Monte Carlo methods in finance:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Chichester [u.a.]
Wiley
2006
|
| Ausgabe: | Reprint. with corr. |
| Schriftenreihe: | Wiley finance series
|
| Schlagworte: | |
| Online-Zugang: | Klappentext Inhaltsverzeichnis |
| Beschreibung: | XVI, 222 S. zahlr. graph. Darst. 1 CD-ROM (12 cm) |
| ISBN: | 047149741X 9780471497417 |
Internformat
MARC
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| 100 | 1 | |a Jaeckel, Peter |e Verfasser |0 (DE-588)1216571805 |4 aut | |
| 245 | 1 | 0 | |a Monte Carlo methods in finance |c Peter Jäckel |
| 250 | |a Reprint. with corr. | ||
| 264 | 1 | |a Chichester [u.a.] |b Wiley |c 2006 | |
| 300 | |a XVI, 222 S. |b zahlr. graph. Darst. |e 1 CD-ROM (12 cm) | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Wiley finance series | |
| 650 | 0 | 7 | |a Monte-Carlo-Simulation |0 (DE-588)4240945-7 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
| _version_ | 1804136711829061632 |
|---|---|
| adam_text | Contents
Xl
Preface
xiii
Acknowledgements
xv
Mathematical Notation
1
1
Introduction
5
2
The Mathematics Behind Monte Carlo Methods
5
2.1
A Few Basic Terms in Probability and Statistics
η
2.2
Monte Carlo Simulations
8
2.2.1
Monte Carlo Supremacy
8
2.2.2
Multi-dimensional Integration
9
2.3
Some Common Distributions
18
2.4
Kolmogorov s Strong Law
18
2.5
The Central Limit Theorem
19
2.6
The Continuous Mapping Theorem
20
2.7
Error Estimation for Monte Carlo Methods
21
2.8
The Feynman-Kac Theorem
21
2.9
The Moore-Penrose Pseudo-inverse
23
3
Stochastic Dynamics
23
3.1
Brownian Motion
24
3.2
Itô s
Lemma
25
3.3
Normal Processes
26
3.4 Lognormal
Processes
26
3.5
The Markovian Wiener Process Embedding Dimension
27
3.6
Bessel Processes
28
3.7
Constant Elasticity Of Variance Processes
29
3.8
Displaced Diffusion
31
4
Process-driven Sampling
31
4.1
Strong versus Weak Convergence
32
4.2
Numerical Solutions
Contents
4.2.1 The Euler
Scheme
32
4.2.2
The Milstein Scheme
33
4.2.3
Transformations
33
4.2.4
Predictor-Corrector
35
4.3
Spurious Paths
36
4.4
Strong Convergence for
Euler
and Milstein
37
5
Correlation and Co-movement
41
5.1
Measures for Co-dependence
42
5.2
Copula
45
5.2.1
The Gaussian Copula
46
5.2.2
The f-Copula
49
5.2.3
Archimedean Copulae
51
6
Salvaging a Linear Correlation Matrix
59
6.1
Hypersphere Decomposition
60
6.2
Spectral Decomposition
61
6.3
Angular Decomposition of Lower Triangular Form
62
6.4
Examples
63
6.5
Angular Coordinates on a Hypersphere of Unit Radius
65
7
Pseudo-random Numbers
67
7.1
Chaos
68
7.2
The Mid-square Method
72
7.3
Congraential Generation
72
7.4
RanO To Ran3
74
7.5
The Mersenne Twister
74
7.6
Which One to Use?
