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

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Copyright 2004 Riccardo Rebonato

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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

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CONTENTS

1.5

1.6

Across-Markets Comparison of Pricing and Modelling Practices

Using Models

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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

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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

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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

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CONTENTS

4.9

Hedging Plain-Vanilla Options: Finite Re-Hedging Intervals Again

4.9.1 The Crouhy–Galai Set-Up

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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

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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

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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

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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

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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)

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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

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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

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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

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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

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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

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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

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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

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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

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III Interest Rates – Deterministic Volatilities

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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

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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

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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

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