Mathematical finance: theory, modeling, implementation
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| Format: | Buch |
| Sprache: | English |
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Hoboken, NJ
Wiley-Interscience
2007
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| Schlagworte: | |
| Online-Zugang: | Table of contents only Contributor biographical information Publisher description Inhaltsverzeichnis |
| Beschreibung: | XXII, 520 S. Ill., graph. Darst. |
| ISBN: | 9780470047224 0470047224 |
Internformat
MARC
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| 015 | |a GBA766147 |2 dnb | ||
| 020 | |a 9780470047224 |c hbk |9 978-0-470-04722-4 | ||
| 020 | |a 0470047224 |c cloth : alk. paper |9 0-470-04722-4 | ||
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| 084 | |a WIR 651f |2 stub | ||
| 084 | |a MAT 902f |2 stub | ||
| 084 | |a WIR 160f |2 stub | ||
| 100 | 1 | |a Fries, Christian |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Mathematical finance |b theory, modeling, implementation |c Christian Fries |
| 264 | 1 | |a Hoboken, NJ |b Wiley-Interscience |c 2007 | |
| 300 | |a XXII, 520 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Instruments dérivés (Finances) - Prix - Modèles mathématiques | |
| 650 | 4 | |a Investissements - Modèles mathématiques | |
| 650 | 4 | |a Valeurs mobilières - Modèles mathématiques | |
| 650 | 4 | |a Mathematisches Modell | |
| 650 | 4 | |a Derivative securities |x Prices |x Mathematical models | |
| 650 | 4 | |a Investments |x Mathematical models | |
| 650 | 4 | |a Securities |x Mathematical models | |
| 650 | 0 | 7 | |a Finanzmathematik |0 (DE-588)4017195-4 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Finanzmathematik |0 (DE-588)4017195-4 |D s |
| 689 | 0 | |5 DE-604 | |
| 856 | 4 | |u http://www.loc.gov/catdir/toc/ecip0713/2007011325.html |3 Table of contents only | |
| 856 | 4 | |u http://www.loc.gov/catdir/enhancements/fy0739/2007011325-b.html |3 Contributor biographical information | |
| 856 | 4 | |u http://www.loc.gov/catdir/enhancements/fy0739/2007011325-d.html |3 Publisher description | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016247746&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
Datensatz im Suchindex
| _version_ | 1805083704538693632 |
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| adam_text |
Contents
1
Introduction
1
1.1
Theory, Modeling, and Implementation
. 1
1.2
Interest Rate Models and Interest Rate Derivatives
. 1
1.3
About This Book
. 3
1.3.1
How to Read This Book
. 3
1.3.2
Abridged Versions
. 3
1.3.3
Special Sections
. 4
1.3.4
Notation
. 4
1.3.5
Feedback
. 5
1.3.6
Resources
. 5
I Foundations
Foundations
9
2.1
Probability Theory
. 9
2.2
Stochastic Processes
. 18
2.3
Filtration
. 20
2.4
Brownian Motion
. 22
2.5
Wiener Measure, Canonical Setup
. 24
2.6
Ito
Calculus
. 25
2.6.1
Ito
Integral
. 28
2.6.2
Ito
Process
. 30
2.6.3
Ito
Lemma and Product Rule
. 32
2.7
Brownian Motion with Instantaneous Correlation
. 36
xi
2.8
Martingales
. 38
2.8.1
Martingale
Representation
Theorem. 38
2.9
Change
of Measure
. 39
2.10
Stochastic Integration
. 44
2.11
Partial Differential Equations (PDEs)
. 46
2.11.1
Feynman-Kač
Theorem
. 46
2.12
List of Symbols
. 48
Replication
49
3.1
Replication Strategies
. 49
3.1.1
Introduction
. 49
3.1.2
Replication in a Discrete Model
. 53
3.2
Foundations: Equivalent Martingale Measure
. 58
3.2.1
Challenge and Solution Outline
. 58
3.2.2
Steps toward the Universal Pricing Theorem
. 61
3.3
Excursus: Relative Prices and Risk-Neutral Measures
. 70
3.3.1
Why relative prices?
. 70
3.3.2
Risk-Neutral Measure
. 72
II First Applications
73
4
Pricing of a European Stock Option under the Black-Scholes
Model
75
5
Excursus: The Density of the Underlying of a European Call
Option
81
6
Excursus: Interpolation of European Option Prices
83
6.1
No-Arbitrage Conditions for Interpolated Prices
. 83
6.2
Arbitrage Violations through Interpolation
. 85
6.2.1
Example
1 :
Interpolation of Four Prices
. 85
6.2.2
Example
2:
Interpolation of Two Prices
. 87
6.3
Arbitrage-Free Interpolation of European Option Prices
. 89
7
Hedging in Continuous and Discrete Time and the Greeks
93
7.1
Introduction
. 93
7.2
Deriving the Replications Strategy from Pricing Theory
. 94
7.2.1
Deriving the Replication Strategy under the Assumption
of a Locally Riskless Product
. 96
хи
7.2.2
Black-Scholes Differential Equation
. 97
7.2.3
Derivative V(t) as a Function of Its Underlyings
S¡(t)
. . 97
7.2.4
Example: Replication Portfolio and PDE under a Black-
Scholes Model
. 99
7.3
Greeks
. 102
7.3.1
Greeks of a European Call-Option under the Black-
Scholes Model
. 103
7.4
Hedging in Discrete Time: Delta and Delta-Gamma Hedging
. 103
7.4.1
Delta Hedging
. 105
7.4.2
Error Propagation
. 106
7.4.3
Delta-Gamma Hedging
. 109
7.4.4 Vega
Hedging
. 113
7.5
Hedging in Discrete Time: Minimizing the Residual Error
(Bouchaud-Sornette Method)
. 113
7.5.)
