Volatility and correlation: the perfect hedger and the fox
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Chichester [u.a.]
Wiley
2007
|
| Edition: | 2. ed., |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Rev. ed. of: Volatility and correlation in the pricing of equity. 1999. Includes bibliographical references (p. 805-812) and index |
| Physical Description: | XXVII, 836 S. graph. Darst. 25 cm |
| ISBN: | 0470091398 9780470091395 |
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| 245 | 1 | 0 | |a Volatility and correlation |b the perfect hedger and the fox |c Riccardo Rebonato |
| 250 | |a 2. ed., | ||
| 264 | 1 | |a Chichester [u.a.] |b Wiley |c 2007 | |
| 300 | |a XXVII, 836 S. |b graph. Darst. |c 25 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Rev. ed. of: Volatility and correlation in the pricing of equity. 1999. | ||
| 500 | |a Includes bibliographical references (p. 805-812) and index | ||
| 650 | 4 | |a Marchés à terme de taux d'intérêt - Modèles mathématiques | |
| 650 | 4 | |a Options (Finances) - Modèles mathématiques | |
| 650 | 4 | |a Valeurs mobilières - Prix - Modèles mathématiques | |
| 650 | 4 | |a Mathematisches Modell | |
| 650 | 4 | |a Options (Finance) |x Mathematical models | |
| 650 | 4 | |a Interest rate futures |x Mathematical models | |
| 650 | 4 | |a Securities |x Prices |x Mathematical models | |
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| adam_text | Contents
Preface
xxi
0.1
Why
a Second
Edition? xxi
0.2
What This Book Is Not About
xxiii
0.3
Structure of the Book
xxiv
0.4
The New Subtitle
xxiv
Acknowledgements
xxvii
I Foundations
1
1
Theory and Practice of Option Modelling
3
1.1
The Role of Models in Derivatives Pricing
3
1.1.1
What Are Models For?
3
1.1.2
The Fundamental Approach
5
1.1.3
The Instrumental Approach
7
1.1.4
A Conundrum (or, What is
Vega
Hedging For? )
8
1.2
The Efficient Market Hypothesis and Why It Matters for Option Pricing
9
1.2.1
The Three Forms of the EMH
9
1.2.2
Pseudo-
Arbitrageurs in Crisis
10
1.2.3
Model Risk for Traders and Risk Managers
11
1.2.4
The Parable of the Two Volatility Traders
12
1.3
Market Practice
14
1.3.1
Different Users of Derivatives Models
14
1.3.2
In-Model and Out-of-Model Hedging
15
1.4
The Calibration Debate
17
1.4.1
Historical vs Implied Calibration
18
1.4.2
The Logical Underpinning of the Implied Approach
19
1.4.3
Are Derivatives Markets Informationally Efficient?
21
1.4.4
Back to Calibration
26
1.4.5
A Practical Recommendation
27
vii
viii CONTENTS
1.5
Across-Markets Comparison of Pricing and Modelling Practices
27
1.6
Using Models
30
2
Option Replication
31
2.1
The Bedrock of Option Pricing
31
2.2
The Analytic (PDE) Approach
32
2.2.1
The Assumptions
32
2.2.2
The Portfolio-Replication Argument (Deterministic Volatility)
32
2.2.3
The Market Price of Risk with Deterministic Volatility
34
2.2.4
Link with Expectations
-
the Feynman-Kac Theorem
36
2.3
Binomial Replication
38
2.3.1
First Approach
-
Replication Strategy
39
2.3.2
Second Approach
-
Naive Expectation
41
2.3.3
Third Approach
-
Market Price of Risk
42
2.3.4
A Worked-Out Example
45
2.3.5
Fourth Approach
-
Risk-Neutral Valuation
46
2.3.6
Pseudo-Probabilities
48
2.3.7
Are the Quantities
π
and
7Г2
Really Probabilities?
