Mathematical methods for financial markets:
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| Language: | English |
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[2009]
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| Series: | Springer finance textbook
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| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xxv, 732 Seiten Illustrationen |
| ISBN: | 9781852333768 9781846287374 |
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| 100 | 1 | |a Jeanblanc, Monique |d 1947- |0 (DE-588)171430689 |4 aut | |
| 245 | 1 | 0 | |a Mathematical methods for financial markets |c Monique Jeanblanc ; Marc Yor ; Marc Chesney |
| 264 | 1 | |a London [u.a.] |b Springer |c [2009] | |
| 264 | 4 | |c © 2009 | |
| 300 | |a xxv, 732 Seiten |b Illustrationen | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Springer finance textbook | |
| 650 | 4 | |a Finance - Mathematical models | |
| 650 | 7 | |a Finanzmarkt |2 stw | |
| 650 | 7 | |a Mathematische Ökonomie |2 stw | |
| 650 | 7 | |a Optionspreistheorie |2 stw | |
| 650 | 7 | |a Stochastischer Prozess |2 stw | |
| 650 | 0 | 7 | |a Stochastisches Modell |0 (DE-588)4057633-4 |2 gnd |9 rswk-swf |
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| 689 | 1 | |5 DE-604 | |
| 700 | 1 | |a Yor, Marc |d 1949-2014 |0 (DE-588)120628635 |4 aut | |
| 700 | 1 | |a Chesney, Marc |d 1959- |0 (DE-588)170190595 |4 aut | |
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Contents
Part I Continuous Path Processes
Continuous-Path Random Processes: Mathematical
Prerequisites
. 3
1.1
Some Definitions
. 3
1.1.1
Measurability
. 3
1.1.2
Monotone Class Theorem
. 4
1.1.3
Probability Measures
. 5
1.1.4
Filtration
. 5
1.1.5
Law of a Random Variable, Expectation
. 6
1.1.6
Independence
. 6
1.1.7
Equivalent Probabilities and
Radon-Nikodým
Densities
7
1.1.8
Construction of Simple Probability Spaces
. 8
1.1.9
Conditional Expectation
. 9
1.1.10
Stochastic Processes
. 10
1.1.11
Convergence
. 12
1.1.12
Laplace Transform
. 13
1.1.13
Gaussian Processes
. 15
1.1.14
Markov Processes
. 15
1.1.15
Uniform Integrability
. 18
1.2
Martingales
. 19
1.2.1
Definition and Main Properties
. 19
1.2.2
Spaces of Martingales
. 21
1.2.3
Stopping Times
. 21
1.2.4
Local Martingales
. 25
1.3
Continuous Semi-martingales
. 27
1.3.1
Brackets of Continuous Local Martingales
. 27
1.3.2
Brackets of Continuous Semi-martingales
. 29
1.4
Brownian Motion
. 30
1.4.1
One-dimensional Brownian Motion
. 30
1.4.2
d-dimensional Brownian Motion
. 34
Contents
1.4.3
Correlated Brownian Motions
. 34
1.5
Stochastic Calculus
. 35
1.5.1
Stochastic Integration
. 36
1.5.2
Integration by Parts
. 38
1.5.3
Itô's
Formula: The Fundamental Formula of Stochastic
Calculus
. 38
1.5.4
Stochastic Differential Equations
. 43
1.5.5
Stochastic Differential Equations: The One-
dimensional Case
. 47
1.5.6
Partial Differential Equations
. 51
1.5.7
Doléans-Dade
Exponential
. 52
1.6
Predictable Representation Property
. 55
1.6.1
Brownian Motion Case
. 55
1.6.2
Towards a General Definition of the Predictable
Representation Property
. 57
1.6.3
Dudley's Theorem
. 60
1.6.4
Backward Stochastic Differential Equations
. 61
1.7
Change of Probability and Girsanov's Theorem
. 66
1.7.1
Change of Probability
. 66
1.7.2
Decomposition of P-Martingales as Q-semi-martingales
. 68
1.7.3
Girsanov's Theorem: The One-dimensional Brownian
Motion Case
. 69
1.7.4
Multidimensional Case
. 72
1.7.5
Absolute Continuity
. 73
1.7.6
Condition for Martingale Property of Exponential
Local Martingales
. 74
1.7.7
Predictable Representation Property under a Change
of Probability
. 77
1.7.8
An Example of
Invariance
