Nonlinear option pricing:
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| Format: | Buch |
| Sprache: | English |
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Boca Raton [u.a.]
CRC Press
2014
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| Schriftenreihe: | Chapman & Hall/CRC financial mathematics series
A Chapman & Hall Book |
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| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XXXVIII, 445 S. graph. Darst. |
| ISBN: | 9781466570337 1466570334 |
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| 020 | |a 9781466570337 |9 978-1-4665-7033-7 | ||
| 020 | |a 1466570334 |9 1-4665-7033-4 | ||
| 035 | |a (OCoLC)873379563 | ||
| 035 | |a (DE-599)BVBBV040932043 | ||
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| 100 | 1 | |a Guyon, Julien |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Nonlinear option pricing |c Julien Guyon ; Pierre Henry-Labordère |
| 264 | 1 | |a Boca Raton [u.a.] |b CRC Press |c 2014 | |
| 300 | |a XXXVIII, 445 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Chapman & Hall/CRC financial mathematics series | |
| 490 | 0 | |a A Chapman & Hall Book | |
| 650 | 0 | 7 | |a Stochastische Differentialgleichung |0 (DE-588)4057621-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Nichtlineare partielle Differentialgleichung |0 (DE-588)4128900-6 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Optionspreistheorie |0 (DE-588)4135346-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Finanzmathematik |0 (DE-588)4017195-4 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Optionspreistheorie |0 (DE-588)4135346-8 |D s |
| 689 | 0 | 1 | |a Finanzmathematik |0 (DE-588)4017195-4 |D s |
| 689 | 0 | 2 | |a Nichtlineare partielle Differentialgleichung |0 (DE-588)4128900-6 |D s |
| 689 | 0 | 3 | |a Stochastische Differentialgleichung |0 (DE-588)4057621-8 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Henry-Labordère, Pierre |e Verfasser |4 aut | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025910972&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
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Datensatz im Suchindex
| _version_ | 1804150231192829952 |
|---|---|
| adam_text | Contents
Preface
xvii
1 Option
Pricing in a Nutshell
1
1.1
The super-replication paradigm
................ 1
1.1.1
Models of financial markets
.............. 1
1.1.2
Self-financing portfolios
................ 2
1.1.3
Arbitrage and arbitrage-free models
......... 3
1.1.4
Super-replication
................... 5
1.1.5
Complete models versus incomplete models
..... 11
1.1.6
Pricing in practice
................... 13
1.2
Stochastic representation of solutions of linear PDEs
.... 14
1.2.1
The Cauchy problem
................. 14
1.2.2
The Dirichlet problem
................. 16
1.2.3
The Neumann problem
................ 19
1.3
Exercises
............................ 24
1.3.1
Local martingales and
supermartingales
....... 24
1.3.2
Timer options on two assets
............. 24
1.3.3
Azéma-Yor
martingales
................ 24
1.3.4
Azéma-Yor
martingales and the Skorokhod
embedding problem
.................. 25
2
Monte Carlo
27
2.1
The Monte Carlo method
................... 27
2.1.1
Principle of the method
................ 27
2.1.2
Sampling error, reduction of variance
........ 28
2.1.3
Discretization error, reduction of bias
........ 30
2.2
Euler
discretization error
................... 32
2.2.1
Weak error
....................... 33
2.2.2
Results concerning smooth or bounded payoffs
... 33
2.2.3
Results for general payoffs
.............. 34
2.2.4
Application to option pricing and hedging
..... 36
2.3
Romberg extrapolation
..................... 38
2.3.1
Standard Romberg extrapolation
.......... 38
2.3.2
Iterated Romberg extrapolation
........... 40
2.3.3
Numerical experiments with vanilla options
..... 41
2.3.4
Numerical experiments with path-dependent options
44
ix
χ
Contents
3
Some Excursions in Option Pricing
49
3.1
Complete market models
................... 49
3.1.1
Modeling the dynamics of implied volatility:
Schönbucher s
market model
............. 50
3.1.2
Modeling the dynamics of log-contract prices:
Bergomi s variance swap model
........... 51
3.1.3
Modeling the dynamics of power-payoff prices: an
HJM-like model
.................... 53
3.1.4
The case of a family of power-payoffs
........ 55
3.1.5
Modeling the dynamics of local volatility:
Carmona
and Nadtochiy s market model
............ 57
3.2
Beyond replication and super-replication
........... 60
3.2.1
The utility indifference price
............. 60
3.2.2
Quantile hedging
................... 61
3.2.3
Minimum variance hedging
.............. 62
3.3
Exercises
............................ 63
3.3.1
Super-replication and quantile hedging
....... 63
3.3.2
Duality for the utility indifference price
....... 63
3.3.3
Utility indifference price for exponential utility
... 63
3.3.4
Minimum variance hedging for the double
lognormal
stochastic volatility model
.............. 63
4
Nonlinear PDEs: A Bit of Theory
65
4.1
Nonlinear second order parabolic PDEs: Some generalities
. 65
4.1.1
Parabolic PDEs
.................... 66
4.1.2
What do we mean by a well-posed PDE?
