Volatility and Correlation The Perfect Hedger and the Fox

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ISBN: 9780470091395 | 0470091398
Cover: Hardcover
Copyright: 9/3/2004

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The new edition of Volatility and Correlation has been thoroughly updated and expanded with over 80% new or reworked material, reflecting the changes and developments that have taken place in the field. The new and updated material includes: empirical and theoretical analysis of the smile dynamics; examination of the perfect-replication model in relation to exotic options; treatment of additional important models, namely, Variance Gamma, displaced diffusion, CEV, stochastic volatility for interest-rate smiles and equity/FX options; questioning of the informational efficiency of markets in commonly-used calibration and hedging practices.

Preface

xxi

0.1 Why a Second Edition?

xxi

0.2 What This Book Is Not About

xxiii

0.3 Structure of the Book

xxiv

0.4 The New Subtitle

xxiv

Acknowledgements

xxvii

I Foundations

(164)

1 Theory and Practice of Option Modelling

(28)

1.1 The Role of Models in Derivatives Pricing

(6)

1.1.1 What Are Models For?

(2)

1.1.2 The Fundamental Approach

(2)

1.1.3 The Instrumental Approach

(1)

1.1.4 A Conundrum (or, 'What is Vega Hedging For?')

(1)

1.2 The Efficient Market Hypothesis and Why It Matters for Option Pricing

(5)

1.2.1 The Three Forms of the EMH

(1)

1.2.2 Pseudo-Arbitrageurs in Crisis

(1)

1.2.3 Model Risk for Traders and Risk Managers

(1)

1.2.4 The Parable of the Two Volatility Traders

(2)

1.3 Market Practice

(3)

1.3.1 Different Users of Derivatives Models

(1)

1.3.2 In-Model and Out-of-Model Hedging

(2)

1.4 The Calibration Debate

(10)

1.4.1 Historical vs Implied Calibration

(1)

1.4.2 The Logical Underpinning of the Implied Approach

(2)

1.4.3 Are Derivatives Markets Informationally Efficient?

(5)

1.4.4 Back to Calibration

(1)

1.4.5 A Practical Recommendation

(1)

1.5 Across-Markets Comparison of Pricing and Modelling Practices

(3)

1.6 Using Models

(1)

2 Option Replication

(44)

2.1 The Bedrock of Option Pricing

(1)

2.2 The Analytic (PDE) Approach

(6)

2.2.1 The Assumptions

(1)

2.2.2 The Portfolio-Replication Argument (Deterministic Volatility)

(2)

2.2.3 The Market Price of Risk with Deterministic Volatility

(2)

2.2.4 Link with Expectations - the Feynman-Kac Theorem

(2)

2.3 Binomial Replication

(1)

2.3.1 First Approach - Replication Strategy

(2)

2.3.2 Second Approach - 'Naive Expectation'

(1)

2.3.3 Third Approach - 'Market Price of Risk'

(3)

2.3.4 A Worked-Out Example

(1)

2.3.5 Fourth Approach Risk-Neutral Valuation

(2)

2.3.6 Pseudo-Probabilities

(1)

2.3.7 Are the Quantities π1 andπ2 Really Probabilities?

(2)

2.3.8 Introducing Relative Prices

(2)

2.3.9 Moving to a Multi-Period Setting

(3)

2.3.10 Fair Prices as Expectations

(2)

2.3.11 Switching Numeraires and Relating Expectations Under Different Measures

(3)

2.3.12 Another Worked-Out Example

(3)

2.3.13 Relevance of the Results

(1)

2.4 Justifying the Two-State Branching Procedure

(4)

2.4.1 How To Recognize a Jump When You See One

(4)

2.5 The Nature of the Transformation between Measures: Girsanov's Theorem

(4)

2.5.1 An Intuitive Argument

(1)

2.5.2 A Worked-Out Example

(3)

2.6 Switching Between the PDE, the Expectation and the Binomial Replication Approaches

(2)

3 The Building Blocks

(26)