75
8
Low-discrepancy Numbers
77
8.1
Discrepancy
78
8.2
Halton
Numbers
79
8.3
Sobol
Numbers
80
8.3.1
Primitive Polynomials Modulo Two
81
8.3.2
The Construction of
Sobol
Numbers
82
8.3.3
The Gray Code
83
8.3.4
The Initialisation of
Sobol
Numbers
85
8.4 Niederreiter (1988)
Numbers
88
8.5
Pairwise Projections
88
8.6
Empirical Discrepancies
91
8.7
The Number of Iterations
96
8.8
Appendix
96
8.8.1
Explicit Formula for the Z^-norm Discrepancy on the
Unit Hypercube
96
8.8.2
Expected
Ьг-погт
Discrepancy of Truly Random Numbers
97
9
Non-uniform
Variâtes
99
9.1
Inversion of the Cumulative Probability Function
99
9.2
Using a Sampler Density
101
Contents
9.2.1
Importance
Sampling
103
9.2.2
Rejection Sampling
104
9.3
Normal
Variâtes
105
9.3.1
The Box-Muller Method
105
9.3.2
The Neave Effect
106
9.4
Simulating Multivariate Copula Draws
109
10
Variance Reduction Techniques 111
10.1
Antithetic Sampling 111
10.2
Variate
Recycling
112
10.3
Control
Variâtes
113
10.4
Stratified Sampling
114
10.5
Importance Sampling
115
10.6
Moment Matching
116
10.7
Latin Hypercube Sampling
119
10.8
Path Construction
120
10.8.1
Incremental
120
10.8.2
Spectral
122
10.8.3
The Brownian Bridge
124
10.8.4
A Comparison of Path Construction Methods
128
10.8.5
Multivariate Path Construction
131
10.9
Appendix
134
10.9.1
Eigenvalues and Eigenvectors
of a Discrete-time Co variance Matrix
134
10.9.2
The Conditional Distribution of the Brownian Bridge
137
11
Greeks
139
11.1
Importance Of Greeks
139
11.2
An Up-Out-Call Option
139
11.3
Finite Differencing with Path Recycling
140
11.4
Finite Differencing with Importance Sampling
143
11.5
Pathwise Differentiation
144
11.6
The Likelihood Ratio Method
145
11.7
Comparative Figures
147
11.8
Summary
153
11.9
Appendix
153
11.9.1
The Likelihood Ratio Formula for
Vega 153
11.9.2
The Likelihood Ratio Formula for Rho
156
12
Monte Carlo in the BGM/J Framework
159
12.1
The Brace-Gatarek-Musiela/Jamshidian Market Model
159
12.2
Factorisation
161
12.3
Bermudán
Swaptions
163
12.4
Calibration to European Swaptions
163
12.5
The Predictor-Corrector Scheme
169
12.6
Heuristics of the Exercise Boundary
171
12.7
Exercise Boundary Parametrisation
174
12.8
The Algorithm
176
Contents
177
182
183
183
184
187
190
191
192
195
196
199
200
14
Miscellanea
201
14.1 Interpolation
of the Term Structure of Implied Volatility
201
14.2
Watch Your CPU Usage
202
14.3
Numerical Overflow and Underflow
205
14.4
A Single Number or a Convergence Diagram?
205
14.5
Embedded Path Creation
206
14.6
How Slow is
Exp
() ? 207
14.7
Parallel Computing And Multi-threading
209
Bibliography
213
Index
219
12.9
Numerical Results
12.10
Summary
13
Non-i
recombining Trees
13.1
Introduction
13.2
Evolving the Forward Rates
13.3
Optimal Simplex Alignment
13.4
Implementation
13.5
Convergence Performance
13.6
Variance Matching
13.7
Exact Martingale Conditioning
13.8
Clustering
13.9
A Simple Example
13.10
1
Summary
Monte
Carlo Methods in Finance is an important
reference for those working in investment banks,
insurance and strategic management consultancy.
Of particular importance are the many known
variance reduction methods, and they are duly
covered, not only in their own right, but also with
respect to their potential combinations, and in
the direct context of realistic applications. Most
notably, the issue of the reliability of low-discrepancy
numbers in high dimensions is discussed in detail.