of BM under Change of
Measure
. 78
Basic Concepts and Examples in Finance
. 79
2.1
A Semi-martingale Framework
. 79
2.1.1
The Financial Market
. 80
2.1.2
Arbitrage Opportunities
. 83
2.1.3
Equivalent Martingale Measure
. 85
2.1.4
Admissible Strategies
. 85
2.1.5
Complete Market
. 87
2.2
A Diffusion Model
. 89
2.2.1
Absence of Arbitrage
. 90
2.2.2
Completeness of the Market
. 90
2.2.3
PDE Evaluation of Contingent Claims in a Complete
Market
. 92
2.3
The Black and Scholes Model
. 93
2.3.1
The Model
. 94
Contents xi
2.3.2
European
Call and Put Options
. 97
2.3.3
The Greeks
.101
2.3.4
General Case
.102
2.3.5
Dividend Paying Assets
.102
2.3.6
Rôle
of Information
.104
2.4
Change of
Numéraire
.105
2.4.1
Change of
Numéraire
and Black-Scholes Formula
.106
2.4.2
Self-financing Strategy and Change of
Numéraire
.107
2.4.3
Change of
Numéraire
and Change of Probability
.108
2.4.4
Forward Measure
.108
2.4.5
Self-financing Strategies: Constrained Strategies
.109
2.5
Feynman-Kac
.112
2.5.1
Feynman-Kac Formula
.112
2.5.2
Occupation Time for a Brownian Motion
.113
2.5.3
Occupation Time for a Drifted Brownian Motion
.114
2.5.4
Cumulative Options
.116
2.5.5
Quantiles
.118
2.6
Ornstein-Uhlenbeck Processes and Related Processes
.119
2.6.1
Definition and Properties
.119
2.6.2
Zero-coupon Bond
.123
2.6.3
Absolute Continuity Relationship for Generalized
Vasicek Processes
.124
2.6.4
Square of a Generalized Vasicek Process
.127
2.6.5
Powers of ^-Dimensional Radial
OU
Processes, Alias
CIR
Processes
.128
2.7
Valuation of European Options
.129
2.7.1
The Garman and Kohlhagen Model for Currency
Options
.129
2.7.2
Evaluation of an Exchange Option
.130
2.7.3
Quanto
Options
.132
Hitting Times: A Mix of Mathematics and Finance
.135
3.1
Hitting Times and the Law of the Maximum for Brownian
Motion
.136
3.1.1
The Law of the Pair of Random Variables {Wu Mt)
■ ■ ■ ■ 136
3.1.2
Hitting Times Process
.138
3.1.3
Law of the Maximum of a Brownian Motion over
[0,ű]
. 139
3.1.4
Laws of Hitting Times
.140
3.1.5
Law of the Infimum
.142
3.1.6
Laplace Transforms of Hitting Times
.143
3.2
Hitting Times for a Drifted Brownian Motion
.145
3.2.1
Joint Laws of (Mx, X) and (mx, X) at Time
t
.145
3.2.2
Laws of Maximum, Minimum, and Hitting Times
.147
3.2.3
Laplace Transforms
.148
3.2.4
Computation of W(v>(l{Tl,<x)<t}
e-XT'{x))
.149
xii Contents
3.2.5 Normal
Inverse
Gaussian Law
.150
3.3
Hitting Times for Geometric Brownian Motion
.151
3.3.1
Laws of the Pairs (Mf, St) and (mf, St)
.151
3.3.2
Laplace Transforms
.152
3.3.3
Computation of E(e~XT'^%{Ta{s)<t})
.153
3.4
Hitting Times in Other Cases
.153
3.4.1
Ornstein-Uhlenbeck Processes
.153
3.4.2
Deterministic Volatility and
Nonconstant
Barrier
.154
3.5
Hitting Time of a Two-sided Barrier for BM and GBM
.156
3.5.1
Brownian Case
.156
3.5.2
Drifted Brownian Motion
.159
3.6
Barrier Options
.160
3.6.1
Put-Call Symmetry
.160
3.6.2
Binary Options and
A's
.163
3.6.3
Barrier Options: General Characteristics
.164
3.6.4
Valuation and Hedging of a Regular Down-and-In Call
Option When the Underlying is a Martingale
.166
3.6.5
Mathematical Results Deduced from the Previous
Approach
.169
3.6.6
Valuation and Hedging of Regular Down-and-In Call
Options: The General Case
.172
3.6.7
Valuation and Hedging of Reverse Barrier Options
.175
3.6.8
The Emerging Calls Method
.177
3.6.9
Closed Form Expressions
.178
3.7
Lookback
Options
.179
3.7.1
Using Binary Options
.179
3.7.2
Traditional Approach
.180
3.8
Double-barrier Options
.182
3.9
Other Options
.183
3.9.1
Options Involving a Hitting Time
.183
3.9.2
Boost Options
.184
3.9.3
Exponential Down Barrier Option
.186
3.10
A Structural Approach to Default Risk
.188
3.10.1
Merton's Model
.188