Minimizing the Residual Error at Maturity
Τ
. 115
7.5.2
Minimizing the Residual Error in Each Time Step
. . 117
III Interest Rate Structures, Interest Rate Products,
and Analytic Pricing Formulas
119
Motivation and Overview
. 12
1
б
Interest Rate Structures
123
8.1
Introduction
. 123
8.1.1
Fixing Times and Tenor Times
. 124
8.2
Definitions
. 124
8.3
Interest Rate Curve Bootstrapping
. 130
8.4
Interpolation of Interest Rate Curves
. 130
8.5
Implementation
. 131
9
Simple Interest Rate Products
133
9.1
Interest Rate Products Part
1 :
Products without Optionality
. 133
9.1.1
Fix, Floating, and Swap
. 133
9.1.2
Money Market Account
. 140
9.2
Interest Rate Products Part
2:
Simple Options
. 142
9.2.1
Cap, Floor, and Swaption
. 142
9.2.2
Foreign Caplet.
Quanto
. 144
10
The Black Model for a Caplet
147
Xlii
11
Pricing of
a Quanto
Caplet (Modeling the FFX)
151
11.1
Choice of
Numéraire
. 151
12
Exotic Derivatives
155
12.1
Prototypical Product Properties
. 155
12.2
Interest Rate Products Part
3:
Exotic Interest Rate Derivatives
. . . 157
12.2.1
Structured Bond, Structured Swap, and Zero Structure
. 158
12.2.2
Bermudán
Option
. 162
12.2.3
Bermudán
Callable and
Bermudán Cancelable
. 164
12.2.4
Compound Options
. 166
12.2.5
Trigger Products
. 167
12.2.6
Structured Coupons
. 168
12.2.7
Shout Options
. 173
12.3
Product Toolbox
. 174
IV Discretization and Numerical Valuation Methods
177
Motivation and Overview
. 179
13
Discretization of Time and State Space
181
13.1
Discretization of Time: The
Euler
and the Milstein Schemes
. 181
13.1.1
Definitions
. 183
13.1.2
Time Discretization of
a
Lognormal
Process
. 185
13.2
Discretization of Paths (Monte Carlo Simulation)
. 186
13.2.1
Monte Carlo Simulation
. 187
13.2.2
Weighted Monte Carlo Simulation
. 187
13.2.3
Implementation
. 188
13.2.4
Review
. 193
13.3
Discretization of State Space
. 195
13.3.1
Definitions
. 195
13.3.2
Backward Algorithm
. 197
13.3.3
Review
. 197
13.4
Path Simulation through a Lattice: Two Layers
. 198
14
Numerical Methods for Partial Differential Equations
199
15
Pricing
Bermudán
Options in a Monte Carlo Simulation
201
15.1
Introduction
. 201
15.2
Bermudán
Options: Notation
. 202
15.2.1
Bermudán
Callable
. 203
xiv
15.2.2 Relative
Prices
. 203
15.3
Bermudán
Option
as
Optimal
Exercise
Problem. 204
15.3.1
Bermudán
Option
Value as Single (Unconditioned) Ex¬
pectation: The Optimal Exercise Value
. 204
15.4
Bermudán
Option Pricing
—
The Backward Algorithm
. 205
15.5
Resimulation
. 207
15.6
Perfect Foresight
. 207
15.7
Conditional Expectation as Functional Dependence
. 209
15.8
Binning
. 210
15.8.1
Binning as a Least-Square Regression
. 212
15.9
Foresight Bias
. 214
15.10
Regression Methods
—
Least-Square Monte Carlo
. 215
15.10.1
Least-Square Approximation of the Conditional Expectation215
15.10.2
Example: Evaluation of
a Bermudán
Option on a Stock
(Backward Algorithm with Conditional Expectation Esti¬
mator)
. 216
15.10.3
Example: Evaluation of
a Bermudán
Callable
. 217
15.10.4
Implementation
. 222
15.10.5
Binning as Linear Least-Square Regression
. 223
15.11
Optimization Methods
. 224
15.11.1
Andersen Algorithm for
Bermudán
Swaptions
. 224
15.11.2
Review of the Threshold Optimization Method
. 225
15.11.3
Optimization of Exercise Strategy: A More General For¬
mulation
. 228
15.11.4
Comparison of Optimization Method and Regression
Method
. 228
15.12
Duality Method: Upper Bound for
Bermudán
Option Prices
. . . . 230
15.12.1
Foundations
. 230
15.12.2
American Option Evaluation as Optimal Stopping Problem232
15.13
Primal-Dual Method: Upper and Lower Bound
. 235
16
Pricing Path-Dependent Options in a Backward Algorithm
237
16.1
State Space Extension
. 237
16.2
Implementation
. 238
16.3
Path-Dependent
Bermudán
Options
. 239
16.4
Examples
. 240
16.4.1
Evaluation of a Snowball in a Backward Algorithm
. . . 240
16.4.2
Evaluation of a Autocap in a Backward Algorithm
. . . . 240
17
Sensitivities (Partial Derivatives) of Monte Carlo Prices
243
17.1
Introduction
. 243
xv
17.2 Problem
Description
. 244
17.2.1
Pricing using
Monte-Carlo
Simulation. 244
17.2.2
Sensitivities from Monte Carlo Pricing
. 245
17.2.3
Example: The Linear and the Discontinuous Payout
. . 245
17.2.4
Example: Trigger Products
. 247
17.3
Generic Sensitivities: Bumping the Model
. 249
17.4
Sensitivities by Finite Differences
. 251
17.4.1
Example: Finite Differences Applied to Smooth and Dis¬
continuous Payout
. 252
17.5
Sensitivities by Pathwise Differentiation
. 254
17.5.1
Example: Delta of a European Option under a Black-
Scholes Model
. 254
17.5.2
Pathwise Differentiation for Discontinuous Payouts
. . . 255
17.6
Sensitivities by Likelihood Ratio Weighting
. 256
17.6.1
Example: Delta of a European Option under a Black-
Scholes Model Using Pathwise Derivative
. 257
17.6.2
Example: Variance Increase of the Sensitivity when using
Likelihood Ratio Method for Smooth Payouts
. 257
17.7
