Volatility correlation, rebonato

Volatility and Correlation
2nd Edition
The Perfect Hedger and the Fox

Riccardo Rebonato



Volatility and Correlation
2nd Edition



Volatility and Correlation
2nd Edition
The Perfect Hedger and the Fox

Riccardo Rebonato


Published 2004

John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester,
West Sussex PO19 8SQ, England
Telephone (+44) 1243 779777

Copyright  2004 Riccardo Rebonato
Email (for orders and customer service enquiries):
Visit our Home Page on www.wileyeurope.com or www.wiley.com
All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in

any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under
the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the
Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in
writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John
Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to
, or faxed to (+44) 1243 770620.
This publication is designed to provide accurate and authoritative information in regard to the subject matter
covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If
professional advice or other expert assistance is required, the services of a competent professional should be
sought.
Other Wiley Editorial Offices
John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA
Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA
Wiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Germany
John Wiley & Sons Australia Ltd, 33 Park Road, Milton, Queensland 4064, Australia
John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809
John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada M9W 1L1
Wiley also publishes its books in a variety of electronic formats. Some content that appears
in print may not be available in electronic books.
Library of Congress Cataloging-in-Publication Data
Rebonato, Riccardo.
Volatility and correlation: the perfect hedger and the fox/Riccardo
Rebonato – 2nd ed.
p. cm.
Rev. ed. of: Volatility and correlation in the pricing of equity. 1999.
Includes bibliographical references and index.
ISBN 0-470-09139-8 (cloth: alk. paper)
1. Options (Finance) – Mathematical models. 2. Interest rate
futures – Mathematical models. 3. Securities – Prices – Mathematical models.
I. Rebonato, Riccardo. Volatility and correlation in the pricing of equity.

II. Title.
HG6024.A3R43 2004
332.64 53 – dc22
2004004223
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
ISBN 0-470-09139-8
Typeset in 10/12 Times by Laserwords Private Limited, Chennai, India
Printed and bound in Great Britain by TJ International, Padstow, Cornwall
This book is printed on acid-free paper responsibly manufactured from sustainable forestry
in which at least two trees are planted for each one used for paper production.


To my parents
To Rosamund



Contents
Preface
0.1
0.2
0.3
0.4

xxi
xxi
xxiii
xxiv
xxiv

Why a Second Edition?
What This Book Is Not About
Structure of the Book
The New Subtitle

Acknowledgements

xxvii

I Foundations

1

1 Theory and Practice of Option Modelling
1.1 The Role of Models in Derivatives Pricing
1.1.1 What Are Models For?
1.1.2 The Fundamental Approach
1.1.3 The Instrumental Approach
1.1.4 A Conundrum (or, ‘What is Vega Hedging For?’)
1.2 The Efficient Market Hypothesis and Why It Matters for Option Pricing
1.2.1 The Three Forms of the EMH
1.2.2 Pseudo-Arbitrageurs in Crisis
1.2.3 Model Risk for Traders and Risk Managers
1.2.4 The Parable of the Two Volatility Traders
1.3 Market Practice
1.3.1 Different Users of Derivatives Models
1.3.2 In-Model and Out-of-Model Hedging
1.4 The Calibration Debate
1.4.1 Historical vs Implied Calibration

1.4.2 The Logical Underpinning of the Implied Approach
1.4.3 Are Derivatives Markets Informationally Efficient?
1.4.4 Back to Calibration
1.4.5 A Practical Recommendation
vii

3
3
3
5
7
8
9
9
10
11
12
14
14
15
17
18
19
21
26
27


viii

CONTENTS
1.5
1.6

Across-Markets Comparison of Pricing and Modelling Practices
Using Models

27
30

2 Option Replication
2.1 The Bedrock of Option Pricing
2.2 The Analytic (PDE) Approach
2.2.1 The Assumptions
2.2.2 The Portfolio-Replication Argument (Deterministic Volatility)
2.2.3 The Market Price of Risk with Deterministic Volatility
2.2.4 Link with Expectations – the Feynman–Kac Theorem
2.3 Binomial Replication
2.3.1 First Approach – Replication Strategy
2.3.2 Second Approach – ‘Na¨ıve Expectation’
2.3.3 Third Approach – ‘Market Price of Risk’
2.3.4 A Worked-Out Example
2.3.5 Fourth Approach – Risk-Neutral Valuation
2.3.6 Pseudo-Probabilities
2.3.7 Are the Quantities π1 and π2 Really Probabilities?
2.3.8 Introducing Relative Prices
2.3.9 Moving to a Multi-Period Setting
2.3.10 Fair Prices as Expectations
2.3.11 Switching Numeraires and Relating Expectations Under
Different Measures