3.10.2
First Passage Time Models
.190
3.11
American Options
.191
3.11.1
American Stock Options
.192
3.11.2
American Currency Options
.193
3.11.3
Perpetual American Currency Options
.195
3.12
Real Options
.198
3.12.1
Optimal Entry with Stochastic Investment Costs
.198
3.12.2
Optimal Entry in the Presence of Competition
.201
3.12.3
Optimal Entry and Optimal Exit
.204
3.12.4
Optimal Exit and Optimal Entry in the Presence of
Competition
.205
Contents xiii
3.12.5 Optimal
Entry and Exit Decisions
.206
Complements on Brownian Motion
.211
4.1
Local Time
.211
4.1.1
A Stochastic
Pubini
Theorem
.211
4.1.2
Occupation Time Formula
.211
4.1.3
An Approximation of Local Time
.213
4.1.4
Local Times for Semi-martingales
.214
4.1.5
Tanaka's Formula
.214
4.1.6
The
Balayage
Formula
.216
4.1.7
Skorokhod's Reflection Lemma
.217
4.1.8
Local Time of a Semi-martingale
.222
4.1.9
Generalized
Itô-Tanaka
Formula
.226
4.2
Applications
.227
4.2.1
Dupire's Formula
.227
4.2.2
Stop-Loss Strategy
.229
4.2.3
Knock-out BOOST
.230
4.2.4
Passport Options
.232
4.3
Bridges, Excursions, and Meanders
.232
4.3.1
Brownian Motion Zeros
.232
4.3.2
Excursions
.232
4.3.3
Laws of Tx, dt and gt
.233
4.3.4
Laws of (Bt,gt,dt)
.236
4.3.5
Brownian Bridge
.237
4.3.6
Slow Brownian Filtrations
.241
4.3.7
Meanders
.242
4.3.8
The Azema
Martingale
.243
4.3.9
Drifted Brownian Motion
.244
4.4
Parisian Options
.246
4.4.1
The Law of
(G¿e(W)
,
WG-,e)
.249
4.4.2
Valuation of a Down-and-In Parisian Option
.252
4.4.3
PDE Approach
.256
4.4.4
American Parisian Options
.257
Complements on Continuous Path Processes
.259
5.1
Time Changes
.259
5.1.1
Inverse of an Increasing Process
.259
5.1.2
Time Changes and Stopping Times
.260
5.1.3
Brownian Motion and Time Changes
.261
5.2
Dual Predictable Projections
.264
5.2.1
Definitions
.264
5.2.2
Examples
.266
5.3
Diffusions
.269
5.3.1
(Time-homogeneous) Diffusions
.270
5.3.2
Scale Function and Speed Measure
.270
xiv Contents
5.3.3
Boundary
Points.273
5.3.4
Change of Time or Change of Space Variable
.275
5.3.5
Recurrence
.277
5.3.6
Resolvent Kernel and Green Function
.277
5.3.7
Examples
.279
5.4
Non-homogeneous Diffusions
.281
5.4.1
Kolmogorov's Equations
.281
5.4.2
Application: Dupire's Formula
.284
5.4.3
Fokker-Planck Equation
.286
5.4.4
Valuation of Contingent Claims
.289
5.5
Local Times for a Diffusion
.290
5.5.1
Various Definitions of Local Times
.290
5.5.2
Some Diffusions Involving Local Time
.291
5.6
Last Passage Times
.294
5.6.1
Notation and Basic Results
.294
5.6.2
Last Passage Time of a Transient Diffusion
.294
5.6.3
Last Passage Time Before Hitting a Level
.297
5.6.4
Last Passage Time Before Maturity
.298
5.6.5
Absolutely Continuous Compensator
.301
5.6.6
Time When the Supremum is Reached
.302
5.6.7
Last Passage Times for Particular Martingales
.303
5.7
Pitman's Theorem about (2Mt -Wt)
.306
5.7.1
Time Reversal of Brownian Motion
.306
5.7.2
Pitman's Theorem
.307
5.8
Filtrations
.309
5.8.1
Strong and Weak Brownian Filtrations
.310
5.8.2
Some Examples
.312
5.9
Enlargements of Filtrations
.315
5.9.1
Immersion of Filtrations
.315
5.9.2
The Brownian Bridge as an Example of Initial
Enlargement
.318
5.9.3
Initial Enlargement: General Results
.319
5.9.4
Progressive Enlargement
.323
5.10
Filtering the Information
.329
5.10.1
Independent Drift
.329
5.10.2
Other Examples of Canonical Decomposition
.330
5.10.3
Innovation Process
.331
6
A Special Family of Diffusions: Bessel Processes
.333
6.1
Definitions and First Properties
.333
6.1.1
The Euclidean Norm of the
n-Dimensional
Brownian
Motion
.333
6.1.2
General Definitions
.334
6.1.3
Path Properties
.337
6.1.4
Infinitesimal Generator
.337
Contents xv
6.1.5 Absolute
Continuity.