Sensitivities by Malliavin Weighting
. 258
17.8
Proxy Simulation Scheme
. 259
18
Proxy Simulation Schemes for Monte Carlo Sensitivities and
Importance Sampling
261
18.1
Full Proxy Simulation Scheme
. 261
18.1.1
Pricing under a Proxy Simulation Scheme
. 262
18.1.2
Calculation of Monte Carlo Weights
. 262
18.1.3
Sensitivities by Finite Differences on a Proxy Simulation
Scheme
. 263
18.1.4
Localization
. 264
18.1.5
Object-Oriented Design
. 265
18.1.6
Importance Sampling
. 265
18.2
Partial Proxy Simulation Schemes
. 268
18.2.1
Linear Proxy Constraint
. 268
18.2.2
Comparison to Full Proxy Scheme Method
. 269
18.2.3
Nonlinear Proxy Constraint
. 269
18.2.4
Transition Probability from a Nonlinear Proxy Constraint
271
18.2.5
Sensitivity with Respect to the Diffusion Coefficients
—Vega274
18.2.6
Example:
LIBOR
Target Redemption Note
. 274
18.2.7
Example: CMS Target Redemption Note
. 276
18.3
Localized Proxy Simulation Schemes
. 279
18.3.1
Problem Description
. 279
xvi
18.3.2
Solution
. 282
18.3.3
Partial Proxy Simulation
Scheme
(revisited)
. 282
18.3.4
Localized
Proxy Simulation
Scheme
. 285
18.3.5
Example:
Euler
Schemes
. 286
18.3.6
Implementation
. 286
18.3.7
Examples and Numerical Results
. 287
V Pricing Models for Interest Rate Derivatives
293
Motivation and Overview
. 295
19
LIBOR
Market Model
297
19.1
Derivation of the Drift Term
. 299
19.1.1
Derivation of the Drift Term under the Terminal Measure
299
19.1.2
Derivation of the Drift Term under the Spot
LIBOR Measure301
19.1.3
Derivation of the Drift Term under the
Гд
-Forward
МеаѕигеЗОЗ
19.2
The Short Period Bond P(Tm{t)+\\t)
. 304
19.2.1
Role of the Short Bond in
a LIBOR
Market Model
. . . 304
19.2.2
Link to Continuous Time Tenors
. 304
19.2.3
Drift of the Short Bond in
a LIBOR
Market Model
. . . 304
19.3
Discretization and (Monte Carlo) Simulation
. 305
19.3.1
Generation of the (Time-Discrete) Forward Rate Process
305
19.3.2
Generation of the Sample Paths
. 306
19.3.3
Generation of the
Numéraire
. 306
19.4
Calibration
—
Choice of the Free Parameters
. 307
19.4.1
Choice of the Initial Conditions
. 308
19.4.2
Choice of the Volatilities
. 308
19.4.3
Choice of the Correlations
. 311
19.4.4
Covariance Structure
. 313
19.4.5
Analytic Evaluation of Caplets.
S
waptions and Swap Rate
Covariance
. 314
19.5
Interpolation of Forward Rates in the
LIBOR
Market Model
. 319
19.5.1
Interpolation of the Tenor Structure
{T¡\
. 319
19.6
Object-Oriented Design
. 323
19.6.1
Reuse of Implementation
. 324
19.6.2
Separation of Product and Model
. 324
19.6.3
Abstraction of Model Parameters
. 324
19.6.4
Abstraction of Calibration
. 325
20
Swap Rate Market Models
329
XVII
20.1
The Swap Measure
. 330
20.2
Derivation of the Drift Term
. 331
20.3
Calibration
—
Choice of the Free Parameters
. 332
20.3.1
Choice of the Initial Conditions
. 332
20.3.2
Choice of the Volatilities
. 332
21
Excursus: Instantaneous Correlation and Terminal Correlation
335
21.1
Definitions
. 335
21.2
Terminal Correlation Examined in
a LIBOR
Market Model Example
336
21.2.1
Decorrelation in a One-Factor Model
. 337
21.2.2
Impact of the Time Structure of the Instantaneous Volatil¬
ity on Caplet and Swaption Prices
. 339
21.2.3
Swaption Value as a Function of Forward Rates
. 340
21.3
Terminal Correlation Is Dependent on the Equivalent Martingale
Measure
. 342
21.3.1
Dependence of the Terminal Density on the Martingale
Measure
. 342
22
Heath-Jarrow-Morton Framework: Foundations
345
22.1
Short-Rate Process in the HJM Framework
. 346
22.2
The HJM Drift Condition
. 347
23
Short-Rate Models
351
23.1
Introduction
. 351
23.2
The Market Price of Risk
. 352
23.3
Overview: Some Common Models
. 354
23.4
Implementations
. 355
23.4.1
Monte Carlo Implementation of Short-Rate Models
. . . 355
23.4.2
Lattice Implementation of Short-Rate Models
. 355
24
Heath-Jarrow-Morton Framework: Immersion of Short-Rate
Models and
LIBOR
Market Model
357
24.1
Short-Rate Models in the HJM Framework
. 357
24.1.1
Example: The
Но
-Lee
Model in the HJM Framework
. . 358
24.1.2
Example: The Hull-White Model in the HJM Framework
359
24.2
LIBOR
Market Model in the HJM Framework
. 360
24.2.1
HJM Volatility Structure of the
LIBOR
Market Model
. 360
24.2.2
LIBOR
Market Model Drift under the QB Measure
. 362
24.2.3
LIBOR
Market Model as a Short Rate Model
. 364
xviii
25
Excursus: Shape of the Interest Rate Curve under Mean Rever¬
sion and a Multifactor Model
365
25.1
Model
. 365
25.2
Interpretation of the Figures
. 366
25.3
Mean Reversion
. 367
25.4
Factors
. 368
25.5
Exponential Volatility Function
. 369
25.6
Instantaneous Correlation
. 371
26
Ritchken-Sakarasubramanian Framework: HJM with Low
Markov Dimension
373
26.1
Introduction
. 373
26.2
Cheyette Model
. 374
26.3
Implementation: PDE
. 375
27
Markov Functional Models
377
27.1
Introduction
. 377
27.1.1
The Markov Functional Assumption (Independent of the
Model Considered)
. 378
27.1.2
Outline of This Chapter
