The SABR, LIBOR market model: pricing, calibrating and hedging for complex, interest-rate derivates
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Chichester
Wiley
2009
|
| Edition: | 1. publ. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | XI, 284 S. graph. Darst. |
| ISBN: | 0470740051 9780470740057 |
Staff View
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| 245 | 1 | 0 | |a The SABR, LIBOR market model |b pricing, calibrating and hedging for complex, interest-rate derivates |c Riccardo Rebonato ; Kenneth Mckay and Richard White |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Chichester |b Wiley |c 2009 | |
| 300 | |a XI, 284 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 7 | |a Finanzderivat |2 stw | |
| 650 | 7 | |a Hedging |2 stw | |
| 650 | 7 | |a LIBOR Market Modell |2 stw | |
| 650 | 7 | |a Optionspreistheorie |2 stw | |
| 650 | 7 | |a Zins |2 stw | |
| 650 | 4 | |a Mathematisches Modell | |
| 650 | 4 | |a Derivative securities |x Accounting | |
| 650 | 4 | |a Hedging (Finance) |x Mathematical models | |
| 650 | 4 | |a Interest rate futures | |
| 650 | 4 | |a Options (Finance) |x Prices |x Mathematical models | |
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| adam_text | Contents
Acknowledgements
xi
1
Introduction
1
1 The Theoretical Set-Up
7
2
The
LIBOR
Market Model
9
2.1
Definitions
10
2.2
The Volatility Functions
11
2.3
Separating the Correlation from the Volatility Term
12
2.4
The Caplet-Pricing Condition Again
14
2.5
The Forward-Rate/Forward-Rate Correlation
16
2.5.1
The Simple Exponential Correlation
16
2.5.2
The Multiplicative Correlation
17
2.6
Possible Shapes of the Doust Correlation Function
19
2.7
The Covariance Integral Again
21
3
The SABR Model
25
3.1
The SABR Model (and Why it is a Good Model)
25
3.2
Description of the Model
26
3.3
The Option Prices Given by the SABR Model
27
3.4
Special Cases
28
3.4.1
ATM Options
28
3.4.2
The Normal Case
(β
= 0) 28
3.4.3
The Log-Normal Case
(β
= 1) 29
3.5
Qualitative Behaviour of the SABR Model
29
3.5.1
Dependence on
σ0Γ
29
3.5.2
Dependence on
β
31
3.5.3
Dependence on
ρ
33
3.5.4
Dependence on
ν
33
3.6
The Link Between the Exponent,
β,
and the Volatility of Volatility,
ν
35
vi
CONTENTS
3.7
Volatility Clustering in the
(LMM)-SABR
Model
37
3.8
The Market
40
3.8.1
Analysis of
σ0Γ (β =
0.5) 40
3.8.2
Analysis of
ντ (β
= 0.5) 41
3.8.3
Analysis of
ρτ (β
= 0.5) 43
3.9
How Do We Know that the Market has Chosen
β
= 0.5? 43
3.10
The Problems with the
SABR
Modely
46
3.10.1
Log-Normality of the Volatility Process
46
3.10.2
Problems with the (Stochastic) CEV Process
47
4
The LMM-SABR Model
51
4.1
The Equations of Motion
52
4.2
The Nature of the Stochasticity Introduced by Our Model
53
4.3
A Simple Correlation Structure
54
4.4
A More General Correlation Structure
55
4.5
Observations on the Correlation Structure
57
4.6
The Volatility Structure
58
4.7
What We Mean by Time Homogeneity
59
4.8
The Volatility Structure in Periods of Market Stress
59
4.9
A More General Stochastic Volatility Dynamics
63
4.10
Calculating the No-Arbitrage Drifts
64
4.10.1
Preliminaries
64
4.10.2
Standard
LIBOR
and
LIBOR
in Arrearsy
70
4.10.3
LIBOR
in Arrears: The Volatility Drift
73
4.10.4
The Drifts in the General Case of Several Forward Ratesy
74
4.10.5
Volatility Drifts in the Swap Measure
75
II Implementation and Calibration
79
5
Calibrating the LMM-SABR Model to Market Caplet Prices
81
5.1
The Caplet-Calibration Problem
81
5.2
Choosing the Parameters of the Function, g(·), and the Initial
Values, kl
83
5.3
Choosing the Parameters of the Function h(-)
84
5.4
Choosing the Exponent,
β,
and the Correlation,
Фѕавк
88
5.5
Results
88
5.6