339
6.2
Properties
.342
6.2.1 Additivity
of BESQ's
.342
6.2.2
Transition Densities
.343
6.2.3
Hitting Times for Bessel Processes
.345
6.2.4
Lamperti's Theorem
.347
6.2.5
Laplace Transforms
.349
6.2.6
BESQ Processes with Negative Dimensions
.353
6.2.7
Squared Radial Ornstein-Uhlenbeck
.356
6.3
Cox-Ingersoll-Ross Processes
.356
6.3.1
GIR
Processes and BESQ
.357
6.3.2
Transition Probabilities for a CIR Process
.358
6.3.3
CIR Processes as Spot Rate Models
.359
6.3.4
Zero-coupon Bond
.361
6.3.5
Inhomogeneous CIR Process
.364
6.4
Constant Elasticity of Variance Process
.365
6.4.1
Particular Case
μ
= 0.366
6.4.2
CEV Processes and CIR Processes
.368
6.4.3
CEV Processes and BESQ Processes
.368
6.4.4
Properties
.370
6.4.5
Scale Functions for CEV Processes
.371
6.4.6
Option Pricing in a CEV Model
.372
6.5
Some Computations on Bessel Bridges
.373
6.5.1
Bessel Bridges
.373
6.5.2
Bessel Bridges and Ornstein-Uhlenbeck Processes
.374
6.5.3
European Bond Option
.376
6.5.4
American Bond Options and the CIR Model
.378
6.6
Asian Options
.381
6.6.1
Parity and Symmetry Formulae
.382
6.6.2
Laws of
Ä%]
and
Ąv)
.383
6.6.3
The Moments of At
.388
6.6.4
Laplace Transform Approach
.389
6.6.5
PDE Approach
.391
6.7
Stochastic Volatility
.392
6.7.1
Black and Scholes Implied Volatility
.392
6.7.2
A General Stochastic Volatility Model
.392
6.7.3
Option Pricing in Presence of Non-normality of
Returns: The Martingale Approach
.393
6.7.4
Hull and White Model
.396
6.7.5
Closed-form Solutions in Some Correlated Cases
.398
6.7.6
PDE Approach
.
401
6.7.7
Heston's Model
.
401
6.7.8
Mellin Transform
.