2.3.12 Another Worked-Out Example
2.3.13 Relevance of the Results
2.4 Justifying the Two-State Branching Procedure
2.4.1 How To Recognize a Jump When You See One
2.5 The Nature of the Transformation between Measures: Girsanov’s Theorem
2.5.1 An Intuitive Argument
2.5.2 A Worked-Out Example
2.6 Switching Between the PDE, the Expectation and the Binomial
Replication Approaches

31
31
32
32
32
34
36
38
39
41
42
45
46
48
49
51
53
56

3 The

3.1
3.2
3.3

Building Blocks
Introduction and Plan of the Chapter
Definition of Market Terms
Hedging Forward Contracts Using Spot Quantities
3.3.1 Hedging Equity Forward Contracts
3.3.2 Hedging Interest-Rate Forward Contracts
3.4 Hedging Options: Volatility of Spot and Forward Processes

58
61
64
65
65
69
69
70
73
75
75
75
77
78
79
80

CONTENTS
3.5
3.6

3.7
3.8

3.9

The Link Between Root-Mean-Squared Volatilities and the
Time-Dependence of Volatility
Admissibility of a Series of Root-Mean-Squared Volatilities
3.6.1 The Equity/FX Case
3.6.2 The Interest-Rate Case
Summary of the Definitions So Far
Hedging an Option with a Forward-Setting Strike
3.8.1 Why Is This Option Important? (And Why Is it Difficult
to Hedge?)
3.8.2 Valuing a Forward-Setting Option
Quadratic Variation: First Approach
3.9.1 Definition
3.9.2 Properties of Variations
3.9.3 First and Second Variation of a Brownian Process
T
3.9.4 Links between Quadratic Variation and t σ (u)2 du
3.9.5 Why Quadratic Variation Is So Important (Take 1)

4 Variance and Mean Reversion in the Real and the Risk-Adjusted Worlds
4.1 Introduction and Plan of the Chapter
4.2 Hedging a Plain-Vanilla Option: General Framework

4.2.1 Trading Restrictions and Model Uncertainty:
Theoretical Results
4.2.2 The Setting
4.2.3 The Methodology
4.2.4 Criterion for Success
4.3 Hedging Plain-Vanilla Options: Constant Volatility
4.3.1 Trading the Gamma: One Step and Constant Volatility
4.3.2 Trading the Gamma: Several Steps and Constant Volatility
4.4 Hedging Plain-Vanilla Options: Time-Dependent Volatility
4.4.1 Views on Gamma Trading When the Volatility is Time
Dependent
4.4.2 Which View Is the Correct One? (and the Feynman–Kac
Theorem Again)
4.5 Hedging Behaviour In Practice
4.5.1 Analysing the Replicating Portfolio
4.5.2 Hedging Results: the Time-Dependent Volatility Case
4.5.3 Hedging with the Wrong Volatility
4.6 Robustness of the Black-and-Scholes Model
4.7 Is the Total Variance All That Matters?
4.8 Hedging Plain-Vanilla Options: Mean-Reverting Real-World Drift

ix
84
85
85
86
87
89
90
91

95
95
96
97
97
98
101
101
102
103
104
104
106
106
108
114
116
116
119
121
121
122
125
127
130
131


x

CONTENTS
4.9

Hedging Plain-Vanilla Options: Finite Re-Hedging Intervals Again
4.9.1 The Crouhy–Galai Set-Up

135
135

5 Instantaneous and Terminal Correlation
5.1 Correlation, Co-Integration and Multi-Factor Models
5.1.1 The Multi-Factor Debate
5.2 The Stochastic Evolution of Imperfectly Correlated Variables
5.3 The Role of Terminal Correlation in the Joint Evolution of Stochastic
Variables
5.3.1 Defining Stochastic Integrals
5.3.2 Case 1: European Option, One Underlying Asset
5.3.3 Case 2: Path-Dependent Option, One Asset
5.3.4 Case 3: Path-Dependent Option, Two Assets
5.4 Generalizing the Results
5.5 Moving Ahead