The book also contains an introduction to the
theory of copulae as an extension to the modelling
of correlation of financial securities. An entire
chapter is dedicated to the evaluation of interest rate
derivatives in the Brace-Gatarek-Musiela/Jamshidian
framework by the aid of fast-convergence Monte
Carlo simulations. What s more, for the first time,
this book also gives a description of the construction
of non-recombining trees. Whilst non-recombining
trees are usually not viable in a production envi¬
ronment, they often are the very tool of last resort
when Monte Carlo approximations to problems
such as
Bermudán
swaptions are to be tested, and
the tricks for the construction of non-recombining
trees presented in this book are invaluable for that
purpose.
PETER
JÄCKEL
received his
D.
Phil, in Physics from
Oxford University in
1995.
After a short period in
academic research, he moved into quantitative
analysis and financial modelling in
1997,
when he
joined Nikko Securities. Following that he worked as
a quant at
NatWest,
which later became part of the
Royal Bank of Scotland group. In December
2000,
he
joined
Commerzbank
Securities as a Financial
Engineer in their front office product development
and derivatives modelling unit, and jointly with his
co-head ran the team from May
2003.
Since
September
2004,
he has been with
ABN AMR0
as
Global Head of Credit, Hybrid, and Commodity
Derivative Analytics.
|
| adam_txt |
Contents
Xl
Preface
xiii
Acknowledgements
xv
Mathematical Notation
1
1
Introduction
5
2
The Mathematics Behind Monte Carlo Methods
5
2.1
A Few Basic Terms in Probability and Statistics
η
2.2
Monte Carlo Simulations
8
2.2.1
Monte Carlo Supremacy
8
2.2.2
Multi-dimensional Integration
9
2.3
Some Common Distributions
18
2.4
Kolmogorov's Strong Law
18
2.5
The Central Limit Theorem
19
2.6
The Continuous Mapping Theorem
20
2.7
Error Estimation for Monte Carlo Methods
21
2.8
The Feynman-Kac Theorem
21
2.9
The Moore-Penrose Pseudo-inverse
23
3
Stochastic Dynamics
23
3.1
Brownian Motion
24
3.2
Itô's
Lemma
25
3.3
Normal Processes
26
3.4 Lognormal
Processes
26
3.5
The Markovian Wiener Process Embedding Dimension
27
3.6
Bessel Processes
28
3.7
Constant Elasticity Of Variance Processes
29
3.8
Displaced Diffusion
31
4
Process-driven Sampling
31
4.1
Strong versus Weak Convergence
32
4.2
Numerical Solutions
Contents
4.2.1 The Euler
Scheme
32
4.2.2
The Milstein Scheme
33
4.2.3
Transformations
33
4.2.4
Predictor-Corrector
35
4.3
Spurious Paths
36
4.4
Strong Convergence for
Euler
and Milstein
37
5
Correlation and Co-movement
41
5.1
Measures for Co-dependence
42
5.2
Copula
45
5.2.1
The Gaussian Copula
46
5.2.2
The f-Copula
49
5.2.3
Archimedean Copulae
51
6
Salvaging a Linear Correlation Matrix
59
6.1
Hypersphere Decomposition
60
6.2
Spectral Decomposition
61
6.3
Angular Decomposition of Lower Triangular Form
62
6.4
Examples
63
6.5
Angular Coordinates on a Hypersphere of Unit Radius
65
7
Pseudo-random Numbers
67
7.1
Chaos
68
7.2
The Mid-square Method
72
7.3
Congraential Generation
72
7.4
RanO To Ran3
74
7.5
The Mersenne Twister
74
7.6
Which One to Use?