. 379
27.2
Equity Markov Functional Model
. 379
27.2.1
Markov Functional Assumption
. 379
27.2.2
Example: The Black-Scholes Model
. 380
27.2.3
Numerical Calibration to a Full Two-Dimensional Euro¬
pean Option Smile Surface
. 381
27.2.4
Interest Rates
. 383
27.2.5
Model Dynamics
. 384
27.2.6
Implementation
. 390
27.3
LIBOR
Markov Functional Model
. 390
27.3.1
LIBOR
Markov Functional Model in Terminal Measure
390
27.3.2
LIBOR
Markov Functional Model in Spot Measure
. 396
27.3.3
Remark on Implementation
. 400
27.3.4
Change of
Numéraire
in a Markov Functional Model
. . 401
27.4
Implementation: Lattice
. 403
27.4.1
Convolution with the Normal Probability Density
. 404
27.4.2
State Space Discretization
. 407
VI Extended Models
409
XIX
28
Credit Spreads
411
28.1
Introduction
—
Different Types of Spreads
. 411
28.1.1
Spread
on a
Coupon
. 411
28.1.2
Credit Spread
. 411
28.2
Defaultable
Bonds
. 412
28.3
Integrating Deterministic
Credit Spread
into a Pricing
Model
. 414
28.3.1
Deterministic Credit Spread
. 415
28.3.2
Implementation
. 416
28.4
Receiver's and Payer's Credit Spreads
. 418
28.4.1
Example: Defaultable Forward Starting Coupon Bond
. 419
28.4.2
Example: Option on a Defaultable Coupon Bond
. 420
29
Hybrid Models
421
29.1
Cross-Currency
LIBOR
Market Model
. 421
29.1.1
Derivation of the Drift Term under Spot Measure
. 422
29.1.2
Implementation
. 426
29.2
Equity Hybrid
LIBOR
Market Model
. 426
29.2.1
Derivation of the Drift Term under Spot-Measure
. 426
29.2.2
Implementation
. 428
29.3
Equity Hybrid Cross-Currency
LIBOR
Market Model
. 428
29.3.1
Dynamic of the Foreign Stock under Spot Measure
. . . 429
29.3.2
Summary
. 430
29.3.3
Implementation
. 431
VII
Implementation
433
30
Object-Oriented Implementation in Java™
435
30.1
Elements of Object-Oriented Programming: Class and Objects
. . 435
30.1.1
Example: Class of a Binomial Distributed Random Variable436
30.1.2
Constructor
. 438
30.1.3
Methods: Getter, Setter, and Static Methods
. 438
30.2
Principles of Object Oriented Programming
. 440
30.2.1
Encapsulation and Interfaces
. 440
30.2.2
Abstraction and Inheritance
. 444
30.2.3
Polymorphism
. 447
30.3
Example: A Class Structure for One-Dimensional Root Finders
. . 449
30.3.1
Root Finder for General Functions
. 449
30.3.2
Root Finder for Functions with Analytic Derivative: New¬
ton's Method
. 451
xx
30.3.3
Root Finder for Functions with Derivative Estimation:
Secant Method
. 452
30.4
Anatomy of a Java™ Class
. 458
30.5
Libraries
. 460
30.5.1
Java™
2
Platform, Standard Edition (j2se)
. 460
30.5.2
Java™
2
Platform, Enterprise Edition (j2ee)
. 460
30.5.3
Colt
. 460
30.5.4
Commons-Math: The Jakarta Mathematics Library
. 461
30.6
Some Final Remarks
. 461
30.6.1
Object-Oriented Design (OOD)/Unined Modeling Lan¬
guage (UML)
. 461
VIII
Appendices
463
A A Small Collection of Common Misconceptions
465
В
Tools (Selection)
467
B.I Generation of Random Numbers
. 467
B.I.I Uniform Distributed Random Variables
. 467
B.
1.2
Transformation of the Random Number Distribution via
the Inverse Distribution Function
. 468
B.1.3 Normal Distributed Random Variables
. 468
B.1.4
Poisson
Distributed Random Variables
. 468
B.
1.5
Generation of Paths of an
«-Dimensional
Brownian Motion469
B.2 Factor Decomposition
—
Generation of Correlated Brownian Motion
471
B.3 Factor Reduction
. 472
B.4 Optimization (One-Dimensional): Golden Section Search
. 475
B.5 Linear Regression
. 476
B.6 Convolution with Normal Density
. 477
С
Exercises
479
D
Java™ Source Code (Selection)
487
D.I Java™ Classes for Chapter
30 . 487
List of Symbols
493
List of Figures
495
List of Tables
499
XXI
List of Listings
500
Bibliography
503
Index 511
XXII |
| adam_txt |
Contents
1
Introduction
1
1.1
Theory, Modeling, and Implementation
. 1
1.2
Interest Rate Models and Interest Rate Derivatives
. 1
1.3
About This Book
. 3
1.3.1
How to Read This Book
. 3
1.3.2
Abridged Versions
. 3
1.3.3
Special Sections
. 4
1.3.4
Notation
. 4
1.3.5
Feedback
. 5
1.3.6
Resources
. 5
I Foundations
Foundations
9
2.1
Probability Theory
. 9
2.2
Stochastic Processes
. 18
2.3
Filtration
. 20
2.4
Brownian Motion
. 22
2.5
Wiener Measure, Canonical Setup
. 24
2.6
Ito
Calculus
. 25
2.6.1
Ito
Integral
. 28
2.6.2
Ito
Process
. 30
2.6.3
Ito
Lemma and Product Rule
. 32
2.7
Brownian Motion with Instantaneous Correlation
. 36
xi
2.8
Martingales
. 38
2.8.1
Martingale
Representation
Theorem. 38
2.9
Change
of Measure
. 39
2.10
Stochastic Integration
. 44
2.11
Partial Differential Equations (PDEs)
. 46
2.11.1
Feynman-Kač
Theorem
. 46
2.12
List of Symbols
. 48
Replication
49
3.1
Replication Strategies
. 49
3.1.1
Introduction
. 49
3.1.2
Replication in a Discrete Model
. 53
3.2
Foundations: Equivalent Martingale Measure
. 58
3.2.1
Challenge and Solution Outline
. 58
3.2.2
Steps toward the Universal Pricing Theorem
. 61
3.3
Excursus: Relative Prices and Risk-Neutral Measures
. 70
3.3.1
Why relative prices?