141
141
144
146
151
151
153
155

156
162
164

II Smiles – Equity and FX

165

6 Pricing Options in the Presence of Smiles
6.1 Plan of the Chapter
6.2 Background and Definition of the Smile
6.3 Hedging with a Compensated Process: Plain-Vanilla and Binary Options
6.3.1 Delta- and Vega-Hedging a Plain-Vanilla Option
6.3.2 Pricing a European Digital Option
6.4 Hedge Ratios for Plain-Vanilla Options in the Presence of Smiles
6.4.1 The Relationship Between the True Call Price Functional
and the Black Formula
6.4.2 Calculating the Delta Using the Black Formula and the
Implied Volatility
6.4.3 Dependence of Implied Volatilities on the Strike and the
Underlying
6.4.4 Floating and Sticky Smiles and What They Imply about Changes
in Option Prices
6.5 Smile Tale 1: ‘Sticky’ Smiles
6.6 Smile Tale 2: ‘Floating’ Smiles
6.6.1 Relevance of the Smile Story for Floating Smiles
6.7 When Does Risk Aversion Make a Difference?
6.7.1 Motivation
6.7.2 The Importance of an Assessment of Risk Aversion
for Model Building

6.7.3 The Principle of Absolute Continuity

167
167
168
169
169
172
173
174
175
176
178
180
182
183
184
184
185
186


CONTENTS
6.7.4
6.7.5
6.7.6
6.7.7
6.7.8
6.7.9

xi
The Effect of Supply and Demand
A Stylized Example: First Version
A Stylized Example: Second Version
A Stylized Example: Third Version
Overall Conclusions
The EMH Again

187
187
194
196
196
199

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

201
201
201
202
203
203
204
204
205
206
206
206
222
222
224
227
227
227
235

8 General Features of Smile-Modelling Approaches
8.1 Fully-Stochastic-Volatility Models
8.2 Local-Volatility (Restricted-Stochastic-Volatility) Models
8.3 Jump–Diffusion Models
8.3.1 Discrete Amplitude

8.3.2 Continuum of Jump Amplitudes
8.4 Variance–Gamma Models
8.5 Mixing Processes
8.5.1 A Pragmatic Approach to Mixing Models
8.6 Other Approaches
8.6.1 Tight Bounds with Known Quadratic Variation
8.6.2 Assigning Directly the Evolution of the Smile Surface
8.7 The Importance of the Quadratic Variation (Take 2)

237
237
239
241
241
242
243
243
244
245
245
246
246


xii

CONTENTS

9 The
9.1

9.2
9.3
9.4
9.5
9.6
9.7

Input Data: Fitting an Exogenous Smile Surface
What is This Chapter About?
Analytic Expressions for Calls vs Process Specification
Direct Use of Market Prices: Pros and Cons
Statement of the Problem
Fitting Prices
Fitting Transformed Prices
Fitting the Implied Volatilities
9.7.1 The Problem with Fitting the Implied Volatilities
9.8 Fitting the Risk-Neutral Density Function – General
9.8.1 Does It Matter if the Price Density Is Not Smooth?
9.8.2 Using Prior Information (Minimum Entropy)
9.9 Fitting the Risk-Neutral Density Function: Mixture of Normals
9.9.1 Ensuring the Normalization and Forward Constraints
9.9.2 The Fitting Procedure
9.10 Numerical Results
9.10.1 Description of the Numerical Tests
9.10.2 Fitting to Theoretical Prices: Stochastic-Volatility Density
9.10.3 Fitting to Theoretical Prices: Variance–Gamma Density
9.10.4 Fitting to Theoretical Prices: Jump–Diffusion Density
9.10.5 Fitting to Market Prices
9.11 Is the Term ∂C
∂S Really a Delta?

9.12 Fitting the Risk-Neutral Density Function:
The Generalized-Beta Approach
9.12.1 Derivation of Analytic Formulae
9.12.2 Results and Applications
9.12.3 What Does This Approach Offer?