49
2.3.8
Introducing Relative Prices
51
2.3.9
Moving to a Multi-Period Setting
53
2.3.10
Fair Prices as Expectations
56
2.3.11
Switching Numeraires and Relating Expectations Under
Different Measures
58
2.3.12
Another Worked-Out Example
61
2.3.13
Relevance of the Results
64
2.4
Justifying the Two-State Branching Procedure
65
2.4.1
How To Recognize a Jump When You See One
65
2.5
The Nature of the Transformation between Measures: Girsanov s Theorem
69
2.5.1
An Intuitive Argument
69
2.5.2
A Worked-Out Example
70
2.6
Switching Between the PDE, the Expectation and the Binomial
Replication Approaches
73
3
The Building Blocks
75
3.1
Introduction and Plan of the Chapter
75
3.2
Definition of Market Terms
75
3.3
Hedging Forward Contracts Using Spot Quantities
77
3.3.1
Hedging Equity Forward Contracts
78
3.3.2
Hedging Interest-Rate Forward Contracts
79
3.4
Hedging Options: Volatility of Spot and Forward Processes
80
CONTENTS ix
3.5
The Link Between Root-Mean-Squared Volatilities and the
Time-Dependence of Volatility
84
3.6
Admissibility of a Series of Root-Mean-Squared Volatilities
85
3.6.1
The Equity/FX Case
85
3.6.2
The Interest-Rate Case
86
3.7
Summary of the Definitions So Far
87
3.8
Hedging an Option with a Forward-Setting Strike
89
3.8.1
Why Is This Option Important? (And Why Is it Difficult
to Hedge?)
90
3.8.2
Valuing a Forward-Setting Option
91
3.9
Quadratic Variation: First Approach
95
3.9.1
Definition
95
3.9.2
Properties of Variations
96
3.9.3
First and Second Variation of a Brownian Process
97
3.9.4
Links between Quadratic Variation and fr
a(u)2áu
97
3.9.5
Why Quadratic Variation Is So Important (Take
1) 98
4
Variance and Mean Reversion in the Real and the Risk-Adjusted Worlds
101
4.1
Introduction and Plan of the Chapter
101
4.2
Hedging a Plain-Vanilla Option: General Framework
102
4.2.1
Trading Restrictions and Model Uncertainty:
Theoretical Results
103
4.2.2
The Setting
104
4.2.3
The Methodology
104
4.2.4
Criterion for Success
106
4.3
Hedging Plain-Vanilla Options: Constant Volatility
106
4.3.1
Trading the Gamma: One Step and Constant Volatility
108
4.3.2
Trading the Gamma: Several Steps and Constant Volatility
114
4.4
Hedging Plain-Vanilla Options: Time-Dependent Volatility
116
4.4.1
Views on Gamma Trading When the Volatility is Time
Dependent
116
4.4.2
Which View Is the Correct One? (and the Feynman-Kac
Theorem Again)
119
4.5
Hedging Behaviour In Practice
121
4.5.1
Analysing the Replicating Portfolio
121
4.5.2
Hedging Results: the Time-Dependent Volatility Case
122
4.5.3
Hedging with the Wrong Volatility
125
4.6
Robustness of the Black-and-Scholes Model
127
4.7
Is the Total Variance All That Matters?
130
4.8
Hedging Plain-Vanilla Options: Mean-Reverting Real-World Drift
131
χ
CONTENTS
4.9
Hedging Plain-Vanilla Options: Finite Re-Hedging Intervals Again
135
4.9.1
The Crouhy-Galai Set-Up
135
5
Instantaneous and Terminal Correlation
141
5.1
Correlation, Co-Integration and Multi-Factor Models
141
5.1.1
The Multi-Factor Debate
144
5.2
The Stochastic Evolution of Imperfectly Correlated Variables
146
5.3
The Role of Terminal Correlation in the Joint Evolution of Stochastic
Variables
151
5.3.1
Defining Stochastic Integrals
151
5.3.2
Case
1:
European Option, One Underlying Asset
153
5.3.3
Case
2:
Path-Dependent Option, One Asset
155
5.3.4
Case
3:
Path-Dependent Option, Two Assets
156
5.4
Generalizing the Results
162
5.5
Moving Ahead
164
II Smiles
-
Equity and FX
165
6
Pricing Options in the Presence of Smiles
167
6.1
Plan of the Chapter
167
6.2
Background and Definition of the Smile
168
6.3
Hedging with a Compensated Process: Plain-Vanilla and Binary Options
169
6.3.1
Delta- and Vega-Hedging a Plain-Vanilla Option
169
6.3.2
Pricing a European Digital Option
172
6.4
Hedge Ratios for Plain-Vanilla Options in the Presence of Smiles
173
6.4.1
The Relationship Between the True Call Price Functional
and the Black Formula
174
6.4.2
Calculating the Delta Using the Black Formula and the
Implied Volatility
175
6.4.3
Dependence of Implied Volatilities on the Strike and the
Underlying
176
6.4.4
Floating and Sticky Smiles and What They Imply about Changes
in Option Prices
178
6.5
Smile Tale
1:
Sticky Smiles
180
6.6
Smile Tale
2:
Floating Smiles
182
6.6.1
Relevance of the Smile Story for Floating Smiles
183
6.7
When Does Risk Aversion Make a Difference?