. 70
3.3.2
Risk-Neutral Measure
. 72
II First Applications
73
4
Pricing of a European Stock Option under the Black-Scholes
Model
75
5
Excursus: The Density of the Underlying of a European Call
Option
81
6
Excursus: Interpolation of European Option Prices
83
6.1
No-Arbitrage Conditions for Interpolated Prices
. 83
6.2
Arbitrage Violations through Interpolation
. 85
6.2.1
Example
1 :
Interpolation of Four Prices
. 85
6.2.2
Example
2:
Interpolation of Two Prices
. 87
6.3
Arbitrage-Free Interpolation of European Option Prices
. 89
7
Hedging in Continuous and Discrete Time and the Greeks
93
7.1
Introduction
. 93
7.2
Deriving the Replications Strategy from Pricing Theory
. 94
7.2.1
Deriving the Replication Strategy under the Assumption
of a Locally Riskless Product
. 96
хи
7.2.2
Black-Scholes Differential Equation
. 97
7.2.3
Derivative V(t) as a Function of Its Underlyings
S¡(t)
. . 97
7.2.4
Example: Replication Portfolio and PDE under a Black-
Scholes Model
. 99
7.3
Greeks
. 102
7.3.1
Greeks of a European Call-Option under the Black-
Scholes Model
. 103
7.4
Hedging in Discrete Time: Delta and Delta-Gamma Hedging
. 103
7.4.1
Delta Hedging
. 105
7.4.2
Error Propagation
. 106
7.4.3
Delta-Gamma Hedging
. 109
7.4.4 Vega
Hedging
. 113
7.5
Hedging in Discrete Time: Minimizing the Residual Error
(Bouchaud-Sornette Method)
. 113
7.5.)
Minimizing the Residual Error at Maturity
Τ
. 115
7.5.2
Minimizing the Residual Error in Each Time Step
. . 117
III Interest Rate Structures, Interest Rate Products,
and Analytic Pricing Formulas
119
Motivation and Overview
. 12
1
б
Interest Rate Structures
123
8.1
Introduction
. 123
8.1.1
Fixing Times and Tenor Times
. 124
8.2
Definitions
. 124
8.3
Interest Rate Curve Bootstrapping
. 130
8.4
Interpolation of Interest Rate Curves
. 130
8.5
Implementation
. 131
9
Simple Interest Rate Products
133
9.1
Interest Rate Products Part
1 :
Products without Optionality
. 133
9.1.1
Fix, Floating, and Swap
. 133
9.1.2
Money Market Account
. 140
9.2
Interest Rate Products Part
2:
Simple Options
. 142
9.2.1
Cap, Floor, and Swaption
. 142
9.2.2
Foreign Caplet.
Quanto
. 144
10
The Black Model for a Caplet
147
Xlii
11
Pricing of
a Quanto
Caplet (Modeling the FFX)
151
11.1
Choice of
Numéraire
. 151
12
Exotic Derivatives
155
12.1
Prototypical Product Properties
. 155
12.2
Interest Rate Products Part
3:
Exotic Interest Rate Derivatives
. . . 157
12.2.1
Structured Bond, Structured Swap, and Zero Structure
. 158
12.2.2
Bermudán
Option
. 162
12.2.3
Bermudán
Callable and
Bermudán Cancelable
. 164
12.2.4
Compound Options
. 166
12.2.5
Trigger Products
. 167
12.2.6
Structured Coupons
. 168
12.2.7
Shout Options
. 173
12.3
Product Toolbox
. 174
IV Discretization and Numerical Valuation Methods
177
Motivation and Overview
. 179
13
Discretization of Time and State Space
181
13.1
Discretization of Time: The
Euler
and the Milstein Schemes
. 181
13.1.1
Definitions
. 183
13.1.2
Time Discretization of
a
Lognormal
Process
. 185
13.2
Discretization of Paths (Monte Carlo Simulation)
. 186
13.2.1
Monte Carlo Simulation
. 187
13.2.2
Weighted Monte Carlo Simulation
. 187
13.2.3
Implementation
. 188
13.2.4
Review
. 193
13.3
Discretization of State Space
. 195
13.3.1
Definitions
. 195
13.3.2
Backward Algorithm
. 197
13.3.3
Review
. 197
13.4
Path Simulation through a Lattice: Two Layers
. 198
14
Numerical Methods for Partial Differential Equations
199
15
Pricing
Bermudán
Options in a Monte Carlo Simulation
201
15.1
Introduction
. 201
15.2
Bermudán
Options: Notation
. 202
15.2.1
Bermudán
Callable
. 203
xiv
15.2.2 Relative
Prices
. 203
15.3
Bermudán
Option
as
Optimal
Exercise
Problem. 204
15.3.1
Bermudán
Option
Value as Single (Unconditioned) Ex¬
pectation: The Optimal Exercise Value
. 204
15.4
Bermudán
Option Pricing
—
The Backward Algorithm
. 205
15.5
Resimulation
. 207
15.6
Perfect Foresight
. 207
15.7
Conditional Expectation as Functional Dependence
. 209
15.8
Binning
. 210
15.8.1
Binning as a Least-Square Regression
. 212
15.9
Foresight Bias
. 214
15.10
Regression Methods
—
Least-Square Monte Carlo
. 215
15.10.1
Least-Square Approximation of the Conditional Expectation215
15.10.2
Example: Evaluation of
a Bermudán
Option on a Stock
(Backward Algorithm with Conditional Expectation Esti¬