10 Quadratic Variation and Smiles
10.1 Why This Approach Is Interesting
10.2 The BJN Framework for Bounding Option Prices
10.3 The BJN Approach – Theoretical Development
10.3.1 Assumptions and Definitions
10.3.2 Establishing Bounds
10.3.3 Recasting the Problem
10.3.4 Finding the Optimal Hedge
10.4 The BJN Approach: Numerical Implementation
10.4.1 Building a ‘Traditional’ Tree
10.4.2 Building a BJN Tree for a Deterministic Diffusion
10.4.3 Building a BJN Tree for a General Process
10.4.4 Computational Results

249
249
249
250
251
252
254
255
255
256

257
258
259
261
264
265
265
265
268
270
272
275
277
280
287
291
293
293
293
294
294
297
298
299
300
301
301
304
307

CONTENTS
10.4.5 Creating Asymmetric Smiles
10.4.6 Summary of the Results
10.5 Discussion of the Results
10.5.1 Resolution of the Crouhy–Galai Paradox
10.5.2 The Difference Between Diffusions and Jump–Diffusion
Processes: the Sample Quadratic Variation
10.5.3 How Can One Make the Approach More Realistic?
10.5.4 The Link with Stochastic-Volatility Models
10.5.5 The Link with Local-Volatility Models
10.5.6 The Link with Jump–Diffusion Models
10.6 Conclusions (or, Limitations of Quadratic Variation)
11 Local-Volatility Models: the Derman-and-Kani Approach
11.1 General Considerations on Stochastic-Volatility Models
11.2 Special Cases of Restricted-Stochastic-Volatility Models
11.3 The Dupire, Rubinstein and Derman-and-Kani Approaches
11.4 Green’s Functions (Arrow–Debreu Prices) in the DK Construction
11.4.1 Definition and Main Properties of Arrow–Debreu Prices
11.4.2 Efficient Computation of Arrow–Debreu Prices
11.5 The Derman-and-Kani Tree Construction
11.5.1 Building the First Step
11.5.2 Adding Further Steps
11.6 Numerical Aspects of the Implementation of the DK Construction
11.6.1 Problem 1: Forward Price Greater Than S(up) or Smaller
Than S(down)
11.6.2 Problem 2: Local Volatility Greater Than 12 |S(up) − S(down)|
11.6.3 Problem 3: Arbitrariness of the Choice of the Strike
11.7 Implementation Results
11.7.1 Benchmarking 1: The No-Smile Case

11.7.2 Benchmarking 2: The Time-Dependent-Volatility Case
11.7.3 Benchmarking 3: Purely Strike-Dependent Implied Volatility
11.7.4 Benchmarking 4: Strike-and-Maturity-Dependent Implied
Volatility
11.7.5 Conclusions
11.8 Estimating Instantaneous Volatilities from Prices as an Inverse Problem
12 Extracting the Local Volatility from Option Prices
12.1 Introduction
12.1.1 A Possible Regularization Strategy
12.1.2 Shortcomings
12.2 The Modelling Framework
12.3 A Computational Method

xiii
309
311
312
312
312
314
314
315
315
316
319
319
321
321
322
322

324
326
327
329
331
331
332
332
334
334
335
336
337
338
343
345
345
346
346
347
349


xiv

CONTENTS

12.4
12.5

12.6

12.7
12.8

12.9

12.3.1 Backward Induction
12.3.2 Forward Equations
12.3.3 Why Are We Doing Things This Way?
12.3.4 Related Approaches
Computational Results
12.4.1 Are We Looking at the Same Problem?
The Link Between Implied and Local-Volatility Surfaces
12.5.1 Symmetric (‘FX’) Smiles
12.5.2 Asymmetric (‘Equity’) Smiles
12.5.3 Monotonic (‘Interest-Rate’) Smile Surface
Gaining an Intuitive Understanding
12.6.1 Symmetric Smiles
12.6.2 Asymmetric Smiles: One-Sided Parabola
12.6.3 Asymmetric Smiles: Monotonically Decaying
What Local-Volatility Models Imply about Sticky and Floating Smiles
No-Arbitrage Conditions on the Current Implied Volatility Smile Surface
12.8.1 Constraints on the Implied Volatility Surface
12.8.2 Consequences for Local Volatilities
Empirical Performance

12.10 Appendix I: Proof that

∂ 2 Call(St ,K,T ,t)