184
6.7.1
Motivation
184
6.7.2
The Importance of an Assessment of Risk Aversion
for Model Building
185
6.7.3
The Principle of Absolute Continuity
186
CONTENTS
6.7.4
The Effect of Supply and Demand
6.7.5
A Stylized Example: First Version
6.7.6
A Stylized Example: Second Version
6.7.7
A Stylized Example: Third Version
6.7.8
Overall Conclusions
6.7.9
The EMH Again
7
Empirical Facts About Smiles
7.1
What is
this Chapter About?
7.1.1
Fundamental and Derived Analyses
7.1.2
A Methodological Caveat
7.2
Market
Information About Smiles
7.2.1
Direct Static Information
7.2.2
Semi-Static Information
7.2.3
Direct Dynamic Information
7.2.4
Indirect Information
7.3
Equities
7.3.1
Basic Facts
7.3.2
Subtler Effects
7.4
Interest
Rates
7.4.1
Basic Facts
7.4.2
Subtler Effects
7.5
FX Rates
7.5.1
Basic Facts
7.5.2
Subtler Effects
7.6
Conclusions
Xl
187
187
194
196
196
199
201
201
201
202
203
203
204
204
205
206
206
206
222
222
224
227
227
227
235
General Features of Smile-Modelling Approaches
237
8.1
Fully-Stochastic-Volatility Models
237
8.2
Local-Volatility (Restricted-Stochastic-Volatility) Models
239
8.3
Jump-Diffusion Models
241
8.3.1
Discrete Amplitude
241
8.3.2
Continuum of Jump Amplitudes
242
8.4
Variance-Gamma Models
243
8.5
Mixing Processes
243
8.5.1
A Pragmatic Approach to Mixing Models
244
8.6
Other Approaches
245
8.6.1
Tight Bounds with Known Quadratic Variation
245
8.6.2
Assigning Directly the Evolution of the Smile Surface
246
8.7
The Importance of the Quadratic Variation (Take
2) 246
xii CONTENTS
9
The Input Data: Fitting an Exogenous Smile Surface
249
9.1
What is This Chapter About?
249
9.2
Analytic Expressions for Calls vs Process Specification
249
9.3
Direct Use of Market Prices: Pros and Cons
250
9.4
Statement of the Problem
251
9.5
Fitting Prices
252
9.6
Fitting Transformed Prices
254
9.7
Fitting the Implied Volatilities
255
9.7.1
The Problem with Fitting the Implied Volatilities
255
9.8
Fitting the Risk-Neutral Density Function
-
General
256
9.8.1
Does It Matter if the Price Density Is Not Smooth?
257
9.8.2
Using Prior Information (Minimum Entropy)
258
9.9
Fitting the Risk-Neutral Density Function: Mixture of Normals
259
9.9.1
Ensuring the Normalization and Forward Constraints
261
9.9.2
The Fitting Procedure
264
9.10
Numerical Results
265
9.10.1
Description of the Numerical Tests
265
9.10.2
Fitting to Theoretical Prices: Stochastic-Volatility Density
265
9.10.3
Fitting to Theoretical Prices: Variance-Gamma Density
268
9.10.4
Fitting to Theoretical Prices: Jump-Diffusion Density
270
9.10.5
Fitting to Market Prices
272
9.
H
Is the Term |§ Really a Delta?
275
9.12
Fitting the Risk-Neutral Density Function:
The Generalized-Beta Approach
277
9.12.1
Derivation of Analytic Formulae
280
9.12.2
Results and Applications
287
9.12.3
What Does This Approach Offer?