75
8
Low-discrepancy Numbers
77
8.1
Discrepancy
78
8.2
Halton
Numbers
79
8.3
Sobol'
Numbers
80
8.3.1
Primitive Polynomials Modulo Two
81
8.3.2
The Construction of
Sobol'
Numbers
82
8.3.3
The Gray Code
83
8.3.4
The Initialisation of
Sobol'
Numbers
85
8.4 Niederreiter (1988)
Numbers
88
8.5
Pairwise Projections
88
8.6
Empirical Discrepancies
91
8.7
The Number of Iterations
96
8.8
Appendix
96
8.8.1
Explicit Formula for the Z^-norm Discrepancy on the
Unit Hypercube
96
8.8.2
Expected
Ьг-погт
Discrepancy of Truly Random Numbers
97
9
Non-uniform
Variâtes
99
9.1
Inversion of the Cumulative Probability Function
99
9.2
Using a Sampler Density
101
Contents
9.2.1
Importance
Sampling
103
9.2.2
Rejection Sampling
104
9.3
Normal
Variâtes
105
9.3.1
The Box-Muller Method
105
9.3.2
The Neave Effect
106
9.4
Simulating Multivariate Copula Draws
109
10
Variance Reduction Techniques 111
10.1
Antithetic Sampling 111
10.2
Variate
Recycling
112
10.3
Control
Variâtes
113
10.4
Stratified Sampling
114
10.5
Importance Sampling
115
10.6
Moment Matching
116
10.7
Latin Hypercube Sampling
119
10.8
Path Construction
120
10.8.1
Incremental
120
10.8.2
Spectral
122
10.8.3
The Brownian Bridge
124
10.8.4
A Comparison of Path Construction Methods
128
10.8.5
Multivariate Path Construction
131
10.9
Appendix
134
10.9.1
Eigenvalues and Eigenvectors
of a Discrete-time Co variance Matrix
134
10.9.2
The Conditional Distribution of the Brownian Bridge
137
11
Greeks
139
11.1
Importance Of Greeks
139
11.2
An Up-Out-Call Option
139
11.3
Finite Differencing with Path Recycling
140
11.4
Finite Differencing with Importance Sampling
143
11.5
Pathwise Differentiation
144
11.6
The Likelihood Ratio Method
145
11.7
Comparative Figures
147
11.8
Summary
153
11.9
Appendix
153
11.9.1
The Likelihood Ratio Formula for
Vega 153
11.9.2
The Likelihood Ratio Formula for Rho
156
12
Monte Carlo in the BGM/J Framework
159
12.1
The Brace-Gatarek-Musiela/Jamshidian Market Model
159
12.2
Factorisation
161
12.3
Bermudán
Swaptions
163
12.4
Calibration to European Swaptions
163
12.5
The Predictor-Corrector Scheme
169
12.6
Heuristics of the Exercise Boundary
171
12.7
Exercise Boundary Parametrisation
174
12.8
The Algorithm
176
Contents
177
182
183
183
184
187
190
191
192
195
196
199
200
14
Miscellanea
201
14.1 Interpolation
of the Term Structure of Implied Volatility
201
14.2
Watch Your CPU Usage
202
14.3
Numerical Overflow and Underflow
205
14.4
A Single Number or a Convergence Diagram?
205
14.5
Embedded Path Creation
206
14.6
How Slow is
Exp
() ? 207
14.7
Parallel Computing And Multi-threading
209
Bibliography
213
Index
219
12.9
Numerical Results
12.10
Summary
13
Non-i
recombining Trees
13.1
Introduction
13.2
Evolving the Forward Rates
13.3
Optimal Simplex Alignment
13.4
Implementation
13.5
Convergence Performance
13.6
Variance Matching
13.7
Exact Martingale Conditioning
13.8
Clustering
13.9
A Simple Example
13.10
1
Summary
Monte
Carlo Methods in Finance is an important
reference for those working in investment banks,
insurance and strategic management consultancy.
Of particular importance are the many known
variance reduction methods, and they are duly
covered, not only in their own right, but also with
respect to their potential combinations, and in
the direct context of realistic applications. Most
notably, the issue of the reliability of low-discrepancy
numbers in high dimensions is discussed in detail.
The book also contains an introduction to the
theory of copulae as an extension to the modelling
of correlation of financial securities. An entire
chapter is dedicated to the evaluation of interest rate
derivatives in the Brace-Gatarek-Musiela/Jamshidian
framework by the aid of fast-convergence Monte
Carlo simulations. What's more, for the first time,
this book also gives a description of the construction
of non-recombining trees. Whilst non-recombining
trees are usually not viable in a production envi¬
ronment, they often are the very tool of last resort
when Monte Carlo approximations to problems
such as
Bermudán
swaptions are to be tested, and
the tricks for the construction of non-recombining
trees presented in this book are invaluable for that
purpose.