...... 67
4.1.3
What do we mean by a solution?
.......... 67
4.1.4
Comparison principle and uniqueness
........ 67
4.2
Why is a pricing equation a parabolic PDE?
......... 71
4.3
Finite difference schemes
.................... 73
4.3.1
Introduction
...................... 73
4.3.2
Consistency
...................... 74
4.3.3
Stability
........................ 75
4.3.4
Convergence
...................... 75
4.4
Stochastic control and the Hamilton-Jacobi-Bellman PDE
. 77
4.4.1
Introduction
...................... 77
4.4.2
Standard form
..................... 78
4.4.3
Bellman s principle
.................. 79
4.4.4
Formal derivation of the HJB PDE
......... 80
4.5
Optimal stopping problems
.................. 83
4.6
Viscosity solutions
....................... 84
4.6.1
Motivation
....................... 84
4.6.2
Definition
........................ 86
4.6.3
Viscosity solutions to HJB equations
........ 88
Contents xi
4.6.4
Numerical scheme: The Barles-Souganadis
framework
....................... 89
4.7
Exercises
............................ 90
4.7.1
The American timer put
............... 90
4.7.2
Super-replication price in a stochastic volatility
model
.......................... 91
4.7.3
The robust utility indifference price
......... 92
5
Examples of Nonlinear Problems in Finance
93
5.1
American options
........................ 93
5.2
The uncertain volatility model
................ 94
5.2.1
The model
....................... 94
5.2.2
Pricing vanilla options
................ 95
5.2.3
Robust super-replication
............... 95
5.2.4
A finite difference scheme
............... 97
5.3
Transaction costs: Leland s model
.............. 97
5.4
Illiquid markets
......................... 99
5.4.1
Feedback effects
.................... 99
5.4.2
A simple model of a limit order book
........ 102
5.4.3
Continuous-time limit: Some intuition
........ 104
5.4.4
Admissible trading strategies
............. 105
5.4.5
Perfect replication paradigm
............. 106
5.4.6
The parabolic envelope of the Hamiltonian
..... 107
5.5
Super-replication under delta and gamma constraints
.... 110
5.5.1
Super-replication under delta constraints
...... 110
5.5.2
HJB and variational inequality
............ 112
5.5.3
Adding gamma constraints
.............. 112
5.5.4
Numerical algorithm
................. 113
5.5.5
Numerical experiments
................ 116
5.6
The uncertain mortality model for reinsurance deals
.... 116
5.7
Different rates for borrowing and lending
.......... 119
5.8
Credit valuation adjustment
.................. 120
5.9
The passport option
...................... 122
5.10
Exercises
............................ 124
5.10.1
Transaction costs
................... 124
5.10.2
Super-replication under delta constraints
...... 124
6
Early Exercise Problems
125
6.1
Super-replication of American options
............ 125
6.2
American options and
semilinear
PDEs
........... 129
6.3
The dual method for American options
........... 131
6.4
On the ownership of the exercise right
............ 133
6.5
On the
finiteness
of exercise dates
.............. 133
6.6
On the accounting of multiple coupons
............ 135
6.7
Finite difference methods for American options
....... 137
xii Contents
6.8 Monte Carlo
methods for
American
options .........
138
6.8.1
Why is it difficult?
................... 138
6.8.2
Estimating Vti: The Tsitsiklis-Van Roy algorithm
. 139
6.8.3
Estimating
τ*:
The Longstaff-Schwartz algorithm
. 140
6.8.4
Estimating M*: The Andersen-Broadie algorithm
. 142
6.8.5
Conditional expectation approximation
....... 144
6.8.6
Parametric methods
.................. 145
6.8.7
Quantization methods
................. 146
6.8.8
Mesh methods
..................... 147
6.8.9
The case of continuous exercise dates
........ 147
6.9
Case study: Pricing and hedging of a multi-asset convertible
bond
............................... 148
6.9.1
Introduction
...................... 148
6.9.2
Modeling assumptions
................. 149
6.9.3
Pricing parameters and regressors
.......... 149
6.9.4
Price and exercise strategy
.............. 150
6.9.5
Hedge ratios
...................... 151
6.9.6
First order ratios need no reoptimization of exercise
strategy
......................... 153
6.9.7
Why exercise the convertible bond?