3.1 Introduction and Plan of the Chapter

(1)

3.2 Definition of Market Terms

(2)

3.3 Hedging Forward Contracts Using Spot Quantities

(3)

3.3.1 Hedging Equity Forward Contracts

(1)

3.3.2 Hedging Interest-Rate Forward Contracts

(1)

3.4 Hedging Options: Volatility of Spot and Forward Processes

(4)

3.5 The Link Between Root-Mean-Squared Volatilities and the Time-Dependence of Volatility

(1)

3.6 Admissibility of a Series of Root-Mean-Squared Volatilities

(2)

3.6.1 The Equity/FX Case

(1)

3.6.2 The Interest-Rate Case

(1)

3.7 Summary of the Definitions So Far

(2)

3.8 Hedging an Option with a Forward-Setting Strike

(6)

3.8.1 Why Is This Option Important? (And Why Is it Difficult to Hedge?)

(1)

3.8.2 Valuing a Forward-Setting Option

(4)

3.9 Quadratic Variation: First Approach

(6)

3.9.1 Definition

(1)

3.9.2 Properties of Variations

(1)

3.9.3 First and Second Variation of a Brownian Process

(1)

3.9.4 Links between Quadratic Variation and ƒT/t σ (u)²du

(1)

3.9.5 Why Quadratic Variation Is So Important (Take 1)

(3)

4 Variance and Mean Reversion in the Real and the Risk-Adjusted Worlds

101

(40)

4.1 Introduction and Plan of the Chapter

101

(1)

4.2 Hedging a Plain-Vanilla Option: General Framework

102

(4)

4.2.1 Trading Restrictions and Model Uncertainty: Theoretical Results

103

(1)

4.2.2 The Setting

104

(1)

4.2.3 The Methodology

104

(2)

4.2.4 Criterion for Success

106

(1)

4.3 Hedging Plain-Vanilla Options: Constant Volatility

106

(10)

4.3.1 Trading the Gamma: One Step and Constant Volatility

108

(6)

4.3.2 Trading the Gamma: Several Steps and Constant Volatility

114

(2)

4.4 Hedging Plain-Vanilla Options: Time-Dependent Volatility

116

(5)

4.4.1 Views on Gamma Trading When the Volatility is Time Dependent

116

(3)

4.4.2 Which View Is the Correct One? (and the Feynman-Kac Theorem Again)

119

(2)

4.5 Hedging Behaviour In Practice

121

(6)

4.5.1 Analysing the Replicating Portfolio

121

(1)

4.5.2 Hedging Results: the Time-Dependent Volatility Case

122

(3)

4.5.3 Hedging with the Wrong Volatility

125

(2)

4.6 Robustness of the Black-and-Scholes Model

127

(3)

4.7 Is the Total Variance All That Matters?

130

(1)

4.8 Hedging Plain-Vanilla Options: Mean-Reverting Real-World Drift

131

(4)

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

135

(6)

4.9.1 The Crouhy-Galai Set-Up

135

(6)

5 Instantaneous and Terminal Correlation

141

(26)

5.1 Correlation, Co-Integration and Multi-Factor Models

141

(5)

5.1.1 The Multi-Factor Debate

144

(2)

5.2 The Stochastic Evolution of Imperfectly Correlated Variables

146

(5)

5.3 The Role of Terminal Correlation in the Joint Evolution of Stochastic Variables

151

(11)

5.3.1 Defining Stochastic Integrals

151

(2)

5.3.2 Case 1: European Option, One Underlying Asset

153

(2)

5.3.3 Case 2: Path-Dependent Option, One Asset

155

(1)

5.3.4 Case 3: Path-Dependent Option, Two Assets

156

(6)

5.4 Generalizing the Results

162

(2)

5.5 Moving Ahead

164

(1)

II Smiles - Equity and FX

165

(436)

6 Pricing Options in the Presence of Smiles

167

(34)

6.1 Plan of the Chapter

167

(1)