403
Contents
Part II Jump Processes
Default Risk: An Enlargement of Filtration Approach
.407
7.1
A Toy Model
.407
7.1.1
Defaultable Zero-coupon with Payment at Maturity
. 408
7.1.2
Defaultable Zero-coupon with Payment at Hit
.410
7.2
Toy Model and Martingales
.412
7.2.1
Key Lemma
.412
7.2.2
The Fundamental Martingale
.412
7.2.3
Hazard Function
.413
7.2.4
Incompleteness of the Toy Model,
non
Arbitrage Prices
415
7.2.5
Predictable Representation Theorem
.415
7.2.6
Risk-neutral Probability Measures
.416
7.2.7
Partial Information: Duffie and Lando's Model
.418
7.3
Default Times with a Given Stochastic Intensity
.418
7.3.1
Construction of Default Time with a Given Stochastic
Intensity
.418
7.3.2
Conditional Expectation with Respect to Tt
.419
7.3.3
Enlargements of Filtrations
.420
7.3.4
Conditional Expectations with Respect to Qt
.420
7.3.5
Conditional Expectations of ^oo-Measurable Random
Variables
.422
7.3.6
Correlated Defaults: Copula Approach
.423
7.3.7
Correlated Defaults: Jarrow and Yu's Model
.425
7.4
Conditional Survival Probability Approach
.426
7.4.1
Conditional Expectations
.427
7.5
Conditional Survival Probability Approach and Immersion
_428
7.5.1
(ft)-Hypothesis and Arbitrages
.429
7.5.2
Pricing Contingent Claims
.430
7.5.3
Correlated Defaults: Kusuoka's Example
.431
7.5.4
Stochastic Barrier
.432
7.5.5
Predictable Representation Theorems
.432
7.5.6
Hedging Contingent Claims with DZC
.434
7.6
General Case: Without the
(Tť)-Hypothesis
.437
7.6.1
An Example of Partial Observation
.437
7.6.2
Two Defaults, Trivial Reference Filtration
.440
7.6.3
Initial Times
.442
7.6.4
Explosive Defaults
.444
7.7
Intensity Approach
.445
7.7.1
Definition
.445
7.7.2
Valuation Formula
.446
7.8
Credit Default Swaps
.446
7.8.1
Dynamics of the CDS's Price in a single name setting
. 447
7.8.2
Dynamics of the CDS's Price in a multi-name setting
. 448
Contents xvii
7.9 PDE
Approach for Hedging Defaultable Claims
.449
7.9.1
Defaultable Asset with Total Default
.449
7.9.2
PDE for Valuation
.450
7.9.3
General Case
.454
Poisson
Processes and Ruin Theory
.457
8.1
Counting Processes and Stochastic Integrals
.457
8.2
Standard
Poisson
Process
.459
8.2.1
Definition and First Properties
.459
8.2.2
Martingale Properties
.461
8.2.3
Infinitesimal Generator
.464
8.2.4
Change of Probability Measure: An Example
.465
8.2.5
Hitting Times
.466
8.3
Inhomogeneous
Poisson
Processes
.467
8.3.1
Definition
.467
8.3.2
Martingale Properties
.467
8.3.3
Watanabe's Characterization of Inhomogeneous
Poisson
Processes
.468
8.3.4
Stochastic Calculus
.469
8.3.5
Predictable Representation Property
.473
8.3.6
Multidimensional
Poisson
Processes
.474
8.4
Stochastic Intensity Processes
.475
8.4.1
Doubly Stochastic
Poisson
Processes
.475
8.4.2
Inhomogeneous
Poisson
Processes with Stochastic
Intensity
.476
8.4.3
ItÔ's
Formula
.476
8.4.4
Exponential Martingales
.477
8.4.5
Change of Probability Measure
.478
8.4.6
An Elementary Model of Prices Involving Jumps
.479
8.5
Poisson
Bridges
.480
8.5.1
Definition of the
Poisson
Bridge
.480
8.5.2
Harness Property
.
481
8.6
Compound
Poisson
Processes
.483
8.6.1
Definition and Properties
.483
8.6.2
Integration Formula
.484
8.6.3
Martingales
.485
8.6.4
Itô's
Formula
.492
8.6.5
Hitting Times
.
492
8.6.6
Change of Probability Measure
.
494
8.6.7
Price Process
.
495
8.6.8
Martingale Representation Theorem
.496
8.6.9
Option Pricing
.
497
8.7
Ruin Process
.
497
8.7.1
Ruin Probability
.
497
8.7.2
Integral Equation
.
498
xviii Contents
8.7.3 An
Example.
498
8.8
Marked
Point
Processes
.501
8.8.1
Random Measure
.501
8.8.2
Definition
.501
8.8.3
An Integration Formula
.503
8.8.4
Marked Point Processes with Intensity and Associated
Martingales
.503
8.8.5
Girsanov's Theorem
.504
8.8.6
Predictable Representation Theorem
.504
8.9
Poisson
Point Processes
.505
8.9.1
Poisson
Measures
.505
8.9.2
Point Processes
.506
8.9.3
Poisson
Point Processes
.506
8.9.4
The
Ito
Measure of Brownian Excursions
.507
9
General Processes: Mathematical Facts
.509
9.1
Some Basic Facts about
càdlàg
Processes
.509
9.1.1
An Illustrative Lemma
.509
9.1.2
Finite Variation Processes, Pure Jump Processes
.510
9.1.3
Some
σ
-algebras
.