mator)
. 216
15.10.3
Example: Evaluation of
a Bermudán
Callable
. 217
15.10.4
Implementation
. 222
15.10.5
Binning as Linear Least-Square Regression
. 223
15.11
Optimization Methods
. 224
15.11.1
Andersen Algorithm for
Bermudán
Swaptions
. 224
15.11.2
Review of the Threshold Optimization Method
. 225
15.11.3
Optimization of Exercise Strategy: A More General For¬
mulation
. 228
15.11.4
Comparison of Optimization Method and Regression
Method
. 228
15.12
Duality Method: Upper Bound for
Bermudán
Option Prices
. . . . 230
15.12.1
Foundations
. 230
15.12.2
American Option Evaluation as Optimal Stopping Problem232
15.13
Primal-Dual Method: Upper and Lower Bound
. 235
16
Pricing Path-Dependent Options in a Backward Algorithm
237
16.1
State Space Extension
. 237
16.2
Implementation
. 238
16.3
Path-Dependent
Bermudán
Options
. 239
16.4
Examples
. 240
16.4.1
Evaluation of a Snowball in a Backward Algorithm
. . . 240
16.4.2
Evaluation of a Autocap in a Backward Algorithm
. . . . 240
17
Sensitivities (Partial Derivatives) of Monte Carlo Prices
243
17.1
Introduction
. 243
xv
17.2 Problem
Description
. 244
17.2.1
Pricing using
Monte-Carlo
Simulation. 244
17.2.2
Sensitivities from Monte Carlo Pricing
. 245
17.2.3
Example: The Linear and the Discontinuous Payout
. . 245
17.2.4
Example: Trigger Products
. 247
17.3
Generic Sensitivities: Bumping the Model
. 249
17.4
Sensitivities by Finite Differences
. 251
17.4.1
Example: Finite Differences Applied to Smooth and Dis¬
continuous Payout
. 252
17.5
Sensitivities by Pathwise Differentiation
. 254
17.5.1
Example: Delta of a European Option under a Black-
Scholes Model
. 254
17.5.2
Pathwise Differentiation for Discontinuous Payouts
. . . 255
17.6
Sensitivities by Likelihood Ratio Weighting
. 256
17.6.1
Example: Delta of a European Option under a Black-
Scholes Model Using Pathwise Derivative
. 257
17.6.2
Example: Variance Increase of the Sensitivity when using
Likelihood Ratio Method for Smooth Payouts
. 257
17.7
Sensitivities by Malliavin Weighting
. 258
17.8
Proxy Simulation Scheme
. 259
18
Proxy Simulation Schemes for Monte Carlo Sensitivities and
Importance Sampling
261
18.1
Full Proxy Simulation Scheme
. 261
18.1.1
Pricing under a Proxy Simulation Scheme
. 262
18.1.2
Calculation of Monte Carlo Weights
. 262
18.1.3
Sensitivities by Finite Differences on a Proxy Simulation
Scheme
. 263
18.1.4
Localization
. 264
18.1.5
Object-Oriented Design
. 265
18.1.6
Importance Sampling
. 265
18.2
Partial Proxy Simulation Schemes
. 268
18.2.1
Linear Proxy Constraint
. 268
18.2.2
Comparison to Full Proxy Scheme Method
. 269
18.2.3
Nonlinear Proxy Constraint
. 269
18.2.4
Transition Probability from a Nonlinear Proxy Constraint
271
18.2.5
Sensitivity with Respect to the Diffusion Coefficients
—Vega274
18.2.6
Example:
LIBOR
Target Redemption Note
. 274
18.2.7
Example: CMS Target Redemption Note
. 276
18.3
Localized Proxy Simulation Schemes
. 279
18.3.1
Problem Description
. 279
xvi
18.3.2
Solution
. 282
18.3.3
Partial Proxy Simulation
Scheme
(revisited)
. 282
18.3.4
Localized
Proxy Simulation
Scheme
. 285
18.3.5
Example:
Euler
Schemes
. 286
18.3.6
Implementation
. 286
18.3.7
Examples and Numerical Results
. 287
V Pricing Models for Interest Rate Derivatives
293
Motivation and Overview
. 295
19
LIBOR
Market Model
297
19.1
Derivation of the Drift Term
. 299
19.1.1
Derivation of the Drift Term under the Terminal Measure
299
19.1.2
Derivation of the Drift Term under the Spot
LIBOR Measure301
19.1.3
Derivation of the Drift Term under the
Гд
-Forward
МеаѕигеЗОЗ
19.2
The Short Period Bond P(Tm{t)+\\t)
. 304
19.2.1
Role of the Short Bond in
a LIBOR
Market Model
. . . 304
19.2.2
Link to Continuous Time Tenors
. 304
19.2.3
Drift of the Short Bond in
a LIBOR
Market Model
. . . 304
19.3
Discretization and (Monte Carlo) Simulation
. 305
19.3.1
Generation of the (Time-Discrete) Forward Rate Process
305
19.3.2
Generation of the Sample Paths
. 306
19.3.3
Generation of the
Numéraire
. 306
19.4
Calibration
—
Choice of the Free Parameters
. 307
19.4.1
Choice of the Initial Conditions
. 308
19.4.2
Choice of the Volatilities
. 308
19.4.3
Choice of the Correlations
. 311
19.4.4
Covariance Structure
. 313
19.4.5
Analytic Evaluation of Caplets.