......... 154
6.10
Introduction to chooser options
................ 157
6.11
Regression methods for chooser options
........... 158
6.11.1
The naive primal algorithm
............ 158
6.11.2
Efficient Monte Carlo valuation of chooser options
. 160
6.12
The dual algorithm for chooser options
............ 163
6.12.1
The case of two tokens
................ 163
6.12.2
The general case
.................... 165
6.13
Numerical examples of pricing of chooser options
...... 166
6.13.1
The multi-times restrikable put
........... 166
6.13.2
The passport option
.................. 170
6.14
Exercise: Bounds on prices of American call options
.... 171
7
Backward Stochastic Differential Equations
173
7.1
First order BSDEs
....................... 173
7.1.1
Introduction
...................... 173
7.1.2
First order BSDEs provide an extension of Feynman-
Kac s formula for
semilinear
PDEs
.......... 177
7.1.3
Numerical simulation
................. 178
7.2
Reflected first order BSDEs
.................. 180
7.3
Second order BSDEs
...................... 181
7.3.1
Introduction
...................... 181
7.3.2
Second order BSDEs provide an extension of Feynman-
Kac s formula for fully nonlinear PDEs
....... 182
7.3.3
Numerical simulation
................. 183
7.4
Exercise: An example of semilinear PDE
........... 185
Contents xiii
8
The Uncertain Lapse and Mortality Model
187
8.1
Reinsurance deals
........................ 187
8.2
The deterministic lapse and mortality model
........ 188
8.3
The uncertain lapse and mortality model
.......... 191
8.4
Path-dependent payoffs
.................... 194
8.4.1
Payoffs depending on realized volatility
....... 194
8.4.2
Payoffs depending on the running maximum
.... 195
8.5
Pricing the option on the up-and-out barrier
........ 196
8.6
An example of PDE implementation
............. 196
8.7
Monte Carlo pricing
...................... 198
8.8
Monte Carlo pricing of the option on the up-and-out barrier
201
8.9
Link with first order BSDEs
.................. 202
8.10
Numerical results using PDEs
................. 205
8.10.1
Vanilla GMxB deal
.................. 205
8.10.2
GMxB with
vari ab] e
fees
............... 209
8.10.3
GMxB with variable put nominal
.......... 210
8.10.4
GMxB with cliqueting strikes
............ 211
8.11
Numerical results using Monte Carlo
............. 211
8.11.1
Vanilla GMxB deal
.................. 213
8.11.2
GMxB with variable fee and variable put nominal
. 216
9
The Uncertain Volatility Model
225
9.1
Introduction
........................... 225
9.2
The model
............................ 227
9.3
The parametric approach
................... 231
9.3.1
The ideas behind the algorithm
........... 231
9.3.2
The algorithm
..................... 232
9.3.3
Choice of the parameterization
............ 233
9.3.4
The single-asset case
................. 234
9.3.5
The two-asset case
................... 234
9.4
Solving the UVM with BSDEs
................ 235
9.4.1
A new numerical scheme
............... 236
9.4.2
First example: At-the-money call option
...... 237
9.4.3
The algorithm
..................... 238
9.4.4
About the generation of the first
Лгі
paths
..... 239
9.5
Numerical experiments
..................... 240
9.5.1
Options with one underlying
............. 240
9.5.2
Options with two imderlyings and no uncertainty on
correlation
....................... 243
9.5.3
Options with two imderlyings and uncertainty on
correlation
....................... 246
9.6
Exercise: UVM with penalty term
.............. 248
xiv Contents
10
McKean Nonlinear Stochastic Differential Equations
249
10.1
Definition
............................ 249
10.2
The particle method in a nutshell
............... 253
10.3
Propagation of chaos and convergence of the particle method
254
10.3.1
Building intuition: The BBGKY hierarchy
..... 254
10.3.2
Propagation of chaos and convergence of the particle
method
......................... 259
10.3.3
The McKean-
Vlasov SDE
propagates the chaos
. . 261
10.4
Exercise: Random matrices, Dyson Brownian motion, and
McKean SDEs
......................... 265
10.