Calibration in Practice: Implications for the SABR Model
91
5.6.1
Looking at Caplets in Isolation
91
5.6.2
Looking at Caplets and Swaptions Together
95
5.7
Implications for Model Choice
99
6
Calibrating the LMM-SABR Model to Market Swaption Prices
101
6.1
The Swaption Calibration Problem
101
CONTENTS
vii
6.2
Swap Rate and Forward Rate Dynamics
102
6.3
Approximating the Instantaneous Swap Rate Volatility, S,
104
6.4
Approximating the Initial Value of the Swap Rate Volatility,
Σο
(First
Route)
105
6.5
Approximating
Σο
(Second Route) and the Volatility of Volatility of the
Swap Rate, V
106
6.6
Approximating the Swap-Rate/Swap-Rate-Volatility Correlation,
Äsabr 108
6.7
Approximating the Swap Rate Exponent,
В
108
6.8
Results
109
6.8.1
Comparison between Approximated and Simulation Prices
109
6.8.2
Comparison between Parameters from the Approximations and the
Simulations
117
6.9
Conclusions and Suggestions for Future Work
118
6.10
Appendix: Derivation of Approximate Swap Rate Volatility
118
6.11
Appendix: Derivation of Swap-Rate/Swap-Rate-Volatility Correlation,
ÄSABR 120
6.12
Appendix: Approximation of dSt/S,
122
7
Calibrating the Correlation Structure
125
7.1
Statement of the Problem
125
7.2
Creating a Valid Model Matrix
126
7.2.1
First Strategy, Stage
1:
Diagonalize
Ρ
128
7.2.2
First Strategy, Stage
2:
Analytic Optimization of c
128
7.2.3
Second Strategy: Optimizing over Angles
129
7.3
A Case Study: Calibration Using the Hypersphere Method
131
7.4
Which Method Should One Choose?
137
7.5
Appendix
138
III Empirical Evidence
141
8
The Empirical Problem
143
8.1
Statement of the Empirical Problem
143
8.2
What Do We Know from the Literature?
145
8.3
Data Description
148
8.4
Distributional Analysis and Its Limitations
150
8.5
What is the True Exponent
β?
153
8.6
Appendix: Some Analytic Results
155
9
Estimating the Volatility of the Forward Rates
159
9.1
Expiry Dependence of Volatility of Forward Rates
160
9.2
Direct Estimation
162
9.3
Looking at the Normality of the Residuals
164
9.4
Maximum-Likelihood and Variations on the Theme
171
viii CONTENTS
9.5
Information
About the Volatility from the Options Market
175
9.6
Overall Conclusions
178
10
Estimating the Correlation Structure
181
10.1
What We are Trying to Do
181
10.2
Some Results from Random Matrix Theory
182
10.3
Empirical Estimation
185
10.4
Descriptive Statistics
185
10.4.1
The Forward-Rate/Forward-Rate Correlation Matrix
185
10.4.2
The Forward-Rate/Volatility Correlation Block
187
10.4.3
The Volatility/Volatility Correlation Matrix
188
10.5
Signal and Noise in the Empirical Correlation Blocks
188
10.5.1
The Forward-Rate/Forward-Rate Correlation Matrix
188
10.5.2
The Volatility/Volatility Correlation Matrix
190
10.5.3
The Forward-Rate/Volatility Correlation Block
190
10.6
What Does Random Matrix Theory Really Tell Us?
190
10.7
Calibrating the Correlation Matrices
191
10.7.1
The Fitting Procedure
192
10.7.2
Results
192
10.8
How Much Information Do the Proposed Models Retain?
195
10.8.1
Eigenvalues of the Correlation Blocks
195
10.8.2
Eigenvalues of Differences in the Correlation Blocks
196
10.8.3
Entropy Measures
198
10.8.4
The Forward-Rate/Volatility Correlation Block
202
IV Hedging
203
11
Various Types of Hedging
205
11.1
Statement of the Problem
205
11.2
Three Types of Hedging
206
11.2.1
In- and Out-of-Model Hedging
206
11.2.2
Functional-Dependence Hedging
207
11.3
Definitions
210
11.4
First-Order Derivatives with Respect to the Underlyings
211
11.4.1
Delta Hedging
211
11.4.2 Vega
Hedging
213
11.5
Second-Order Derivatives with Respect to the Underlyings
214
11.5.1
Vanna
and Volga
214
11.6
Generalizing Functional-Dependence Hedging
215
11.7
How Does the Model Know about
Vanna
and Volga?