∂K 2

= φ(ST )|K

13 Stochastic-Volatility Processes
13.1 Plan of the Chapter
13.2 Portfolio Replication in the Presence of Stochastic Volatility
13.2.1 Attempting to Extend the Portfolio Replication Argument
13.2.2 The Market Price of Volatility Risk
13.2.3 Assessing the Financial Plausibility of λσ
13.3 Mean-Reverting Stochastic Volatility
13.3.1 The Ornstein–Uhlenbeck Process
13.3.2 The Functional Form Chosen in This Chapter
13.3.3 The High-Reversion-Speed, High-Volatility Regime
13.4 Qualitative Features of Stochastic-Volatility Smiles
13.4.1 The Smile as a Function of the Risk-Neutral Parameters
13.5 The Relation Between Future Smiles and Future Stock Price Levels
13.5.1 An Intuitive Explanation
13.6 Portfolio Replication in Practice: The Stochastic-Volatility Case
13.6.1 The Hedging Methodology
13.6.2 A Numerical Example
13.7 Actual Fitting to Market Data
13.8 Conclusions

349
350
352
354
355
356

357
358
361
368
368
369
370
372
373
375
375
381
385
386
389
389
389
389
396
398
401
402
403
404
405
406
416
417
418
418

420
427
436


CONTENTS
14 Jump–Diffusion Processes
14.1 Introduction
14.2 The Financial Model: Smile Tale 2 Revisited
14.3 Hedging and Replicability in the Presence of Jumps: First
Considerations
14.3.1 What Is Really Required To Complete the Market?
14.4 Analytic Description of Jump–Diffusions
14.4.1 The Stock Price Dynamics
14.5 Hedging with Jump–Diffusion Processes
14.5.1 Hedging with a Bond and the Underlying Only
14.5.2 Hedging with a Bond, a Second Option and the Underlying
14.5.3 The Case of a Single Possible Jump Amplitude
14.5.4 Moving to a Continuum of Jump Amplitudes
14.5.5 Determining the Function g Using the Implied Approach
14.5.6 Comparison with the Stochastic-Volatility Case (Again)
14.6 The Pricing Formula for Log-Normal Amplitude Ratios
14.7 The Pricing Formula in the Finite-Amplitude-Ratio Case
14.7.1 The Structure of the Pricing Formula for Discrete Jump
Amplitude Ratios
14.7.2 Matching the Moments
14.7.3 Numerical Results
14.8 The Link Between the Price Density and the Smile Shape
14.8.1 A Qualitative Explanation
14.9 Qualitative Features of Jump–Diffusion Smiles

14.9.1 The Smile as a Function of the Risk-Neutral Parameters
14.9.2 Comparison with Stochastic-Volatility Smiles
14.10 Jump–Diffusion Processes and Market Completeness Revisited
14.11 Portfolio Replication in Practice: The Jump–Diffusion Case
14.11.1 A Numerical Example
14.11.2 Results
14.11.3 Conclusions
15 Variance–Gamma
15.1 Who Can Make Best Use of the Variance–Gamma Approach?
15.2 The Variance–Gamma Process
15.2.1 Definition
15.2.2 Properties of the Gamma Process
15.2.3 Properties of the Variance–Gamma Process
15.2.4 Motivation for Variance–Gamma Modelling
15.2.5 Properties of the Stock Process
15.2.6 Option Pricing

xv
439
439
441
444
445
449
449
455
455
457
460
465

465
470
470
472
474
475
476
485
491
494
494
499
500
502
503
504
509
511
511
513
513
514
514
517
518
519


xvi

CONTENTS
15.3 Statistical Properties of the Price Distribution
15.3.1 The Real-World (Statistical) Distribution
15.3.2 The Risk-Neutral Distribution
15.4 Features of the Smile
15.5 Conclusions

522
522
522
523
527

16 Displaced Diffusions and Generalizations
16.1 Introduction
16.2 Gaining Intuition
16.2.1 First Formulation
16.2.2 Second Formulation
16.3 Evolving the Underlying with Displaced Diffusions
16.4 Option Prices with Displaced Diffusions
16.5 Matching At-The-Money Prices with Displaced Diffusions
16.5.1 A First Approximation
16.5.2 Numerical Results with the Simple Approximation
16.5.3 Refining the Approximation
16.5.4 Numerical Results with the Refined Approximation
16.6 The Smile Produced by Displaced Diffusions
16.6.1 How Quickly is the Normal-Diffusion Limit Approached?
16.7 Extension to Other Processes