A The Monge-Kantorovich distance and its financial
interpretation
.......................... 267
11
Calibration of Local Stochastic Volatility Models to Market
Smiles
271
11.1
Introduction
........................... 271
11.2
The calibration condition
................... 273
11.3
Existence of the calibrated local stochastic volatility model
274
11.4
The PDE method
....................... 274
11.5
The Markovian projection method
.............. 276
11.6
The particle method
...................... 280
11.6.1
The particle method for the calibration of LSVMs to
market smiles
..................... 280
11.6.2
Regularizing kernel
.................. 281
11.6.3
Acceleration techniques
................ 281
11.6.4
The algorithm
..................... 281
11.7
Adding stochastic interest rates
................ 282
11.7.1
The calibration condition
............... 283
11.7.2
The PDE method
................... 285
11.7.3
The particle method
.................. 285
11.7.4
First example: The particle method for the hybrid
Ho-Lee/Dupire model
................. 286
11.7.5
Second example: The particle method for the hybrid
Hull-White/Dupire model
.............. 287
11.7.6
Malliavin representation of the local volatility
. . . 290
11.7.7
Local stochastic volatility combined with
Libor
Market Models
..................... 294
11.8
The particle method: Numerical tests
............ 296
11.8.1
Hybrid Ho-Lee/Dupire model
............ 296
11.8.2
Bergomi s local stochastic volatility model
..... 299
11.8.3
Existence under question
............... 301
11.8.4
Hybrid Ho-Lee/Local Bergomi model
........ 302
11.9
Exercise: Dynamics of forward variance swaps for the double
lognormal SVM......................... 304
ll.A Proof of Proposition
11.2 ................... 304
Contents xv
ll.B
Proof of
Formula
(11.57).................... 308
ll.C
Including (discrete) dividends
................. 309
11.
C.I Calibration of the Dupire local volatility
...... 310
U.C.2
Calibration of the SIR-LSVM
............ 311
11.C.3 Approximate formula for the call option in the Black-
Scholes model with discrete dividends
........ 311
12
Calibration of Local Correlation Models to Market Smiles
315
12.1
Introduction
........................... 315
12.2
The FX triangle smile calibration problem
.......... 320
12.3
A new representation of admissible correlations
....... 322
12.4
The particle method for local correlation
........... 325
12.5
Some examples of pairs of functions (a,b)
.......... 326
12.6
Some links between local correlations
............. 328
12.7
Joint extrapolation of local volatilities
............ 330
12.8
Price impact of correlation
.................. 331
12.8.1
The price impact formula
............... 331
12.8.2
Equivalent local correlation
.............. 333
12.8.3
Implied correlation
.................. 334
12.8.4
Impact of correlation on price
............ 336
12.8.5
Uncertain correlation model
............. 337
12.9
The equity index smile calibration problem
......... 338
12.10
Numerical experiments on the FX triangle problem
..... 342
12.10.1
Calibration
....................... 342
12.10.2
Pricing
......................... 345
12.11
Generalization to stochastic volatility, stochastic interest rates,
and stochastic dividend yield
................. 364
12.11.1
The FX triangle smile calibration problem
..... 364
12.11.2
The equity index smile calibration problem
..... 368
12.12
Path-dependent volatility
................... 370
12.13
Exercises
............................ 375
12.13.1
Arbitrage-freeness condition on cross-currency
smiles: The case of one maturity
........... 375
12.13.2
Arbitrage-freeness condition on cross-currency
smiles: The case of finitely many maturities
.... 376
12.
A Calibration of an LSVM with stochastic interest rates and
stochastic dividend yield
.................... 377
13
Marked Branching Diffusions
379
13.1
Nonlinear Monte Carlo algorithms for some
semilinear
PDEs
379
13.1.1
A brute force algorithm
................ 380
13.1.2
Backward stochastic differential equations
..... 380
13.1.3
Gradient representation
................ 381
13.2
Branching diffusions
...................... 386
xvi Contents
13.2.1
Branching diffusions provide a stochastic representa¬
tion of solutions of some semilinear PDEs
...... 386
13.2.2
Superdiffusions
..................... 388
13.3
Marked branching diffusions
.................. 390
13.3.1
Definition and main result
.............. 390
13.3.2
Diagrammatic interpretation
............. 393
13.3.3
When is
Û
bounded?