6.2 Background and Definition of the Smile

168

(1)

6.3 Hedging with a Compensated Process: Plain-Vanilla and Binary Options

169

(4)

6.3.1 Delta- and Vega-Hedging a Plain-Vanilla Option

169

(3)

6.3.2 Pricing a European Digital Option

172

(1)

6.4 Hedge Ratios for Plain-Vanilla Options in the Presence of Smiles

173

(7)

6.4.1 The Relationship Between the True Call Price Functional and the Black Formula

174

(1)

6.4.2 Calculating the Delta Using the Black Formula and the Implied Volatility

175

(1)

6.4.3 Dependence of Implied Volatilities on the Strike and the Underlying

176

(2)

6.4.4 Floating and Sticky Smiles and What They Imply about Changes in Option Prices

178

(2)

6.5 Smile Tale 1: 'Sticky' Smiles

180

(2)

6.6 Smile Tale 2: 'Floating' Smiles

182

(2)

6.6.1 Relevance of the Smile Story for Floating Smiles

183

(1)

6.7 When Does Risk Aversion Make a Difference?

184

(17)

6.7.1 Motivation

184

(1)

6.7.2 The Importance of an Assessment of Risk Aversion for Model Building

185

(1)

6.7.3 The Principle of Absolute Continuity

186

(1)

6.7.4 The Effect of Supply and Demand

187

(1)

6.7.5 A Stylized Example: First Version

187

(7)

6.7.6 A Stylized Example: Second Version

194

(2)

6.7.7 A Stylized Example: Third Version

196

(1)

6.7.8 Overall Conclusions

196

(3)

6.7.9 The EMH Again

199

(2)

7 Empirical Facts About Smiles

201

(36)

7.1 What is this Chapter About?

201

(2)

7.1.1 'Fundamental' and 'Derived' Analyses

201

(1)

7.1.2 A Methodological Caveat

202

(1)

7.2 Market Information About Smiles

203

(3)

7.2.1 Direct Static Information

203

(1)

7.2.2 Semi-Static Information

204

(1)

7.2.3 Direct Dynamic Information

204

(1)

7.2.4 Indirect Information

205

(1)

7.3 Equities

206

(16)

7.3.1 Basic Facts

206

(1)

7.3.2 Subtler Effects

206

(16)

7.4 Interest Rates

222

(5)

7.4.1 Basic Facts

222

(2)

7.4.2 Subtler Effects

224

(3)

7.5 FX Rates

227

(8)

7.5.1 Basic Facts

227

(1)

7.5.2 Subtler Effects

227

(8)

7.6 Conclusions

235

(2)

8 General Features of Smile-Modelling Approaches

237

(12)

8.1 Fully-Stochastic-Volatility Models

237

(2)

8.2 Local-Volatility (Restricted-Stochastic-Volatility) Models

239

(2)

8.3 Jump-Diffusion Models

241

(2)

8.3.1 Discrete Amplitude

241

(1)

8.3.2 Continuum of Jump Amplitudes

242

(1)

8.4 Variance-Gamma Models

243

(1)

8.5 Mixing Processes

243

(2)

8.5.1 A Pragmatic Approach to Mixing Models

244

(1)

8.6 Other Approaches

245

(1)

8.6.1 Tight Bounds with Known Quadratic Variation

245

(1)

8.6.2 Assigning Directly the Evolution of the Smile Surface

246

(1)

8.7 The Importance of the Quadratic Variation (Take 2)

246

(3)

9 The Input Data: Fitting an Exogenous Smile Surface

249

(44)

9.1 What is This Chapter About?

249

(1)

9.2 Analytic Expressions for Calls vs Process Specification

249

(1)

9.3 Direct Use of Market Prices: Pros and Cons

250

(1)

9.4 Statement of the Problem

251

(1)

9.5 Fitting Prices

252

(2)

9.6 Fitting Transformed Prices

254

(1)

9.7 Fitting the Implied Volatilities

255

(1)

