Volatility & Correlation

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In the ~ri~~n~ of Equity, FX and ~n~~r~~~-~~t~ Options

Risk ~anage~ent andAnalysis. Vol. l: ~easuring and ~odelling Financial Risk

Carol Alexander (ed.)

Risk ~anage~ent andAnalysis. Vol. 2: New ~arkets and Products CaroE Alexander (ed.)

I~plem~nting Value at Risk

Philip Best

Derivat~ves De~yst~ed: Using Structured Financial Products

John C. Braddock

Imple~enting Derivatives ~odels

Les Clewlow and Chris Strickland

Advanced Credit Risk Analysis: ~inancial Approaches and ~athe~atical ~odels to Asses

Price and an age Credit Risk

Didier Cossin and Hugues Pirotte

Derivatives for Decision ~akers: Strate~ic ~anage~ent Issues

George Crawford and Bidyut Sen

Currency Derivatives: Pricing Theo~, Exotic Options, and edging plications

David F. DeRosa

Options on Foreign Exch~nge (rev~sedition)

David F. DeRosa

The andb boo^ of Equity ~eri~atives (revised edition)

Jack Francis, William Toy and J. Gregg Whittaker

Interest-Rate ~odelling

Jessica James and Nick Webber

Dictiona~ of Financial Eng~neering

John F. Marshall

andb book of ~ybrid ~nstruments: Convertible Bonds, Preferred Shares, Lyon~~, ELKS,

DECS and Other ~andato~ ~onverti~le Notes

Izzy Nelken (ed.)

Interest- ate Option ~odels: understanding^ Analysing and Using ~odels for Exotic

Interest-Rate Options (second edition)

Riccardo Rebonato

Volatility and Correlati~n in the Pricing of Equi~, FX and ~nterest-Rate Opt~~ns

Riccardo Rebonato

Derivatives andb book: Risk ~anagement and Control

Robert J. Schwartz and Clifford W. Smith. Jr

Dyna~ic edging: an aging Vanilla and Exotic Options

Nassim Taleb

Credit Derivatives: A Guide to instru~ents and Ap~licat~ons

Janet Tavakoli

Pricing Finan~ial Derivatives: The Finite Di~erence et hod

Doming0 A. Tavella and Art Owen

Chichester * New York * ~einhei~ * Brisbane 0 Singapore * Toronto

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Copyright 0 1999 Riccardo Rebonato

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~i~ra~ of Congress CataLogin~in~~u~Lica~on Dtu

Rebonato, Riccardo,

Volatility and correlation in the pricing of equity, FX and interest-rate options/~iccardo Rebonato.

p. cm.

Includes bibliographical references and index.

ISBN 0-47 1-89998-4 (alk. paper)

l. Options (~inance)-~athematical models. 2. Interest rate futures-Mathematical models. 3. Secu￾rities-Prices- ath he ma tical models. I. Title. 11. Title: Volatility and correlation.

HG6024.A3R43 1999

332.63’23-dc21 99-35 173

CIP

~~sh ~ibra~ CataLog~~~g in Publicu~on Datu

A catalogue record for this book is available from the British Library

ISBN 0-47 1-89998-4

Typeset in 10112pt Times by Laser Words, Madras, India

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1.1

1.2

1.3

1.4

1 .S

1.6

1.7

1.8

~ntroduction and Plan of the Chapter 3

Funda~ental Concepts and Definitions 4

Hedging Forward Contracts Using Spot ~uantities 6

Hedging Options: Volatilities of Spot and Forward Processes 8

Definitions 14

A Series of Options on Futures Contracts 18

Hedging an Option with a Forward-Setting Strike l8

S~itching from the Real World to the Pricing Measure 22

2.1 ~ntroduction and Plan of the Chapter 29

2.2 Hedging a Plain-Vanilla Option in the Presence of Constant

2.3 Hedging a Plain-Vanilla Option in the Presence of Ti~e-~ependent

V~latilit~ 30

Volatility 34

2.3.3 First View 35

2.3.2 Second View 36

2.3.3 Third View 36

2.4 Hedging a Plain-Vanilla Option When the Real-World Process is

Mean Reverting 41

viii ~O~t~~tS

2.5 Hedging a Plain-Vanilla Option With Finite Re-Hedging

Intervals 44

stant~neo~s and Termina~ ~orre~ations

3.1 Introduction 5 1

3.2 The Stochastic Evolution of Imperfectly Correlated Variables

3.3 The Role of Terminal Correlation in the Joint Evolution of Stochastic

52

Variables 57

3.3.1 Case 1 : European Option, One Underlying Asset 58

3.3.2 Case 2: Path-Dependent Option, One Asset 61

3.3.3 Case 3: Path-Dependent Option, Two Assets

3.4 Generalising the Rsults 68

4.1 Introduction 73

4.2 Hedging With a Compensated Process: Plain-Vanilla and Binary

Options 74

4.3 Smile Tale 1: ‘Sticky’ Smiles 78

4.4 Smile Tale 2:‘Floating’ Smiles 80

4.5 Stylised

4.5.1

4.5.2

4.5.3

4.6.1

4.6.2

4.6.3

4.6.4

4.6 General

Empirical Facts About Smiles 83

Equities 83

Interest Rates 85

Foreign. Exchange Rates 87

Features of the Smile-Modelling Approaches 87

Fully Stochastic Volatility Models 88

Complete-M~kets Jump-Diffusion Models 89

Random~Amplitude Jump -Diffusion Models 90

Stochastic Volatility Functionally Dependent on the Under￾lying (Restricted-Stochasti~~Volatility) Models 9 l