529

529
530
530
531
531
532
533
533
534
534
544
553
553
560

17 No-Arbitrage Restrictions on the Dynamics of Smile Surfaces
17.1 A Worked-Out Example: Pricing Continuous Double Barriers
17.1.1 Money For Nothing: A Degenerate Hedging Strategy
for a Call Option
17.1.2 Static Replication of a Continuous Double Barrier
17.2 Analysis of the Cost of Unwinding
17.3 The Trader’s Dream
17.4 Plan of the Remainder of the Chapter
17.5 Conditions of No-Arbitrage for the Stochastic Evolution of Future Smile
Surfaces
17.5.1 Description of the Market
17.5.2 The Building Blocks
17.6 Deterministic Smile Surfaces
17.6.1 Equivalent Descriptions of a State of the World
17.6.2 Consequences of Deterministic Smile Surfaces

17.6.3 Kolmogorov-Compatible Deterministic Smile Surfaces
17.6.4 Conditions for the Uniqueness of Kolmogorov-Compatible
Densities

563
564
564
566
571
575
581
582
582
584
585
585
587
588
589


CONTENTS
17.6.5 Floating Smiles
17.7 Stochastic Smiles
17.7.1 Stochastic Floating Smiles
17.7.2 Introducing Equivalent Deterministic Smile Surfaces
17.7.3 Implications of the Existence of an Equivalent
Deterministic Smile Surface
17.7.4 Extension to Displaced Diffusions
17.8 The Strength of the Assumptions

17.9 Limitations and Conclusions

xvii
591
593
594
595
596
597
597
598

III Interest Rates – Deterministic Volatilities

601

18 Mean Reversion in Interest-Rate Models
18.1 Introduction and Plan of the Chapter
18.2 Why Mean Reversion Matters in the Case of Interest-Rate Models
18.2.1 What Does This Mean for Forward-Rate Volatilities?
18.3 A Common Fallacy Regarding Mean Reversion
18.3.1 The Grain of Truth in the Fallacy
18.4 The BDT Mean-Reversion Paradox
18.5 The Unconditional Variance of the Short Rate in BDT – the
Discrete Case
18.6 The Unconditional Variance of the Short Rate in BDT–the
Continuous-Time Equivalent
18.7 Mean Reversion in Short-Rate Lattices: Recombining vs Bushy Trees
18.8 Extension to More General Interest-Rate Models
18.9 Appendix I: Evaluation of the Variance of the Logarithm of the

Instantaneous Short Rate

603
603
604
606
608
609
610

19 Volatility and Correlation in the LIBOR Market Model
19.1 Introduction
19.2 Specifying the Forward-Rate Dynamics in the LIBOR Market Model
19.2.1 First Formulation: Each Forward Rate in Isolation
19.2.2 Second Formulation: The Covariance Matrix
19.2.3 Third Formulation: Separating the Correlation from the
Volatility Term
19.3 Link with the Principal Component Analysis
19.4 Worked-Out Example 1: Caplets and a Two-Period Swaption
19.5 Worked-Out Example 2: Serial Options
19.6 Plan of the Work Ahead

612
616
617
620
622
625
625
626

626
628
630
631
632
635
636


xviii

CONTENTS

20 Calibration Strategies for the LIBOR Market Model
20.1 Plan of the Chapter
20.2 The Setting
20.2.1 A Geometric Construction: The Two-Factor Case
20.2.2 Generalization to Many Factors
20.2.3 Re-Introducing the Covariance Matrix
20.3 Fitting an Exogenous Correlation Function
20.4 Numerical Results
20.4.1 Fitting the Correlation Surface with a Three-Factor Model
20.4.2 Fitting the Correlation Surface with a Four-Factor Model
20.4.3 Fitting Portions of the Target Correlation Matrix
20.5 Analytic Expressions to Link Swaption and Caplet Volatilities
20.5.1 What Are We Trying to Achieve?
20.5.2 The Set-Up
20.6 Optimal Calibration to Co-Terminal Swaptions
20.6.1 The Strategy

639
639
639
640
642
642
643
646
646
650
654
659
659
659
662
662

21 Specifying the Instantaneous Volatility of Forward Rates
21.1 Introduction and Motivation
21.2 The Link between Instantaneous Volatilities
and the Future Term Structure of Volatilities
21.3 A Functional Form for the Instantaneous Volatility Function
21.3.1 Financial Justification for a Humped Volatility
21.4 Ensuring Correct Caplet Pricing
21.5 Fitting the Instantaneous Volatility Function: Imposing Time
Homogeneity of the Term Structure of Volatilities
21.6 Is a Time-Homogeneous Solution Always Possible?
21.7 Fitting the Instantaneous Volatility Function: The Information from the
Swaption Market
21.8 Conclusions