291
10
Quadratic Variation and Smiles
293
10.1
Why This Approach Is Interesting
293
10.2
The BJN Framework for Bounding Option Prices
293
10.3
The BJN Approach
-
Theoretical Development
294
10.3.1
Assumptions and Definitions
294
10.3.2
Establishing Bounds
297
10.3.3
Recasting the Problem
298
10.3.4
Finding the Optimal Hedge
299
10.4
The BJN Approach: Numerical Implementation
300
10.4.1
Building a Traditional Tree
301
10.4.2
Building a BJN Tree for a Deterministic Diffusion
301
10.4.3
Building a BJN Tree for a General Process
304
10.4.4
Computational Results
307
CONTENTS xiii
10.4.5
Creating
Asymmetrie
Smiles
309
10.4.6
Summary of the Results
311
10.5
Discussion of the Results
312
10.5.1
Resolution of the Crouhy-Galai Paradox
312
10.5.2
The Difference Between Diffusions and Jump-Diffusion
Processes: the Sample Quadratic Variation
312
10.5.3
How Can One Make the Approach More Realistic?
314
10.5.4
The Link with Stochastic-Volatility Models
314
10.5.5
The Link with Local-Volatility Models
315
10.5.6
The Link with Jump-Diffusion Models
315
10.6
Conclusions (or, Limitations of Quadratic Variation)
316
11
Local-Volatility Models: the Derman-and-Kani Approach
319
11.1
General Considerations on Stochastic-Volatility Models
319
11.2
Special Cases of Restricted-Stochastic-Volatility Models
321
11.3
The Dupire, Rubinstein and Derman-and-Kani Approaches
321
11.4
Green s Functions (Arrow-Debreu Prices) in the DK Construction
322
11.4.1
Definition and Main Properties of Arrow-Debreu Prices
322
11.4.2
Efficient Computation of Arrow-Debreu Prices
324
11.5
The Derman-and-Kani Tree Construction
326
11.5.1
Building the First Step
327
11.5.2
Adding Further Steps
329
11.6
Numerical Aspects of the Implementation of the DK Construction
331
11.6.1
Problem
1 :
Forward Price Greater Than 5(up) or Smaller
Than S(down)
331
11.6.2
Problem
2:
Local Volatility Greater Than ^|5(up)
-
S(down)|
332
11.6.3
Problem
3:
Arbitrariness of the Choice of the Strike
332
11.7
Implementation Results
334
11.7.1
Benchmarking
1:
The No-Smile Case
334
11.7.2
Benchmarking
2:
The Time-Dependent-Volatility Case
335
11.7.3
Benchmarking
3:
Purely Strike-Dependent Implied Volatility
336
11.7.4
Benchmarking
4:
Strike-and-Maturity-Dependent Implied
Volatility
337
11.7.5
Conclusions
338
11.8
Estimating Instantaneous Volatilities from Prices as an Inverse Problem
343
12
Extracting the Local Volatility from Option Prices
345
12.1
Introduction
345
12.1.1
A Possible Regularization Strategy
346
12.1.2
Shortcomings
346
12.2
The Modelling Framework
347
12.3
A Computational Method
349
xiv CONTENTS
12.3.1 Backward
Induction
349
12.3.2
Forward
Equations
350
12.3.3
Why Are We Doing Things This Way?
352
12.3.4
Related Approaches
354
12.4
Computational Results
355
12.4.1
Are We Looking at the Same Problem?