................. 394
13.3.4
Optimal probabilities Pk
............... 398
13.3.5
Marked
superdiffusions
................ 401
13.3.6
Numerical experiments
................ 402
13.4
Application: Credit valuation adjustment
.......... 405
13.4.1
Introduction
...................... 405
13.4.2
Algorithm: Final recipe
................ 408
13.4.3
Complexity
....................... 409
13.4.4
Numerical examples
.................. 409
13.4.5
CVA formulas
..................... 410
13.4.6
Building polynomial approximations of the positive
part
........................... 413
13.5
System of semilinear PDEs
.................. 415
13.5.1
Introduction
...................... 415
13.5.2
Stochastic representation using multi-species marked
branching diffusions
.................. 417
13.6
Fully nonlinear PDEs
..................... 418
13.6.1
A toy example: The
Kardar-Parisi-Zhang PDE ... 418
13.6.2
Bootstrapping
..................... 418
13.6.3
The uncertain volatility model
............ 419
13.6.4
Exact simulation of one-dimensional SDEs
..... 420
13.7
Exercises
............................ 422
13.7.1
The law of the number of branchings
........ 422
13.7.2
A high order PDE
................... 424
13.7.3
Hyperbolic (nonlinear) PDEs
............. 425
References
427
Index
441
|
| any_adam_object | 1 |
| author | Guyon, Julien Henry-Labordère, Pierre |
| author_facet | Guyon, Julien Henry-Labordère, Pierre |
| author_role | aut aut |
| author_sort | Guyon, Julien |
| author_variant | j g jg p h l phl |
| building | Verbundindex |
| bvnumber | BV040932043 |
| classification_rvk | QK 600 QK 620 SK 980 |
| ctrlnum | (OCoLC)873379563 (DE-599)BVBBV040932043 |
| discipline | Mathematik Wirtschaftswissenschaften |
| format | Book |
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| id | DE-604.BV040932043 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:35:38Z |
| institution | BVB |
| isbn | 9781466570337 1466570334 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025910972 |
| oclc_num | 873379563 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-703 DE-83 DE-824 DE-11 |
| owner_facet | DE-355 DE-BY-UBR DE-703 DE-83 DE-824 DE-11 |
| physical | XXXVIII, 445 S. graph. Darst. |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | CRC Press |
| record_format | marc |
| series2 | Chapman & Hall/CRC financial mathematics series A Chapman & Hall Book |
| spelling | Guyon, Julien Verfasser aut Nonlinear option pricing Julien Guyon ; Pierre Henry-Labordère Boca Raton [u.a.] CRC Press 2014 XXXVIII, 445 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Chapman & Hall/CRC financial mathematics series A Chapman & Hall Book Stochastische Differentialgleichung (DE-588)4057621-8 gnd rswk-swf Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd rswk-swf Optionspreistheorie (DE-588)4135346-8 gnd rswk-swf Finanzmathematik (DE-588)4017195-4 gnd rswk-swf Optionspreistheorie (DE-588)4135346-8 s Finanzmathematik (DE-588)4017195-4 s Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 s Stochastische Differentialgleichung (DE-588)4057621-8 s DE-604 Henry-Labordère, Pierre Verfasser aut Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025910972&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Guyon, Julien Henry-Labordère, Pierre Nonlinear option pricing Stochastische Differentialgleichung (DE-588)4057621-8 gnd Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd Optionspreistheorie (DE-588)4135346-8 gnd Finanzmathematik (DE-588)4017195-4 gnd |
| subject_GND | (DE-588)4057621-8 (DE-588)4128900-6 (DE-588)4135346-8 (DE-588)4017195-4 |
| title | Nonlinear option pricing |
| title_auth | Nonlinear option pricing |
| title_exact_search | Nonlinear option pricing |
| title_full | Nonlinear option pricing Julien Guyon ; Pierre Henry-Labordère |
| title_fullStr | Nonlinear option pricing Julien Guyon ; Pierre Henry-Labordère |
| title_full_unstemmed | Nonlinear option pricing Julien Guyon ; Pierre Henry-Labordère |
| title_short | Nonlinear option pricing |
| title_sort | nonlinear option pricing |
| topic | Stochastische Differentialgleichung (DE-588)4057621-8 gnd Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd Optionspreistheorie (DE-588)4135346-8 gnd Finanzmathematik (DE-588)4017195-4 gnd |
| topic_facet | Stochastische Differentialgleichung Nichtlineare partielle Differentialgleichung Optionspreistheorie Finanzmathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025910972&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT guyonjulien nonlinearoptionpricing AT henrylaborderepierre nonlinearoptionpricing |