667
667

22 Specifying the Instantaneous Correlation Among Forward Rates
22.1 Why Is Estimating Correlation So Difficult?
22.2 What Shape Should We Expect for the Correlation Surface?
22.3 Features of the Simple Exponential Correlation Function
22.4 Features of the Modified Exponential Correlation Function
22.5 Features of the Square-Root Exponential Correlation Function
22.6 Further Comparisons of Correlation Models
22.7 Features of the Schonmakers–Coffey Approach
22.8 Does It Make a Difference (and When)?

668
671
672
673
677
679
680
686
687
687
688
689
691
694
697
697

698


CONTENTS

xix

IV Interest Rates – Smiles

701

23 How to Model Interest-Rate Smiles
23.1 What Do We Want to Capture? A Hierarchy of Smile-Producing
Mechanisms
23.2 Are Log-Normal Co-Ordinates the Most Appropriate?
23.2.1 Defining Appropriate Co-ordinates
23.3 Description of the Market Data
23.4 Empirical Study I: Transforming the Log-Normal Co-ordinates
23.5 The Computational Experiments
23.6 The Computational Results
23.7 Empirical Study II: The Log-Linear Exponent
23.8 Combining the Theoretical and Experimental Results
23.9 Where Do We Go From Here?

703
703
704
705
706
715

718
719
721
725
725

24 (CEV) Processes in the Context of the LMM
24.1 Introduction and Financial Motivation
24.2 Analytical Characterization of CEV Processes
24.3 Financial Desirability of CEV Processes
24.4 Numerical Problems with CEV Processes
24.5 Approximate Numerical Solutions
24.5.1 Approximate Solutions: Mapping to Displaced Diffusions
24.5.2 Approximate Solutions: Transformation of Variables
24.5.3 Approximate Solutions: the Predictor–Corrector Method
24.6 Problems with the Predictor–Corrector Approximation for the LMM

729
729
730
732
734
735
735
735
736
747

25 Stochastic-Volatility Extensions of the LMM
25.1 Plan of the Chapter

25.2 What is the Dog and What is the Tail?
25.3 Displaced Diffusion vs CEV
25.4 The Approach
25.5 Implementing and Calibrating the Stochastic-Volatility LMM
25.5.1 Evolving the Forward Rates
25.5.2 Calibrating to Caplet Prices
25.6 Suggestions and Plan of the Work Ahead

751
751
753
754
754
756
759
759
764

26 The Dynamics of the Swaption Matrix
26.1 Plan of the Chapter
26.2 Assessing the Quality of a Model
26.3 The Empirical Analysis
26.3.1 Description of the Data
26.3.2 Results

765
765
766
767
767

768


xx

CONTENTS
26.4 Extracting the Model-Implied Principal Components
26.4.1 Results
26.5 Discussion, Conclusions and Suggestions for Future Work

776
778
781

27 Stochastic-Volatility Extension
of the LMM: Two-Regime Instantaneous Volatility
27.1 The Relevance of the Proposed Approach
27.2 The Proposed Extension
27.3 An Aside: Some Simple Properties of Markov Chains
27.3.1 The Case of Two-State Markov Chains
27.4 Empirical Tests
27.4.1 Description of the Test Methodology
27.4.2 Results
27.5 How Important Is the Two-Regime Feature?
27.6 Conclusions

783
783
783
785

787
788
788
790
798
801

Bibliography

805

Index

813


Preface
0.1

Why a Second Edition?

This second edition is, in reality, virtually a whole new book. Approximately 80% of the
material has been added, fully reworked or changed. Let me explain why I have felt that
undertaking such a task was needed.
Some of the messages of the first edition have, to a large extent, become accepted in
the trading community (and perhaps the first edition of this book played a small role in
this process). Let me mention a few. It is now more widely understood, for instance, that
just recovering today’s market prices of plain-vanilla options is a necessary but by no
means sufficient criterion for choosing a good model. As a consequence, the modelling
emphasis has gradually shifted away from the ability of a model to take an accurate snapshot of today’s plain-vanilla option market, towards predicting in a reasonably accurate