356
12.5
The Link Between Implied and Local-Volatility Surfaces
357
12.5.1
Symmetric ( FX ) Smiles
358
12.5.2
Asymmetric ( Equity ) Smiles
361
12.5.3
Monotonie
( Interest-Rate ) Smile Surface
368
12.6
Gaining an Intuitive Understanding
368
12.6.1
Symmetric Smiles
369
12.6.2
Asymmetric Smiles: One-Sided Parabola
370
12.6.3
Asymmetric Smiles: Monotonically Decaying
372
12.7
What Local-Volatility Models Imply about Sticky and Floating Smiles
373
12.8
No-Arbitrage Conditions on the Current Implied Volatility Smile Surface
375
12.8.1
Constraints on the Implied Volatility Surface
375
12.8.2
Consequences for Local Volatilities
381
12.9
Empirical Performance
385
12.10
Appendix I: Proof that ^CaiKSuKjj)
= ф(Ѕт) к
386
13
Stochastic-Volatility Processes
389
13.1
Plan of the Chapter
389
13.2
Portfolio Replication in the Presence of Stochastic Volatility
389
13.2.1
Attempting to Extend the Portfolio Replication Argument
389
13.2.2
The Market Price of Volatility Risk
396
13.2.3
Assessing the Financial Plausibility of
λσ
398
13.3
Mean-Reverting Stochastic Volatility
401
13.3.1
The Ornstein-Uhlenbeck Process
402
13.3.2
The Functional Form Chosen in This Chapter
403
13.3.3
The High-Reversion-Speed, High-Volatility Regime
404
13.4
Qualitative Features of Stochastic-Volatility Smiles
405
13.4.1
The Smile as a Function of the Risk-Neutral Parameters
406
13.5
The Relation Between Future Smiles and Future Stock Price Levels
416
13.5.1
An Intuitive Explanation
417
13.6
Portfolio Replication in Practice: The Stochastic-Volatility Case
418
13.6.1
The Hedging Methodology
418
13.6.2
A Numerical Example
420
13.7
Actual Fitting to Market Data
427
13.8
Conclusions
436
CONTENTS xv
14 Jump-Diffusion
Processes
439
14.1
Introduction
439
14.2
The Financial Model: Smile Tale
2
Revisited
441
14.3
Hedging and Replicability in the Presence of Jumps: First
Considerations
444
14.3.1
What Is Really Required To Complete the Market?
445
14.4
Analytic Description of Jump-Diffusions
449
14.4.1
The Stock Price Dynamics
449
14.5
Hedging with Jump-Diffusion Processes
455
14.5.1
Hedging with a Bond and the Underlying Only
455
14.5.2
Hedging with a Bond, a Second Option and the Underlying
457
14.5.3
The Case of a Single Possible Jump Amplitude
460
14.5.4
Moving to a Continuum of Jump Amplitudes
465
14.5.5
Determining the Function
g
Using the Implied Approach
465
14.5.6
Comparison with the Stochastic-Volatility Case (Again)
470
14.6
The Pricing Formula for Log-Normal Amplitude Ratios
470
14.7
The Pricing Formula in the Finite-Amplitude-Ratio Case
472
14.7.1
The Structure of the Pricing Formula for Discrete Jump
Amplitude Ratios
474
14.7.2
Matching the Moments
475
14.7.3
Numerical Results
476
14.8
The Link Between the Price Density and the Smile Shape
485
14.8.1
A Qualitative Explanation
491
14.9
Qualitative Features of Jump-Diffusion Smiles
494
14.9.1
The Smile as a Function of the Risk-Neutral Parameters
494
14.9.2
Comparison with Stochastic-Volatility Smiles
499
14.10
Jump-Diffusion Processes and Market Completeness Revisited
500
14.11
Portfolio Replication in Practice: The Jump-Diffusion Case
502
14.11.1
A Numerical Example
503
14.11.2
Results
504
14.11.3
Conclusions
509
15
Variance-Gamma
511
15.1
Who Can Make Best Use of the Variance-Gamma Approach?
511
15.2
The Variance-Gamma Process
513
15.2.1
Definition
513
15.2.2
Properties of the Gamma Process
514
15.2.3
Properties of the Variance-Gamma Process
514
15.2.4
Motivation for Variance-Gamma Modelling
517
15.2.5
Properties of the Stock Process
518
15.2.6
Option Pricing
519
xvi CONTENTS
15.3
Statistical Properties of the Price Distribution
522
15.3.1
The Real-World (Statistical) Distribution
522
15.3.2
The Risk-Neutral Distribution
522
15.4
Features of the Smile
523
15.5
Conclusions
527
16
Displaced Diffusions and Generalizations
529
16.1
Introduction
529
16.2
Gaining Intuition
530
16.2.1
First Formulation
530
16.2.2
Second Formulation
531
16.3
Evolving the Underlying with Displaced Diffusions
531
16.4
Option Prices with Displaced Diffusions
532
16.5
Matching At-The-Money Prices with Displaced Diffusions
533
16.5.1
A First Approximation
533
16.5.2
Numerical Results with the Simple Approximation
534
16.5.3
Refining the Approximation
534
16.5.4
Numerical Results with the Refined Approximation
544
16.6
The Smile Produced by Displaced Diffusions
553
16.6.1
How Quickly is the Normal-Diffusion Limit Approached?