219
11.8
Choice of Hedging Instrument
220
CONTENTS ix
12
Hedging against Moves in the Forward Rate and in the Volatility
221
12.1
Delta Hedging in the
SABR-(LMM)
Model
221
12.2 Vega
Hedging in the
SABR-(LMM)
Model
229
13
(LMM)-SABR Hedging in Practice: Evidence from Market Data
231
13.1
Purpose of this Chapter
231
13.2
Notation
231
13.2.1
Estimation of the Unobservable Volatility
232
13.2.2
Tests of the Hedging Performance of the SABR Model
233
13.2.3
Tests of the Hedging Performance of the LMM-SABR Model
233
13.3
Hedging Results for the SABR Model
234
13.4
Hedging Results for the LMM-SABR Model
243
13.5
Conclusions
245
14
Hedging the Correlation Structure
247
14.1
The Intuition Behind the Problem
247
14.2
Hedging the Forward-Rate Block
249
14.3
Hedging the Volatility-Rate Block
251
14.4
Hedging the Forward-Rate/Volatility Block
253
14.5
Final Considerations
254
15
Hedging in Conditions of Market Stress
257
15.1
Statement of the Problem
257
15.2
The Volatility Function
259
15.3
The Case Study
260
15.4
Hedging
261
15.4.1
The Normal-to-Normal State Transition
261
15.4.2
The Normal-to-Excited Transition
263
15.4.3
Normal-to-Unknown Transition
265
15.4.4
Starting from the Excited State
266
15.5
Results
266
15.5.1
Hedging Results for the Normal-to-Normal Transition
267
15.5.2
Hedging Results for the Normal-to-Excited Transition
267
15.6
Are We Getting Something for Nothing?
270
References
271
Index
275
the SABR/LIBOR
Market Model
PRICING, CALIBRATION AND HEDGING FOR COMPLEX INTEREST-BATE DERIVATIVES
The authors take two market standards, the SABR and the
LIBOR
Market Model and produce
a coherent synthesis for the pricing of complex interest-rate derivatives. The SABR model has
become the market standard to recover the price of European options. Its main strengths are
its financial justifiability and its ability to recover the dynamics of the smile evolution when the
underlying changes. However, the SABR model treats each European option in isolation. The
processes for forward rates and swap rates cannot easily be combined to create coherent
dynamics for the entire yield curve.
With their new model, the authors bring the dynamics of the various forward rates and
stochastic volatilities under a single measure, and derive drift adjustments to ensure the
absence of arbitrage and to allow for the pricing of complex derivatives. The credible evolution
of future smiles generated by the model is essential to complex derivatives pricing as it
determines future prices for caplets and swaptions and therefore plausible re-hedging costs.
The authors calibrate their model to hedging instruments in a way that is both accurate and
extremely simple. They also propose a pragmatic hedging approach, inspired by work done
with the two-state Markov-chain approach which relies on the empirical regularities of the
dynamics of the smile surface and the robustness of the fits proposed. The final chapter
considers survival hedging in times of market turmoil. It does so by suggesting a set of
transactions that can protect the value of a complex derivatives book in a stressed market.
The extension of the LMM model provides a valid description of the financial reality while
retaining tractability, computational speed and ease of calibration. The goal for the new model
is to offer the ability to reduce uncertainty in market prices to an acceptable minimum by
making as judicious a use as possible of the econometric information available. The grounding
in empirical information of the modelling approach utilised by the authors differentiates this
title from the stochastic-calculus-heavy, but empirically light, work of others.
The title will be of interest to quantitative analysts, quantitative developers, risk managers and
traders in complex derivatives.