512
9.2
Stochastic Integration for Square
Integrable
Martingales
.513
9.2.1
Square
Integrable
Martingales
.513
9.2.2
Stochastic Integral
.516
9.3
Stochastic Integration for Semi-martingales
.517
9.3.1
Local Martingales
.517
9.3.2
Quadratic Covariation and Predictable Bracket of
Two Local Martingales
.519
9.3.3
Orthogonality
.521
9.3.4
Semi-martingales
.522
9.3.5
Stochastic Integration for Semi-martingales
.524
9.3.6
Quadratic Covariation of Two Semi-martingales
.525
9.3.7
Particular Cases
.525
9.3.8
Predictable Bracket of Two Semi-martingales
.527
9.4
Itô's
Formula and Girsanov's Theorem
.528
9.4.1
Itô's
Formula: Optional and Predictable Forms
.528
9.4.2
Semi-martingale Local Times
.531
9.4.3
Exponential Semi-martingales
.532
9.4.4
Change of Probability, Girsanov's Theorem
.534
9.5
Existence and Uniqueness of the e.m.m
.537
9.5.1
Predictable Representation Property
.537
9.5.2
Necessary Conditions for Existence
.538
9.5.3
Uniqueness Property
.542
9.5.4
Examples
.543
9.6
Self-financing Strategies and Integration by Parts
.544
Contents xix
9.6.1 The Model.545
9.6.2
Self-financing Strategies and Change of
Numéraire
.545
9.7
Valuation in an Incomplete Market
.547
9.7.1
Replication Criteria
.548
9.7.2
Choice of an Equivalent Martingale Measure
.549
9.7.3
Indifference Prices
.550
10
Mixed Processes
.551
10.1
Definition
.551
10.2
Itô's
Formula
.552
10.2.1
Integration by Parts
.552
10.2.2
Itô's
Formula: One-dimensional Case
.553
10.2.3
Multidimensional Case
.555
10.2.4
Stochastic Differential Equations
.556
10.2.5
Feynman-Kac Formula
.557
10.2.6
Predictable Representation Theorem
.558
10.3
Change of Probability
.559
10.3.1
Exponential Local Martingales
.559
10.3.2
Girsanov's Theorem
.560
10.4
Mixed Processes in Finance
.561
10.4.1
Computation of the Moments
.561
10.4.2
Symmetry
.562
10.4.3
Hitting Times
.
563
10.4.4
Affine
Jump-Diffusion Model
.565
10.4.5
General Jump-Diffusion Processes
.569
10.5
Incompleteness
.569
10.5.1
The Set of Risk-neutral Probability Measures
.570
10.5.2
The Range of Prices for European Call Options
.572
10.5.3
General Contingent Claims
.575
10.6
Complete Markets with Jumps
.578
10.6.1
A Three Assets Model
.
578
10.6.2
Structure Equations
.
579
10.7
Valuation of Options
.
582
10.7.1
The Valuation of European Options
.584
10.7.2
American Option
.
586
11
Levy Processes
.
11.1
Infinitely Divisible Random Variables
.
592
11.1.1
Definition
.
J92
11.1.2
Self-decomposable Random Variables
.
11.1.3
Stable Random Variables
.
59°
11.2
Levy Processes
.
11.2.1
Definition and Main Properties
.
11.2.2
Poisson
Point Processes, Levy Measures
.
60J.
11.2.3
Lévy-Khintchine
Formula for a Levy Process
.