553
16.7
Extension to Other Processes
560
17
No-Arbitrage Restrictions on the Dynamics of Smile Surfaces
563
17.1
A Worked-Out Example: Pricing Continuous Double Barriers
564
17.1.1
Money For Nothing: A Degenerate Hedging Strategy
for a Call Option
564
17.1.2
Static Replication of a Continuous Double Barrier
566
17.2
Analysis of the Cost of Unwinding
571
17.3
The Trader s Dream
575
17.4
Plan of the Remainder of the Chapter
581
17.5
Conditions of No-Arbitrage for the Stochastic Evolution of Future Smile
Surfaces
582
17.5.1
Description of the Market
582
17.5.2
The Building Blocks
584
17.6
Deterministic Smile Surfaces
585
17.6.1
Equivalent Descriptions of a State of the World
585
17.6.2
Consequences of Deterministic Smile Surfaces
587
17.6.3
Kolmogorov-Compatible Deterministic Smile Surfaces
588
17.6.4
Conditions for the Uniqueness of Kolmogorov-Compatible
Densities
589
CONTENTS xvii
17.6.5 Floating
Smiles
591
17.7
Stochastic Smiles
593
17.7.1
Stochastic Floating Smiles
594
17.7.2
Introducing Equivalent Deterministic Smile Surfaces
595
17.7.3
Implications of the Existence of an Equivalent
Deterministic Smile Surface
596
17.7.4
Extension to Displaced Diffusions
597
17.8
The Strength of the Assumptions
597
17.9
Limitations and Conclusions
598
III Interest Rates
-
Deterministic Volatilities
601
18
Mean Reversion in Interest-Rate Models
603
18.1
Introduction and Plan of the Chapter
603
18.2
Why Mean Reversion Matters in the Case of Interest-Rate Models
604
18.2.1
What Does This Mean for Forward-Rate Volatilities?
606
18.3
A Common Fallacy Regarding Mean Reversion
608
18.3.1
The Grain of Truth in the Fallacy
609
18.4
The BDT Mean-Reversion Paradox
610
18.5
The Unconditional Variance of the Short Rate in BDT
-
the
Discrete Case
612
18.6
The Unconditional Variance of the Short Rate in BDT-the
Continuous-Time Equivalent
616
18.7
Mean Reversion in Short-Rate Lattices: Recombining vs Bushy Trees
617
18.8
Extension to More General Interest-Rate Models
620
18.9
Appendix I: Evaluation of the Variance of the Logarithm of the
Instantaneous Short Rate
622
19
Volatility and Correlation in the
LIBOR
Market Model
625
19.1
Introduction
625
19.2
Specifying the Forward-Rate Dynamics in the
LIBOR
Market Model
626
19.2.1
First Formulation: Each Forward Rate in Isolation
626
19.2.2
Second Formulation: The Covariance Matrix
628
19.2.3
Third Formulation: Separating the Correlation from the
Volatility Term
630
19.3
Link with the Principal Component Analysis
631
19.4
Worked-Out Example
1:
Caplets and a Two-Period Swaption
632
19.5
Worked-Out Example
2:
Serial Options
635
19.6
Plan of the Work Ahead
636
xviii CONTENTS
20
Calibration Strategies for the
LIBOR
Market Model
639
20.1
Plan of the Chapter
639
20.2
The Setting
639
20.2.1
A Geometric Construction: The Two-Factor Case
640
20.2.2
Generalization to Many Factors
642
20.2.3
Re-Introducing the Covariance Matrix
642
20.3
Fitting an Exogenous Correlation Function
643
20.4
Numerical Results
646
20.4.1
Fitting the Correlation Surface with a Three-Factor Model
646
20.4.2
Fitting the Correlation Surface with a Four-Factor Model
650
20.4.3
Fitting Portions of the Target Correlation Matrix
654
20.5
Analytic Expressions to Link Swaption and Caplet Volatilities
659
20.5.1
What Are We Trying to Achieve?