|
| any_adam_object | 1 |
| author | Rebonato, Riccardo McKay, Kenneth White, Richard |
| author_GND | (DE-588)142802816 (DE-588)133827283 |
| author_facet | Rebonato, Riccardo McKay, Kenneth White, Richard |
| author_role | aut aut aut |
| author_sort | Rebonato, Riccardo |
| author_variant | r r rr k m km r w rw |
| building | Verbundindex |
| bvnumber | BV035286404 |
| callnumber-first | H - Social Science |
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| callnumber-search | HG6024.A3 |
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| callnumber-subject | HG - Finance |
| classification_rvk | QK 622 |
| classification_tum | WIR 160f MAT 902f |
| ctrlnum | (OCoLC)300720214 (DE-599)GBV584051476 |
| dewey-full | 332.63/23 |
| dewey-hundreds | 300 - Social sciences |
| dewey-ones | 332 - Financial economics |
| dewey-raw | 332.63/23 |
| dewey-search | 332.63/23 |
| dewey-sort | 3332.63 223 |
| dewey-tens | 330 - Economics |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV035286404 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:30:28Z |
| institution | BVB |
| isbn | 0470740051 9780470740057 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017091538 |
| oclc_num | 300720214 |
| open_access_boolean | |
| owner | DE-703 DE-91G DE-BY-TUM DE-11 DE-473 DE-BY-UBG |
| owner_facet | DE-703 DE-91G DE-BY-TUM DE-11 DE-473 DE-BY-UBG |
| physical | XI, 284 S. graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Wiley |
| record_format | marc |
| spelling | Rebonato, Riccardo Verfasser (DE-588)142802816 aut The SABR, LIBOR market model pricing, calibrating and hedging for complex, interest-rate derivates Riccardo Rebonato ; Kenneth Mckay and Richard White 1. publ. Chichester Wiley 2009 XI, 284 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Finanzderivat stw Hedging stw LIBOR Market Modell stw Optionspreistheorie stw Zins stw Mathematisches Modell Derivative securities Accounting Hedging (Finance) Mathematical models Interest rate futures Options (Finance) Prices Mathematical models Preisbildung (DE-588)4047103-2 gnd rswk-swf Mathematisches Modell (DE-588)4114528-8 gnd rswk-swf Hedging (DE-588)4123357-8 gnd rswk-swf Derivat Wertpapier (DE-588)4381572-8 gnd rswk-swf Derivat Wertpapier (DE-588)4381572-8 s Preisbildung (DE-588)4047103-2 s Hedging (DE-588)4123357-8 s Mathematisches Modell (DE-588)4114528-8 s DE-604 McKay, Kenneth Verfasser (DE-588)133827283 aut White, Richard Verfasser aut Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017091538&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017091538&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Rebonato, Riccardo McKay, Kenneth White, Richard The SABR, LIBOR market model pricing, calibrating and hedging for complex, interest-rate derivates Finanzderivat stw Hedging stw LIBOR Market Modell stw Optionspreistheorie stw Zins stw Mathematisches Modell Derivative securities Accounting Hedging (Finance) Mathematical models Interest rate futures Options (Finance) Prices Mathematical models Preisbildung (DE-588)4047103-2 gnd Mathematisches Modell (DE-588)4114528-8 gnd Hedging (DE-588)4123357-8 gnd Derivat Wertpapier (DE-588)4381572-8 gnd |
| subject_GND | (DE-588)4047103-2 (DE-588)4114528-8 (DE-588)4123357-8 (DE-588)4381572-8 |
| title | The SABR, LIBOR market model pricing, calibrating and hedging for complex, interest-rate derivates |
| title_auth | The SABR, LIBOR market model pricing, calibrating and hedging for complex, interest-rate derivates |
| title_exact_search | The SABR, LIBOR market model pricing, calibrating and hedging for complex, interest-rate derivates |
| title_full | The SABR, LIBOR market model pricing, calibrating and hedging for complex, interest-rate derivates Riccardo Rebonato ; Kenneth Mckay and Richard White |
| title_fullStr | The SABR, LIBOR market model pricing, calibrating and hedging for complex, interest-rate derivates Riccardo Rebonato ; Kenneth Mckay and Richard White |
| title_full_unstemmed | The SABR, LIBOR market model pricing, calibrating and hedging for complex, interest-rate derivates Riccardo Rebonato ; Kenneth Mckay and Richard White |
| title_short | The SABR, LIBOR market model |
| title_sort | the sabr libor market model pricing calibrating and hedging for complex interest rate derivates |
| title_sub | pricing, calibrating and hedging for complex, interest-rate derivates |
| topic | Finanzderivat stw Hedging stw LIBOR Market Modell stw Optionspreistheorie stw Zins stw Mathematisches Modell Derivative securities Accounting Hedging (Finance) Mathematical models Interest rate futures Options (Finance) Prices Mathematical models Preisbildung (DE-588)4047103-2 gnd Mathematisches Modell (DE-588)4114528-8 gnd Hedging (DE-588)4123357-8 gnd Derivat Wertpapier (DE-588)4381572-8 gnd |
| topic_facet | Finanzderivat Hedging LIBOR Market Modell Optionspreistheorie Zins Mathematisches Modell Derivative securities Accounting Hedging (Finance) Mathematical models Interest rate futures Options (Finance) Prices Mathematical models Preisbildung Derivat Wertpapier |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017091538&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017091538&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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