4.7 Risk Derivatives for Plain-Vanilla Options in the Presence of

Smiles 93

ethodolo~ies for Smiley

5.1 Introduction 97

5.2 General Considerations on Stochastic”~olati1ity Models 97

5.3 The Dupire, Rubinstein and Deman and Kani Approaches 100

5.4 Green’s Function (Arrow-De~reu Prices) in the DK

Const~ction 101

5.5 The Peman and Kani Tree Construction 104

5.6 Numerical Aspects of the Implementation of the PK

5.7 Implementation Results 113

Construction 109

6.1 Introduction 129

6.2 The Computational Framework 130

6.3 Computational Results 135

6.4 The Link Between Implied and Local Volatility Surfaces 139

6.4.1 Sy~met~c (‘FX’) Smiles 140

6.4.2 Asymmetric (‘Equity’) Smile Surface 144

6.4.3 Monotonic (‘Interest-Rate’) Smile Surface 150

6.5 ~aining an Intuitive Understanding 153

6.6 No-Arbitrage Conditions on the Implied Volatility Smile

6.7 A ~or~ed-out Example: Pricing Continuous Pouble Barriers in the

6.8 Analysis of the Cost of Unwinding and Related Considerations

Surface 16 1

Presence of Smiles 174

About Option Pricing in the Presence of Smiles 182

Appendix 6.1: Proof that a2 Call($,, K, T, t)/aK2 = @(S~)llt: 186

7.1 ~ntroduction 189

7.2 Estimatin~ the Risk-Neutral Density Function 195

7.3 Perivation of Analytic Formulae 199

7.4 Results and Applications 206

7.5 on cl us ions and Range of Possible Applications 213

Appendix 7.1 Obtaining the Pensity of the Underlying from Quoted Option

Prices 214

8.1 Introduction 21 5

8.2 The Financial Model: Smile Tale 2Revisited 216

8.3 Analytic Pescription of Mixed Jump-Pi~usion Processes 220

8.4 A General Framework for Option Pricing in Complete or Incomplete

8.5 Finding the Optimal Hedge 235

Markets 229

X ~o~te~ts

8.6 Numerical Implementation of the Britten-Jones-~euberger

8.7 Computational Results 243

8.8 Discussion of the Results and Possible Developments 249

Methodology 236

9. 1

9-2

9.3

9.4

9.5

9.6

Introduction: Why Mean Reversion Matters in the Case of Interest￾Rate Models 253

The BDT Mean-Reversion Paradox 256

The ~nconditional Vlziance of the Short Rate in BDT-The

Discrete Case 259

The ~nconditional Vkriance of the Short Rate in BDT-The

~Ontinuous-Ti~e Equivalent 26 1

Mean Reversion in Short-Rate Lattices: The Equi-Probable Binomial

Versus the Bushy-Tree Approach 263

Extension to More General Interest-Rate Models: The ‘True’ Role

of Mean Reversion 267

Appendix 9.1 : Evaluation of the Variance of the Logarithm of the

Instantaneous Short Rate 269

10.1 Introduction and Statement of the Problem 271

10.2 Constructing the Most General BGM (Market) Model 273

10.3 A ~orked-Out Example: Caplets and a Two-Period

10.4 A ~or~ed-Out Example: Serial Options 280

10.5 Reducing the Di~ensionality of the BGM Model 281

10.6 Numerical Results 286

286

287

Swaption 278

10.6.1 Fitting the Correlation Surface with a Three-Factor Model

10.6.2 Fitting the Correlation Surface with a Four-Factor Model

10.7 Conclusions 298

11.1 The Link Between Instantaneous Volatility and the Future Term

l 1.2 A Functional Fom for the Instantaneous Volatility Function 306

Structure of Volatilities 303

xi

1 1.3 Fitting the Instantaneous Volatility Function: Imposing Time￾Homo~eneity of the Term S~ru~ture of Volatilities 3 1 1

1 l .4 Fitting the Instantaneous Volatility Function: Information from the

Swaption ~ar~~t 3 18

1 1 .S Conclusions 327

33

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Even a cursory visit to the financial section of a good bookshop in the City should

probably be more than enough to test the resolve and optimism of any would-be

author of a financial book: the flow of new titles in virtually every area of finance

can be described by few adjectives other than ‘unprecedented’ , ‘phenomenal’ ,

‘stagge~ng’ and the like. And if the prospective author had considered writing

a book in the option area, her optimism would verge on hubris: with so many

topics, angles and perspectives covered, can one hope to add anything new and

~eaningful? Yet, after several visits to my local bookshop I decided, over a year

ago, that a ‘different’ book on volatility and correl~tion still needed writing. How

could I justify such a claim?

In order to answer this question, let me begin by explaining what this book

does not attempt to be. First of all, the book does not cover the elementa~

aspects of option pricing: the reader is expected to have gathered, from any

of the excellent books available, a simple but clear understanding of the

(and Scholes) formula. The level of the ‘representative reader’ I had in mind

when sitting at my word processor was such that she could understand with

confidence the funda~ental ideas conveyed by a text such as, for instance, Hull’s.

Second, this is not a book on econometric estimation techniques: there are already

literally dozens of works in this area, ranging in quality from the poor to the

excellent; furthermore, statistical esti~ation techniques have never been my main

professional area of expertise and there would therefore be precious little, besides

all the sins I have conmitted in this field over the years, that I could bring to

the table.

Finally, this is not a book on stochastic calculus applied to finance: the topic is

fascinating and can be of great practical relevance, and I have indeed tinkered in

the recent past with the idea of trying to write something in this area; but a few

very good books that have recently appeared have convinced me that there was

very little I could have said differently, and that little certainly not any better.

Despite all of the above, I still think that a ‘different’ book on volatility

and correlation needs writing. The reasons for this conviction are manifold:

to begin with I believe that some fundamental concepts about correlation and