PETER
JÄCKEL
received his
D.
Phil, in Physics from
Oxford University in
1995.
After a short period in
academic research, he moved into quantitative
analysis and financial modelling in
1997,
when he
joined Nikko Securities. Following that he worked as
a quant at
NatWest,
which later became part of the
Royal Bank of Scotland group. In December
2000,
he
joined
Commerzbank
Securities as a Financial
Engineer in their front office product development
and derivatives modelling unit, and jointly with his
co-head ran the team from May
2003.
Since
September
2004,
he has been with
ABN AMR0
as
Global Head of Credit, Hybrid, and Commodity
Derivative Analytics. |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Jaeckel, Peter |
| author_GND | (DE-588)1216571805 |
| author_facet | Jaeckel, Peter |
| author_role | aut |
| author_sort | Jaeckel, Peter |
| author_variant | p j pj |
| building | Verbundindex |
| bvnumber | BV022575926 |
| classification_rvk | QH 239 |
| classification_tum | WIR 522f MAT 629f WIR 170f |
| ctrlnum | (OCoLC)634944739 (DE-599)BVBBV022575926 |
| discipline | Mathematik Wirtschaftswissenschaften |
| discipline_str_mv | Mathematik Wirtschaftswissenschaften |
| edition | Reprint. with corr. |
| format | Book |
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| id | DE-604.BV022575926 |
| illustrated | Illustrated |
| index_date | 2024-07-02T18:15:02Z |
| indexdate | 2024-07-09T21:00:45Z |
| institution | BVB |
| isbn | 047149741X 9780471497417 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015782195 |
| oclc_num | 634944739 |
| open_access_boolean | |
| owner | DE-1102 DE-355 DE-BY-UBR DE-Aug4 DE-739 |
| owner_facet | DE-1102 DE-355 DE-BY-UBR DE-Aug4 DE-739 |
| physical | XVI, 222 S. zahlr. graph. Darst. 1 CD-ROM (12 cm) |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Wiley |
| record_format | marc |
| series2 | Wiley finance series |
| spelling | Jaeckel, Peter Verfasser (DE-588)1216571805 aut Monte Carlo methods in finance Peter Jäckel Reprint. with corr. Chichester [u.a.] Wiley 2006 XVI, 222 S. zahlr. graph. Darst. 1 CD-ROM (12 cm) txt rdacontent n rdamedia nc rdacarrier Wiley finance series Monte-Carlo-Simulation (DE-588)4240945-7 gnd rswk-swf Finanzmathematik (DE-588)4017195-4 gnd rswk-swf Finanzmathematik (DE-588)4017195-4 s Monte-Carlo-Simulation (DE-588)4240945-7 s DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015782195&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015782195&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Jaeckel, Peter Monte Carlo methods in finance Monte-Carlo-Simulation (DE-588)4240945-7 gnd Finanzmathematik (DE-588)4017195-4 gnd |
| subject_GND | (DE-588)4240945-7 (DE-588)4017195-4 |
| title | Monte Carlo methods in finance |
| title_auth | Monte Carlo methods in finance |
| title_exact_search | Monte Carlo methods in finance |
| title_exact_search_txtP | Monte Carlo methods in finance |
| title_full | Monte Carlo methods in finance Peter Jäckel |
| title_fullStr | Monte Carlo methods in finance Peter Jäckel |
| title_full_unstemmed | Monte Carlo methods in finance Peter Jäckel |
| title_short | Monte Carlo methods in finance |
| title_sort | monte carlo methods in finance |
| topic | Monte-Carlo-Simulation (DE-588)4240945-7 gnd Finanzmathematik (DE-588)4017195-4 gnd |
| topic_facet | Monte-Carlo-Simulation Finanzmathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015782195&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015782195&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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