S
waptions and Swap Rate
Covariance
. 314
19.5
Interpolation of Forward Rates in the
LIBOR
Market Model
. 319
19.5.1
Interpolation of the Tenor Structure
{T¡\
. 319
19.6
Object-Oriented Design
. 323
19.6.1
Reuse of Implementation
. 324
19.6.2
Separation of Product and Model
. 324
19.6.3
Abstraction of Model Parameters
. 324
19.6.4
Abstraction of Calibration
. 325
20
Swap Rate Market Models
329
XVII
20.1
The Swap Measure
. 330
20.2
Derivation of the Drift Term
. 331
20.3
Calibration
—
Choice of the Free Parameters
. 332
20.3.1
Choice of the Initial Conditions
. 332
20.3.2
Choice of the Volatilities
. 332
21
Excursus: Instantaneous Correlation and Terminal Correlation
335
21.1
Definitions
. 335
21.2
Terminal Correlation Examined in
a LIBOR
Market Model Example
336
21.2.1
Decorrelation in a One-Factor Model
. 337
21.2.2
Impact of the Time Structure of the Instantaneous Volatil¬
ity on Caplet and Swaption Prices
. 339
21.2.3
Swaption Value as a Function of Forward Rates
. 340
21.3
Terminal Correlation Is Dependent on the Equivalent Martingale
Measure
. 342
21.3.1
Dependence of the Terminal Density on the Martingale
Measure
. 342
22
Heath-Jarrow-Morton Framework: Foundations
345
22.1
Short-Rate Process in the HJM Framework
. 346
22.2
The HJM Drift Condition
. 347
23
Short-Rate Models
351
23.1
Introduction
. 351
23.2
The Market Price of Risk
. 352
23.3
Overview: Some Common Models
. 354
23.4
Implementations
. 355
23.4.1
Monte Carlo Implementation of Short-Rate Models
. . . 355
23.4.2
Lattice Implementation of Short-Rate Models
. 355
24
Heath-Jarrow-Morton Framework: Immersion of Short-Rate
Models and
LIBOR
Market Model
357
24.1
Short-Rate Models in the HJM Framework
. 357
24.1.1
Example: The
Но
-Lee
Model in the HJM Framework
. . 358
24.1.2
Example: The Hull-White Model in the HJM Framework
359
24.2
LIBOR
Market Model in the HJM Framework
. 360
24.2.1
HJM Volatility Structure of the
LIBOR
Market Model
. 360
24.2.2
LIBOR
Market Model Drift under the QB Measure
. 362
24.2.3
LIBOR
Market Model as a Short Rate Model
. 364
xviii
25
Excursus: Shape of the Interest Rate Curve under Mean Rever¬
sion and a Multifactor Model
365
25.1
Model
. 365
25.2
Interpretation of the Figures
. 366
25.3
Mean Reversion
. 367
25.4
Factors
. 368
25.5
Exponential Volatility Function
. 369
25.6
Instantaneous Correlation
. 371
26
Ritchken-Sakarasubramanian Framework: HJM with Low
Markov Dimension
373
26.1
Introduction
. 373
26.2
Cheyette Model
. 374
26.3
Implementation: PDE
. 375
27
Markov Functional Models
377
27.1
Introduction
. 377
27.1.1
The Markov Functional Assumption (Independent of the
Model Considered)
. 378
27.1.2
Outline of This Chapter
. 379
27.2
Equity Markov Functional Model
. 379
27.2.1
Markov Functional Assumption
. 379
27.2.2
Example: The Black-Scholes Model
. 380
27.2.3
Numerical Calibration to a Full Two-Dimensional Euro¬
pean Option Smile Surface
. 381
27.2.4
Interest Rates
. 383
27.2.5
Model Dynamics
. 384
27.2.6
Implementation
. 390
27.3
LIBOR
Markov Functional Model
. 390
27.3.1
LIBOR
Markov Functional Model in Terminal Measure
390
27.3.2
LIBOR
Markov Functional Model in Spot Measure
. 396
27.3.3
Remark on Implementation
. 400
27.3.4
Change of
Numéraire
in a Markov Functional Model
. . 401
27.4
Implementation: Lattice
. 403
27.4.1
Convolution with the Normal Probability Density
. 404
27.4.2
State Space Discretization
. 407
VI Extended Models
409
XIX
28
Credit Spreads
411
28.1
Introduction
—
Different Types of Spreads
. 411
28.1.1
Spread
on a
Coupon
. 411
28.1.2
Credit Spread
. 411
28.2
Defaultable
Bonds
. 412
28.3
Integrating Deterministic
Credit Spread
into a Pricing
Model
. 414
28.3.1
Deterministic Credit Spread
. 415
28.3.2
Implementation
. 416
28.4
Receiver's and Payer's Credit Spreads
. 418
28.4.1
Example: Defaultable Forward Starting Coupon Bond
. 419
28.4.2
Example: Option on a Defaultable Coupon Bond
. 420
29
Hybrid Models
421
29.1
Cross-Currency
LIBOR
Market Model
. 421
29.1.1
Derivation of the Drift Term under Spot Measure
. 422
29.1.2
Implementation
. 426
29.2
Equity Hybrid
LIBOR
Market Model
. 426
29.2.1
Derivation of the Drift Term under Spot-Measure
. 426
29.2.2
Implementation
. 428
29.3
Equity Hybrid Cross-Currency
LIBOR
Market Model
. 428
29.3.1
Dynamic of the Foreign Stock under Spot Measure
. . . 429
29.3.2
Summary
. 430
29.3.3
Implementation
. 431
VII
Implementation
433
30
Object-Oriented Implementation in Java™
435
30.1
Elements of Object-Oriented Programming: Class and Objects
. . 435
30.1.1
Example: Class of a Binomial Distributed Random Variable436
30.1.2
Constructor
. 438
30.1.3
Methods: Getter, Setter, and Static Methods
. 438
30.2
Principles of Object Oriented Programming
. 440
30.2.1
Encapsulation and Interfaces
. 440
30.2.2
Abstraction and Inheritance
. 444
30.2.3
Polymorphism
. 447
30.3
Example: A Class Structure for One-Dimensional Root Finders
. . 449
30.3.1
Root Finder for General Functions
. 449
30.3.2
Root Finder for Functions with Analytic Derivative: New¬
ton's Method
. 451
xx
30.3.3
Root Finder for Functions with Derivative Estimation:
Secant Method
. 452
30.4
Anatomy of a Java™ Class
. 458
30.5
Libraries
. 460
30.5.1
Java™
2
Platform, Standard Edition (j2se)
. 460
30.5.2
Java™
2
Platform, Enterprise Edition (j2ee)
. 460
30.5.3
Colt
. 460
30.5.4
Commons-Math: The Jakarta Mathematics Library
. 461
30.6
Some Final Remarks
. 461
30.6.1
Object-Oriented Design (OOD)/Unined Modeling Lan¬
guage (UML)
. 461
VIII
Appendices
463
A A Small Collection of Common Misconceptions
465
В
Tools (Selection)
467
B.I Generation of Random Numbers
. 467
B.I.I Uniform Distributed Random Variables
. 467
B.
