Pricing derivatives under Lévy models: modern finite-difference and pseudo-differential operators approach
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| Format: | Electronic eBook |
| Language: | English |
| Published: |
New York, NY
Birkhäuser
[2017]
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| Series: | Pseudo-differential operators, theory and applications
vol. 12 |
| Subjects: | |
| Online Access: | BTU01 FHR01 FRO01 FWS01 FWS02 HTW01 TUM01 UBM01 UBT01 UBW01 UEI01 UPA01 Volltext Inhaltsverzeichnis |
| Physical Description: | 1 Online-Ressource (XX, 308 Seiten, 64 illus., 62 illus. in color) |
| ISBN: | 9781493967926 |
| ISSN: | 2297-0355 |
| DOI: | 10.1007/978-1-4939-6792-6 |
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Record in the Search Index
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| adam_text | Titel: Pricing Derivatives Under Lévy Models
Autor: Itkin, Andrey
Jahr: 2017
Part I Modern Tools of Computational Finance
1 Basics of the Finite Difference Method.......................................... 3
1.1 Finite Difference Approximation of Derivatives........................... 3
1.1.1 Construction of Finite Differences................................ 4
1.1.2 Higher-Order Approximations.................................... 5
1.1.3 Higher-Order Derivatives ......................................... 7
1.1.4 Mixed Derivatives................................................. 8
1.2 Finite Difference Method for Solving PDEs............................... 10
1.3 Stability Analysis............................................................ 14
References........................................................................... 19
2 Modern Finite Difference Approach ............................................ 21
2.1 Introduction.................................................................. 21
2.2 Discretization of eAz^ on a Temporal Grid ............................... 24
2.2.1 Examples........................................................... 24
2.3 Discretization of the Operator Jzf on a Spatial Grid ...................... 26
2.3.1 Uniform Grid ...................................................... 26
2.3.2 Nonuniform Grid.................................................. 27
2.4 Requirements of Modern FD Scheines..................................... 34
2.4.1 Order of Approximation........................................... 34
2.4.2 Stability......................,..................................... 36
2.4.3 Nonnegativity ofthe Solution..................................... 38
2.4.4 Complexity......................................................... 40
2.5 Operator Splitting Technique............................................... 41
2.5.1 General Approach................................................. 41
2.5.2 Splitting for a Convection-Diffusion PDE ...................... 46
2.A Appendix: Examples of Some HOC Scheines for Pricing
American Options........................................................... 48
2.A. I Finite Difference Scheme......................................... 50
2.A.2 Higher-Order FD Scheines in Time............................... 52
xv
xv i Contents
2.A.3 L-Stabie Scheine of Fifth Order in Time......................... 54
2.A.4 Boundary Conditions for a High-Order Uniform FD Scheine... 55
References........................................................................... 56
3 An M-Matrix Theory and FD.................................................... 59
3.1 M-Matrices and Metzler Matrices ......................................... 60
3.2 The Operator Jt? as a Generator............................................ 63
3.3 EM-Matrices................................................................. 65
3.3.1 Some Useful Theorems............................................ 66
3.4 Mixed Derivatives and Positivity .......................................... 68
3.4.1 Rate of Convergence of Picard Iterations......................... 75
3.4.2 Second Order of Approximation in Space........................ 76
References........................................................................... 81
Part II Pricing Derivatives Using Levy Processes
4 A Brief Introduction to Levy Processes......................................... 85
4.1 Preliminaries................................................................. 85
4.2 Main Definitions ............................................................ 86
4.3 Levy-Khinchin Formula.................................................... 89
4.4 Levy Measure, Path. and Moments......................................... 92
4.5 Semimartingales and Ito s Lemma......................................... 95
4.6 PIDE for Pricing European Options........................................ 98
References........................................................................... 100
5 Pseudoparabolic and Fractional Equations of Option Pricing............... 101
5.1 Introduction.................................................................. 101
5.2 Levy Models and Backward PIDE......................................... 105
5.3 From PIDE to PDE: A Basic Example..................................... 106
5.4 A More Sophisticated Example: The GTSP Model....................... 109
5.4.1 Transforming a PIDE to a Pseudoparabolic Equation ........... 112
5.5 Solution ofthe Pseudoparabolic Equation................................. 115
5.5.1 Numerical Method When a € I.a 0........................... 116
5.5.2 a e R: Interpolation............................................... 119
5.5.3 Numerical Examples .............................................. 120
5.5.4 TheCaseo^ = 0oraL = 0....................................... 126
5.6 Jump Integral as a Pseudodifferential Operator............................ 130
References........................................................................... 132