9.7.1 The Problem with Fitting the Implied Volatilities

255

(1)

9.8 Fitting the Risk-Neutral Density Function - General

256

(3)

9.8.1 Does It Matter if the Price Density Is Not Smooth?

257

(1)

9.8.2 Using Prior Information (Minimum Entropy)

258

(1)

9.9 Fitting the Risk-Neutral Density Function: Mixture of Normals

259

(6)

9.9.1 Ensuring the Normalization and Forward Constraints

261

(3)

9.9.2 The Fitting Procedure

264

(1)

9.10 Numerical Results

265

(10)

9.10.1 Description of the Numerical Tests

265

(1)

9.10.2 Fitting to Theoretical Prices: Stochastic-Volatility Density

265

(3)

9.10.3 Fitting to Theoretical Prices: Variance-Gamma Density

268

(2)

9.10.4 Fitting to Theoretical Prices: Jump-Diffusion Density

270

(2)

9.10.5 Fitting to Market Prices

272

(3)

9.11 Is the Term ac/as Really a Delta?

275

(2)

9.12 Fitting the Risk-Neutral Density Function: The Generalized-Beta Approach

277

(16)

9.12.1 Derivation of Analytic Formulae

280

(7)

9.12.2 Results and Applications

287

(4)

9.12.3 What Does This Approach Offer?

291

(2)

10 Quadratic Variation and Smiles

293

(26)

10.1 Why This Approach Is Interesting

293

(1)

10.2 The BJN Framework for Bounding Option Prices

293

(1)

10.3 The BJN Approach - Theoretical Development

294

(6)

10.3.1 Assumptions and Definitions

294

(3)

10.3.2 Establishing Bounds

297

(1)

10.3.3 Recasting the Problem

298

(1)

10.3.4 Finding the Optimal Hedge

299

(1)

10.4 The BJN Approach: Numerical Implementation

300

(12)

10.4.1 Building a 'Traditional' Tree

301

(1)

10.4.2 Building a BJN Tree for a Deterministic Diffusion

301

(3)

10.4.3 Building a BJN Tree for a General Process

304

(3)

10.4.4 Computational Results

307

(2)

10.4.5 Creating Asymmetric Smiles

309

(2)

10.4.6 Summary of the Results

311

(1)

10.5 Discussion of the Results

312

(4)

10.5.1 Resolution of the Crouhy-Galai Paradox

312

(1)

10.5.2 The Difference Between Diffusions and Jump-Diffusion Processes: the Sample Quadratic Variation

312

(2)

10.5.3 How Can One Make the Approach More Realistic?

314

(1)

10.5.4 The Link with Stochastic-Volatility Models

314

(1)

10.5.5 The Link with Local-Volatility Models

315

(1)

10.5.6 The Link with Jump-Diffusion Models

315

(1)

10.6 Conclusions (or, Limitations of Quadratic Variation)

316

(3)

11 Local-Volatility Models: the Derman-and-Kani Approach

319

(26)

11.1 General Considerations on Stochastic-Volatility Models

319

(2)

11.2 Special Cases of Restricted-Stochastic-Volatility Models

321

(1)

11.3 The Dupire, Rubinstein and Derman-and-Kani Approaches

321

(1)

11.4 Green's Functions (Arrow-Debreu Prices) in the DK Construction

322

(4)

11.4.1 Definition and Main Properties of Arrow-Debreu Prices

322

(2)

11.4.2 Efficient Computation of Arrow-Debreu Prices

324

(2)

11.5 The Derman-and-Kani Tree Construction

326

(5)

11.5.1 Building the First Step

327

(2)

11.5.2 Adding Further Steps

329

(2)

11.6 Numerical Aspects of the Implementation of the DK Construction

331

(3)

11.6.1 Problem 1: Forward Price Greater Than 1/2 S(up) or Smaller Than S(down)

331

(1)

11.6.2 Problem 2: Local Volatility Greater Than 1/2 S(up) - S(down)

332

(1)

