-27926-1 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015586187&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT seydelrudiger toolsforcomputationalfinance |