6 Pseudoparabolic Equations for Various Levy Models......................... 135
6.1 Introduction.................................................................. 135
6.2 Solution of aPure Jump Equation.......................................... 138
Contents xvü
6.3 Merton Model............................................................... 14!
6.4 Exponential Jumps.......................................................... 143
6.5 Kou Model .................................................................. 145
6.5.1 Numerical Experiments........................................... 146
6.6 CGMY Model............................................................... 148
6.6.1 The Gase aR 0................................................... 151
6.6.2 TheCaseO aR 1.............................................. 153
6.6.3 The Case aR = 1 .................................................. 154
6.6.4 The Case 1 aR 2.............................................. 155
6.6.5 Approximations of Jä?L............................................ 158
6.7 Other Numerical Experiments.............................................. 159
6.8 Pure Jump Models........................................................... 161
6.8.1 Normal Inverse Gaussian Model (NIG).......................... 163
6.8.2 Generalized Hyperbolic Models.................................. 167
6.8.3 Meixner Model .................................................... 175
References........................................................................... 179
7 High-Order Splitting Methods for Forward PDEs and PIDEs............... 183
7.1 Introduction.................................................................. 183
7.2 LSV Model with Jumps..................................................... J 85
7.3 Backward and Forward FD Scheme for the Diffusion Part............... 187
7.3.1 Backward Scheme................................................. 187
7.3.2 Forward Scheme................................................... 188
7.4 Forward Scheme for Jumps................................................. 191
7.4.1 Details of Numerical Implementation............................ 192
7.4.2 Parameters of the Finite Difference Scheme..................... 196
7.5 Construction of the Backward and Forward Evolution Operators........ 199
References........................................................................... 201
Part III 2D and 3D Cases and Correlated Jumps
8 Multidimensional Structural Default Models
and Correlated Jumps............................................................ 205
8.1 Introduction.................................................................. 205
8.2 Interbank Mutual Obligations in a Structural Default Model............. 208
8.3 Correlated Jumps and Structured Default Models......................... 213
8.4 Pseudodifferential Equations and Jump Integrals.......................... 215
8.5 Construction of an FD scheme.............................................. 217
8.6 Benchmark: 1D Structural Default Model with Exponential
Jumps........................................................................ 222
8.6.1 A Generalized Fourier-LapJace Transform Approach........... 223
8.6.2 Inversion ofthe Laplace Transform: No Jumps.................. 225
xviü Contents
8.7 Numerical Experiments..................................................... 228
8.7.1 The One-Dimensional Problem................................... 228
8.7.2 The Two-Dirnensional Problem................................... 230
8.8 The Three-Dimensional Case............................................... 235
8.8.1 Numerical Experiments........................................... 240
References........................................................................... 243
9 LSV Models with Stochastic Interest Rates and Correlated Jumps......... 247
9.1 Introduction.................................................................. 247
9.2 Model........................................................................ 249
9.3 Solution of the PIDE........................................................ 252
9.3.1 Idiosyncratic Jumps ............................................... 252
9.3.2 Common Jumps.................................................... 253
9.4 Numerical Experiments..................................................... 255
References........................................................................... 263
10 Stochastic Skew Model............................................................ 265
10.1 Introduction.................................................................. 265
10.2 Pricing Barrier Options under SSM........................................ 267
10.3 A Sufficient Condition for the Matrix of Second Derivatives
to Be Positive Semidefmite................................................. 270
10.4 Splitting Method ............................................................ 272
10.4.1 Structure of the Numerical Algorithm............................ 276
10.5 Numerical Experiments..................................................... 279
10.5.1 Test 1............................................................... 281
10.5.2 Test 2............................................................... 283
10.5.3 Test 3............................................................... 284
References........................................................................... 295
Giossary.................................................................................. 297
References........................................................................... 302
Index...................................................................................... 305
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| language | English |
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| series2 | Pseudo-differential operators, theory and applications |
| spellingShingle | Itkin, Andrey Pricing derivatives under Lévy models modern finite-difference and pseudo-differential operators approach Mathematics Partial differential equations Economics, Mathematical Computer mathematics Mathematical models Quantitative Finance Mathematical Modeling and Industrial Mathematics Computational Science and Engineering Partial Differential Equations Mathematik Mathematisches Modell |
| title | Pricing derivatives under Lévy models modern finite-difference and pseudo-differential operators approach |
| title_auth | Pricing derivatives under Lévy models modern finite-difference and pseudo-differential operators approach |
| title_exact_search | Pricing derivatives under Lévy models modern finite-difference and pseudo-differential operators approach |
| title_full | Pricing derivatives under Lévy models modern finite-difference and pseudo-differential operators approach Andrey Itkin |
| title_fullStr | Pricing derivatives under Lévy models modern finite-difference and pseudo-differential operators approach Andrey Itkin |
| title_full_unstemmed | Pricing derivatives under Lévy models modern finite-difference and pseudo-differential operators approach Andrey Itkin |
| title_short | Pricing derivatives under Lévy models |
| title_sort | pricing derivatives under levy models modern finite difference and pseudo differential operators approach |
| title_sub | modern finite-difference and pseudo-differential operators approach |
| topic | Mathematics Partial differential equations Economics, Mathematical Computer mathematics Mathematical models Quantitative Finance Mathematical Modeling and Industrial Mathematics Computational Science and Engineering Partial Differential Equations Mathematik Mathematisches Modell |
| topic_facet | Mathematics Partial differential equations Economics, Mathematical Computer mathematics Mathematical models Quantitative Finance Mathematical Modeling and Industrial Mathematics Computational Science and Engineering Partial Differential Equations Mathematik Mathematisches Modell |
| url | https://doi.org/10.1007/978-1-4939-6792-6 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029612404&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT itkinandrey pricingderivativesunderlevymodelsmodernfinitedifferenceandpseudodifferentialoperatorsapproach |