way the future smile.
To give another example, it is now generally recognized that what matters for pricing
is the terminal and not just the instantaneous correlation among the state variables.
Therefore traders now readily acknowledge that time-dependent instantaneous volatilities
can be very effective in creating de-correlation among interest rates. As a corollary, the
once commonly held view that one needs very-high-dimensional models to price complex
interest-rate instruments has been challenged and proven to be, if not wrong, certainly an
overstatement of the truth.
Moving to more general pricing considerations, it is now acknowledged that the marketcompleteness assumption should be invoked to obtain the powerful results it allows (e.g.
the uniqueness of the ‘fair’ price and of the hedging strategy or the replicability of an
arbitrary terminal payoff) only if financially justifiable, not just because it makes the
modelling easy. So, most traders now recognize that claiming that, say, local-volatility
models are desirable because they allow a complete-market framework to be retained
squarely puts the cart before the horse. The relevant question is whether a given market
is truly complete (or completable), not whether a given model assumes it to be so.
As these ideas have become part of the received wisdom as to how models should be
used, I have felt that other issues, perhaps not so relevant when the first edition appeared,
now need to be looked at more carefully. For instance, I think that the distinction between
what I call in my book the fundamental and the instrumental approaches to option pricing
has not received the attention it deserves. Different types of traders use models in different
ways, and for different purposes. The question should therefore at least be asked whether
the same class of models can really simultaneously serve the needs of both types of
xxi


xxii

PREFACE

trader. Is there any such thing as the ‘best’ model for the plain-vanilla trader and for the

exotic trader? Are those features that make a model desirable for the former necessarily
appealing to the latter?
Linked to this is the practical and theoretical importance for option pricing of the joint
practices of vega hedging and daily model re-calibration. I believe that these two nearuniversal practices have not been analysed as carefully as they should be. Yet I think that
they lie at the heart of option pricing, and that they should influence at a very deep level
the choice of a pricing model.
Another reason for updating the original work is that interest-rate smiles were an
interesting second-order effect when I was writing the first edition. They have now become
an essential ingredient of term structure modelling, and the consensus of the trading
community is beginning to crystallize around a sufficiently well-established methodology,
that it makes sense to present a coherent picture of the field.
More generally, outside the interest-rate arena traders have encountered great difficulties in fitting market smiles in a financially-convincing and numerically-robust manner
starting from a specification of the process for the underlying. As a consequence they
have become increasingly interested in trying to model directly the evolution of the smile
surface. Is this a sound practice? Can it be theoretically justified, or is it just a practitioner’s legerdemain? I had only hinted at these issues in the first edition, but they are
given a much fuller treatment in the present work.
Apart from the immediate applications, these developments have given rise to some
important questions, such as: What is more important to model, the dynamics of the
underlying, or the evolution of the smile surface (i.e. of the associated options)?; Can
we (should we) always assume that the changes in option prices can be derived from a
stochastic process that can simultaneously account for the evolution of the underlying?; If
this were not the case, is it really possible (and practicable) to set up arbitrage strategies
to exploit this lack of coherence?
The last question brings me naturally to another aspect of option pricing that I question
more explicitly in this second edition, namely the reliance on the informational efficiency
of markets implicit in the commonly used calibration and hedging practices. This topic
is linked to the popular and, these days, ‘trendy’ topic of behavioural finance. I discuss
at several points in this second edition why I think that one should at least question the
classical rational-investor, efficient-market paradigm when it comes to option pricing.
Another topic that I emphasize more strongly in this second edition is the following.

In the post Black-and-Scholes era the perfect-replication idea has become the bedrock of
option pricing. In a nutshell: ‘If we can replicate perfectly, we don’t have to worry about
aversion to risk.’ All models, of course, are wrong, and the real question is not whether
they are ‘true’ in some metaphysical sense but whether they are useful. Looked at in
this light, the perfect-replication model has been immensely useful for the first-generation
of option products. I feel, however, that, when it comes to some of the products that
are traded today, the dichotomous distinction between complete markets, where payoff
replication should always be possible, and incomplete markets, where no self-financing
hedging strategy can recover with certainty a derivatives payoff, might be fast approaching
its ‘best-before date’. I make an argument as to why this is the case throughout this new
edition, but especially in Part II.
The more one looks into a certain subject, the simpler the overarching structure begins
to appear. I think that, by working with pricing models for close to 15 years, I have come