659
20.5.2
The Set-Up
659
20.6
Optimal Calibration to Co-Terminal Swaptions
662
20.6.1
The Strategy
662
21
Specifying the Instantaneous Volatility of Forward Rates
667
21.1
Introduction and Motivation
667
21.2
The Link between Instantaneous Volatilities
and the Future Term Structure of Volatilities
668
21.3
A Functional Form for the Instantaneous Volatility Function
671
21.3.1
Financial Justification for a Humped Volatility
672
21.4
Ensuring Correct Caplet Pricing
673
21.5
Fitting the Instantaneous Volatility Function: Imposing Time
Homogeneity of the Term Structure of Volatilities
677
21.6
Is a Time-Homogeneous Solution Always Possible?
679
21.7
Fitting the Instantaneous Volatility Function: The Information from the
Swaption Market
680
21.8
Conclusions
686
22
Specifying the Instantaneous Correlation Among Forward Rates
687
22.1
Why Is Estimating Correlation So Difficult?
687
22.2
What Shape Should We Expect for the Correlation Surface?
688
22.3
Features of the Simple Exponential Correlation Function
689
22.4
Features of the Modified Exponential Correlation Function
691
22.5
Features of the Square-Root Exponential Correlation Function
694
22.6
Further Comparisons of Correlation Models
697
22.7
Features of the Schonmakers-Coffey Approach
697
22.8
Does It Make a Difference (and When)?
698
CONTENTS xix
IV
Interest
Rates -
Smiles
701
23
How to Model Interest-Rate Smiles
703
23.1
What Do We Want to Capture? A Hierarchy of Smile-Producing
Mechanisms
703
23.2
Are Log-Normal Co-Ordinates the Most Appropriate?
704
23.2.1
Defining Appropriate Co-ordinates
705
23.3
Description of the Market Data
706
23.4
Empirical Study I: Transforming the Log-Normal Co-ordinates
715
23.5
The Computational Experiments
718
23.6
The Computational Results
719
23.7
Empirical Study II: The Log-Linear Exponent
721
23.8
Combining the Theoretical and Experimental Results
725
23.9
Where Do We Go From Here?
725
24
(CEV) Processes in the Context of the LMM
729
24.1
Introduction and Financial Motivation
729
24.2
Analytical Characterization of CEV Processes
730
24.3
Financial Desirability of CEV Processes
732
24.4
Numerical Problems with CEV Processes
734
24.5
Approximate Numerical Solutions
735
24.5.1
Approximate Solutions: Mapping to Displaced Diffusions
735
24.5.2
Approximate Solutions: Transformation of Variables
735
24.5.3
Approximate Solutions: the Predictor-Corrector Method
736
24.6
Problems with the Predictor-Corrector Approximation for the LMM
747
25
Stochastic-Volatility Extensions of the LMM
751
25.1
Plan of the Chapter
751
25.2
What is the Dog and What is the Tail?
753
25.3
Displaced Diffusion vs CEV
754
25.4
The Approach
754
25.5
Implementing and Calibrating the Stochastic-Volatility LMM
756
25.5.1
Evolving the Forward Rates
759
25.5.2
Calibrating to Caplet Prices
759
25.6
Suggestions and Plan of the Work Ahead
764
26
The Dynamics of the Swaption Matrix
765
26.1
Plan of the Chapter
765
26.2
Assessing the Quality of a Model
766
26.3
The Empirical Analysis
767
26.3.1
Description of the Data
767
26.3.2
Results
768
xx CONTENTS
26.4
Extracting the Model-Implied Principal Components
776
26.4.1
Results
778
26.5
Discussion, Conclusions and Suggestions for Future Work
781
27
Stochastic-Volatility Extension
of the LMM: Two-Regime Instantaneous Volatility
783
27.1
The Relevance of the Proposed Approach
783
27.2
The Proposed Extension
783
27.3
An Aside: Some Simple Properties of Markov Chains
785
27.3.1
The Case of Two-State Markov Chains
787
27.4
Empirical Tests
788
27.4.1
Description of the Test Methodology
788
27.4.2
Results
790
27.5
How Important Is the Two-Regime Feature?