60b
Contents
11.2.4
Itô's
Formulae for a One-dimensional Levy Process
. 612
11.2.5
Itô's
Formula for
Lévy-Itô
Processes
.613
11.2.6
Martingales
.615
11.2.7
Harness Property
.620
11.2.8
Representation Theorem of Martingales in a
Levy Setting
.621
11.3
Absolutely Continuous Changes of Measures
.623
11.3.1
Esscher Transform
.623
11.3.2
Preserving the Levy Property with Absolute Continuity
625
11.3.3
General Case
.627
11.4
Fluctuation Theory
.628
11.4.1
Maximum and Minimum
.628
11.4.2
Pecherskii-Rogozin Identity
.631
11.5
Spectrally Negative Levy Processes
.632
11.5.1
Two-sided Exit Times
.632
11.5.2
Laplace Exponent of the Ladder Process
.633
11.5.3
D. Kendall's Identity
.633
11.6
Subordinators
.634
11.6.1
Definition and Examples
.634
11.6.2
Levy Characteristics of a Subordinated Process
.636
11.7
Exponential Levy Processes as Stock Price Processes
.636
11.7.1
Option Pricing with Esscher Transform
.636
11.7.2
A Differential Equation for Option Pricing
.637
11.7.3
Put-call Symmetry
.638
11.7.4
Arbitrage and Completeness
.639
11.8
Variance-Gamma Model
.639
11.9
Valuation of Contingent Claims
.641
11.9.1
Perpetual American Options
.641
List of Special Features, Probability Laws, and Functions
. 647
A.I Main Formulae
.647
A.
1.1
Absolute Continuity Relationships
.647
A.
1.2
Bessel Processes
.648
A.1.3 Brownian Motion
.649
A.1.4 Diffusions
.650
A.1.5 Finance
.650
A.
1.6
Girsanov's Theorem
.651
A.I.
7
Hitting Times
.651
A.1.8
Itô's
Formulae
.651
A.1.9 Levy Processes
.653
A.I.
10
Semi-martingales
.654
A.2 Processes
.655
A.3 Some Main Models
.655
A.4 Some Important Probability Distributions
.656
A.4.1 Laws with Density
.656
Contents xxi
A.4.2
Some Algebraic Properties for Special r.v.'s
.656
A.4.3
Poisson
Law
.657
A.4.4 Gamma and Inverse Gaussian Law
.658
A.4.5 Generalized Inverse Gaussian and Normal Inverse
Gaussian
.659
A.4.6 Variance Gamma VG(ct,
ν,θ)
.661
A.4.7 Tempered Stable
ТЅ(У±,
C±,M±)
.661
A.5 Special Functions
.662
A.5.1 Gamma and Beta Functions
.662
A.5.2 Bessel Functions
.662
A.5.3 Hermite Functions
.663
A.5.4 Parabolic Cylinder Functions
.663
A.5.5 Airy Function
.663
A.5.6
Kummer
Functions
.664
A.5.7 Whittaker Functions
.664
A.
5.8
Some Laplace Transforms
.665
References
.667
В
Some Papers and Books on Specific Subjects
.709
B.I Theory of Continuous Processes
.709
B.I.I Books
.709
B.1.2 Stochastic Differential Equations
.709
B.1.3 Backward SDE
.709
B.1.4 Martingale Representation Theorems
.710
B.I.
5
Enlargement of Filtrations
.710
B.I.
6
Exponential Functionals
.710
В.1.7
Uniform Integrability of Martingales
.710
B.2 Particular Processes
.710
B.2.1 Ornstein-Uhlenbeck Processes
.710
B.2.2
CIR
Processes
.
710
B.2.3 CEV Processes
.711
B.2.4 Bessel Processes
.711
B.3 Processes with Discontinuous Paths
.711
B.3.1 Some Books
.
7n
B.3.2 Survey Papers
.
7n
B.4 Hitting Times
.
711
B.5 Levy Processes
.
7
B.5.1 Books
.
712
B.5.2 Some Papers
.
712
B.6 Some Books on Finance
.
'^
B.6.1 Discrete Time
.
712
B.6.2 Continuous Time
.
712
B.6.3 Collective Books
.
712
B.6.4 History
.