1.2
Transformation of the Random Number Distribution via
the Inverse Distribution Function
. 468
B.1.3 Normal Distributed Random Variables
. 468
B.1.4
Poisson
Distributed Random Variables
. 468
B.
1.5
Generation of Paths of an
«-Dimensional
Brownian Motion469
B.2 Factor Decomposition
—
Generation of Correlated Brownian Motion
471
B.3 Factor Reduction
. 472
B.4 Optimization (One-Dimensional): Golden Section Search
. 475
B.5 Linear Regression
. 476
B.6 Convolution with Normal Density
. 477
С
Exercises
479
D
Java™ Source Code (Selection)
487
D.I Java™ Classes for Chapter
30 . 487
List of Symbols
493
List of Figures
495
List of Tables
499
XXI
List of Listings
500
Bibliography
503
Index 511
XXII |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Fries, Christian |
| author_facet | Fries, Christian |
| author_role | aut |
| author_sort | Fries, Christian |
| author_variant | c f cf |
| building | Verbundindex |
| bvnumber | BV023044260 |
| callnumber-first | H - Social Science |
| callnumber-label | HG6024 |
| callnumber-raw | HG6024.A3 |
| callnumber-search | HG6024.A3 |
| callnumber-sort | HG 46024 A3 |
| callnumber-subject | HG - Finance |
| classification_rvk | QK 660 QP 890 SK 820 SK 980 |
| classification_tum | WIR 651f MAT 902f WIR 160f |
| ctrlnum | (OCoLC)263711966 (DE-599)BVBBV023044260 |
| dewey-full | 332.601/5195 |
| dewey-hundreds | 300 - Social sciences |
| dewey-ones | 332 - Financial economics |
| dewey-raw | 332.601/5195 |
| dewey-search | 332.601/5195 |
| dewey-sort | 3332.601 45195 |
| dewey-tens | 330 - Economics |
| discipline | Mathematik Wirtschaftswissenschaften |
| discipline_str_mv | Mathematik Wirtschaftswissenschaften |
| format | Book |
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| id | DE-604.BV023044260 |
| illustrated | Illustrated |
| index_date | 2024-07-02T19:21:50Z |
| indexdate | 2024-07-20T07:52:47Z |
| institution | BVB |
| isbn | 9780470047224 0470047224 |
| language | English |
| lccn | 2007011325 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016247746 |
| oclc_num | 263711966 |
| open_access_boolean | |
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| physical | XXII, 520 S. Ill., graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Wiley-Interscience |
| record_format | marc |
| spelling | Fries, Christian Verfasser aut Mathematical finance theory, modeling, implementation Christian Fries Hoboken, NJ Wiley-Interscience 2007 XXII, 520 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Instruments dérivés (Finances) - Prix - Modèles mathématiques Investissements - Modèles mathématiques Valeurs mobilières - Modèles mathématiques Mathematisches Modell Derivative securities Prices Mathematical models Investments Mathematical models Securities Mathematical models Finanzmathematik (DE-588)4017195-4 gnd rswk-swf Finanzmathematik (DE-588)4017195-4 s DE-604 http://www.loc.gov/catdir/toc/ecip0713/2007011325.html Table of contents only http://www.loc.gov/catdir/enhancements/fy0739/2007011325-b.html Contributor biographical information http://www.loc.gov/catdir/enhancements/fy0739/2007011325-d.html Publisher description Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016247746&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fries, Christian Mathematical finance theory, modeling, implementation Instruments dérivés (Finances) - Prix - Modèles mathématiques Investissements - Modèles mathématiques Valeurs mobilières - Modèles mathématiques Mathematisches Modell Derivative securities Prices Mathematical models Investments Mathematical models Securities Mathematical models Finanzmathematik (DE-588)4017195-4 gnd |
| subject_GND | (DE-588)4017195-4 |
| title | Mathematical finance theory, modeling, implementation |
| title_auth | Mathematical finance theory, modeling, implementation |
| title_exact_search | Mathematical finance theory, modeling, implementation |
| title_exact_search_txtP | Mathematical finance theory, modeling, implementation |
| title_full | Mathematical finance theory, modeling, implementation Christian Fries |
| title_fullStr | Mathematical finance theory, modeling, implementation Christian Fries |
| title_full_unstemmed | Mathematical finance theory, modeling, implementation Christian Fries |
| title_short | Mathematical finance |
| title_sort | mathematical finance theory modeling implementation |
| title_sub | theory, modeling, implementation |
| topic | Instruments dérivés (Finances) - Prix - Modèles mathématiques Investissements - Modèles mathématiques Valeurs mobilières - Modèles mathématiques Mathematisches Modell Derivative securities Prices Mathematical models Investments Mathematical models Securities Mathematical models Finanzmathematik (DE-588)4017195-4 gnd |
| topic_facet | Instruments dérivés (Finances) - Prix - Modèles mathématiques Investissements - Modèles mathématiques Valeurs mobilières - Modèles mathématiques Mathematisches Modell Derivative securities Prices Mathematical models Investments Mathematical models Securities Mathematical models Finanzmathematik |
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