798
27.6
Conclusions
801
Bibliography
805
Index
813
|
| any_adam_object | 1 |
| author | Rebonato, Riccardo |
| author_GND | (DE-588)142802816 |
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| dewey-hundreds | 300 - Social sciences |
| dewey-ones | 332 - Financial economics |
| dewey-raw | 332.64/53 332.6323 |
| dewey-search | 332.64/53 332.6323 |
| dewey-sort | 3332.64 253 |
| dewey-tens | 330 - Economics |
| discipline | Wirtschaftswissenschaften |
| edition | 2. ed., |
| format | Book |
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| id | DE-604.BV037400681 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T23:23:31Z |
| institution | BVB |
| isbn | 0470091398 9780470091395 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-022553372 |
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| physical | XXVII, 836 S. graph. Darst. 25 cm |
| publishDate | 2007 |
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| publishDateSort | 2007 |
| publisher | Wiley |
| record_format | marc |
| spelling | Rebonato, Riccardo Verfasser (DE-588)142802816 aut Volatility and correlation the perfect hedger and the fox Riccardo Rebonato 2. ed., Chichester [u.a.] Wiley 2007 XXVII, 836 S. graph. Darst. 25 cm txt rdacontent n rdamedia nc rdacarrier Rev. ed. of: Volatility and correlation in the pricing of equity. 1999. Includes bibliographical references (p. 805-812) and index Marchés à terme de taux d'intérêt - Modèles mathématiques Options (Finances) - Modèles mathématiques Valeurs mobilières - Prix - Modèles mathématiques Mathematisches Modell Options (Finance) Mathematical models Interest rate futures Mathematical models Securities Prices Mathematical models Korrelation (DE-588)4165343-9 gnd rswk-swf Optionspreistheorie (DE-588)4135346-8 gnd rswk-swf Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Volatilität (DE-588)4268390-7 gnd rswk-swf Optionspreistheorie (DE-588)4135346-8 s Volatilität (DE-588)4268390-7 s Korrelation (DE-588)4165343-9 s Mathematisches Modell (DE-588)4114528-8 s DE-604 Digitalisierung UB Bamberg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022553372&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Rebonato, Riccardo Volatility and correlation the perfect hedger and the fox Marchés à terme de taux d'intérêt - Modèles mathématiques Options (Finances) - Modèles mathématiques Valeurs mobilières - Prix - Modèles mathématiques Mathematisches Modell Options (Finance) Mathematical models Interest rate futures Mathematical models Securities Prices Mathematical models Korrelation (DE-588)4165343-9 gnd Optionspreistheorie (DE-588)4135346-8 gnd Mathematisches Modell (DE-588)4114528-8 gnd Volatilität (DE-588)4268390-7 gnd |
| subject_GND | (DE-588)4165343-9 (DE-588)4135346-8 (DE-588)4114528-8 (DE-588)4268390-7 |
| title | Volatility and correlation the perfect hedger and the fox |
| title_auth | Volatility and correlation the perfect hedger and the fox |
| title_exact_search | Volatility and correlation the perfect hedger and the fox |
| title_full | Volatility and correlation the perfect hedger and the fox Riccardo Rebonato |
| title_fullStr | Volatility and correlation the perfect hedger and the fox Riccardo Rebonato |
| title_full_unstemmed | Volatility and correlation the perfect hedger and the fox Riccardo Rebonato |
| title_short | Volatility and correlation |
| title_sort | volatility and correlation the perfect hedger and the fox |
| title_sub | the perfect hedger and the fox |
| topic | Marchés à terme de taux d'intérêt - Modèles mathématiques Options (Finances) - Modèles mathématiques Valeurs mobilières - Prix - Modèles mathématiques Mathematisches Modell Options (Finance) Mathematical models Interest rate futures Mathematical models Securities Prices Mathematical models Korrelation (DE-588)4165343-9 gnd Optionspreistheorie (DE-588)4135346-8 gnd Mathematisches Modell (DE-588)4114528-8 gnd Volatilität (DE-588)4268390-7 gnd |
| topic_facet | Marchés à terme de taux d'intérêt - Modèles mathématiques Options (Finances) - Modèles mathématiques Valeurs mobilières - Prix - Modèles mathématiques Mathematisches Modell Options (Finance) Mathematical models Interest rate futures Mathematical models Securities Prices Mathematical models Korrelation Optionspreistheorie Volatilität |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022553372&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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