713
xxii Contents
B.7 Arbitrage.713
B.8
Exotic
Options
.713
8.8.1
Books
.713
8.8.2
Articles
.713
Index of Authors
.715
Index of Symbols
.723
Subject Index
.725 |
| any_adam_object | 1 |
| author | Jeanblanc, Monique 1947- Yor, Marc 1949-2014 Chesney, Marc 1959- |
| author_GND | (DE-588)171430689 (DE-588)120628635 (DE-588)170190595 |
| author_facet | Jeanblanc, Monique 1947- Yor, Marc 1949-2014 Chesney, Marc 1959- |
| author_role | aut aut aut |
| author_sort | Jeanblanc, Monique 1947- |
| author_variant | m j mj m y my m c mc |
| building | Verbundindex |
| bvnumber | BV019985492 |
| classification_rvk | QH 240 QK 600 QP 890 SK 980 |
| classification_tum | MAT 600f MAT 605f WIR 160f |
| ctrlnum | (OCoLC)475350194 (DE-599)BVBBV019985492 |
| dewey-full | 332.0151 |
| dewey-hundreds | 300 - Social sciences |
| dewey-ones | 332 - Financial economics |
| dewey-raw | 332.0151 |
| dewey-search | 332.0151 |
| dewey-sort | 3332.0151 |
| dewey-tens | 330 - Economics |
| discipline | Mathematik Wirtschaftswissenschaften |
| format | Book |
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| id | DE-604.BV019985492 |
| illustrated | Illustrated |
| indexdate | 2024-08-30T00:15:38Z |
| institution | BVB |
| isbn | 9781852333768 9781846287374 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-013307502 |
| oclc_num | 475350194 |
| open_access_boolean | |
| owner | DE-824 DE-91G DE-BY-TUM DE-384 DE-19 DE-BY-UBM DE-83 DE-1051 DE-355 DE-BY-UBR DE-945 DE-11 DE-188 DE-898 DE-BY-UBR DE-20 DE-M382 |
| owner_facet | DE-824 DE-91G DE-BY-TUM DE-384 DE-19 DE-BY-UBM DE-83 DE-1051 DE-355 DE-BY-UBR DE-945 DE-11 DE-188 DE-898 DE-BY-UBR DE-20 DE-M382 |
| physical | xxv, 732 Seiten Illustrationen |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer finance textbook |
| spelling | Jeanblanc, Monique 1947- (DE-588)171430689 aut Mathematical methods for financial markets Monique Jeanblanc ; Marc Yor ; Marc Chesney London [u.a.] Springer [2009] © 2009 xxv, 732 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Springer finance textbook Finance - Mathematical models Finanzmarkt stw Mathematische Ökonomie stw Optionspreistheorie stw Stochastischer Prozess stw Stochastisches Modell (DE-588)4057633-4 gnd rswk-swf Kreditmarkt (DE-588)4073788-3 gnd rswk-swf Finanzmathematik (DE-588)4017195-4 gnd rswk-swf Finanzmathematik (DE-588)4017195-4 s DE-604 Kreditmarkt (DE-588)4073788-3 s Stochastisches Modell (DE-588)4057633-4 s Yor, Marc 1949-2014 (DE-588)120628635 aut Chesney, Marc 1959- (DE-588)170190595 aut Erscheint auch als Online-Ausgabe 978-1-84628-737-4 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013307502&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Jeanblanc, Monique 1947- Yor, Marc 1949-2014 Chesney, Marc 1959- Mathematical methods for financial markets Finance - Mathematical models Finanzmarkt stw Mathematische Ökonomie stw Optionspreistheorie stw Stochastischer Prozess stw Stochastisches Modell (DE-588)4057633-4 gnd Kreditmarkt (DE-588)4073788-3 gnd Finanzmathematik (DE-588)4017195-4 gnd |
| subject_GND | (DE-588)4057633-4 (DE-588)4073788-3 (DE-588)4017195-4 |
| title | Mathematical methods for financial markets |
| title_auth | Mathematical methods for financial markets |
| title_exact_search | Mathematical methods for financial markets |
| title_full | Mathematical methods for financial markets Monique Jeanblanc ; Marc Yor ; Marc Chesney |
| title_fullStr | Mathematical methods for financial markets Monique Jeanblanc ; Marc Yor ; Marc Chesney |
| title_full_unstemmed | Mathematical methods for financial markets Monique Jeanblanc ; Marc Yor ; Marc Chesney |
| title_short | Mathematical methods for financial markets |
| title_sort | mathematical methods for financial markets |
| topic | Finance - Mathematical models Finanzmarkt stw Mathematische Ökonomie stw Optionspreistheorie stw Stochastischer Prozess stw Stochastisches Modell (DE-588)4057633-4 gnd Kreditmarkt (DE-588)4073788-3 gnd Finanzmathematik (DE-588)4017195-4 gnd |
| topic_facet | Finance - Mathematical models Finanzmarkt Mathematische Ökonomie Optionspreistheorie Stochastischer Prozess Stochastisches Modell Kreditmarkt Finanzmathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013307502&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT jeanblancmonique mathematicalmethodsforfinancialmarkets AT yormarc mathematicalmethodsforfinancialmarkets AT chesneymarc mathematicalmethodsforfinancialmarkets |