11.6.3 Problem 3: Arbitrariness of the Choice of the Strike

332

(2)

11.7 Implementation Results

334

(9)

11.7.1 Benchmarking 1: The No-Smile Case

334

(1)

11.7.2 Benchmarking 2: The Time-Dependent-Volatility Case

335

(1)

11.7.3 Benchmarking 3: Purely Strike-Dependent Implied Volatility

336

(1)

11.7.4 Benchmarking 4: Strike-and-Maturity-Dependent Implied Volatility

337

(1)

11.7.5 Conclusions

338

(5)

11.8 Estimating Instantaneous Volatilities from Prices as an Inverse Problem

343

(2)

12 Extracting the Local Volatility from Option Prices

345

(44)

12.1 Introduction

345

(2)

12.1.1 A Possible Regularization Strategy

346

(1)

12.1.2 Shortcomings

346

(1)

12.2 The Modelling Framework

347

(2)

12.3 A Computational Method

349

(6)

12.3.1 Backward Induction

349

(1)

12.3.2 Forward Equations

350

(2)

12.3.3 Why Are We Doing Things This Way?

352

(2)

12.3.4 Related Approaches

354

(1)

12.4 Computational Results

355

(2)

12.4.1 Are We Looking at the Same Problem?

356

(1)

12.5 The Link Between Implied and Local-Volatility Surfaces

357

(11)

12.5.1 Symmetric ('FX') Smiles

358

(3)

12.5.2 Asymmetric ('Equity') Smiles

361

(7)

12.5.3 Monotonic (Interest-Rate') Smile Surface

368

(1)

12.6 Gaining an Intuitive Understanding

368

(5)

12.6.1 Symmetric Smiles

369

(1)

12.6.2 Asymmetric Smiles: One-Sided Parabola

370

(2)

12.6.3 Asymmetric Smiles: Monotonically Decaying

372

(1)

12.7 What Local-Volatility Models Imply about Sticky and Floating Smiles

373

(2)

12.8 No-Arbitrage Conditions on the Current Implied Volatility Smile Surface

375

(10)

12.8.1 Constraints on the Implied Volatility Surface

375

(6)

12.8.2 Consequences for Local Volatilities

381

(4)

12.9 Empirical Performance

385

(1)

12.10 Appendix 1: Proof that a²call (St,K,T,t)/Ak² = φ(ST)|K

386

(3)

13 Stochastic-Volatility Processes

389

(50)

13.1 Plan of the Chapter

389

(1)

13.2 Portfolio Replication in the Presence of Stochastic Volatility

389

(12)

13.2.1 Attempting to Extend the Portfolio Replication Argument

389

(7)

13.2.2 The Market Price of Volatility Risk

396

(2)

13.2.3 Assessing the Financial Plausibility of λσ

398

(3)

13.3 Mean-Reverting Stochastic Volatility

401

(4)

13.3.1 The Ornstein-Uhlenbeck Process

402

(1)

13.3.2 The Functional Form Chosen in This Chapter

403

(1)

13.3.3 The High-Reversion-Speed, High-Volatility Regime

404

(1)

13.4 Qualitative Features of Stochastic-Volatility Smiles

405

(11)

13.4.1 The Smile as a Function of the Risk-Neutral Parameters

406

(10)

13.5 The Relation Between Future Smiles and Future Stock Price Levels

416

(2)

13.5.1 An Intuitive Explanation

417

(1)

13.6 Portfolio Replication in Practice: The Stochastic-Volatility Case

418

(9)

13.6.1 The Hedging Methodology

418

(2)

13.6.2 A Numerical Example

420

(7)

13.7 Actual Fitting to Market Data

427

(9)

13.8 Conclusions

436

(3)

14 Jump-Diffusion Processes

439

(72)

14.1 Introduction

439

(2)

14.2 The Financial Model: Smile Tale 2 Revisited

441

(3)

14.3 Hedging and Replicability in the Presence of Jumps: First Considerations

444

(5)

14.3.1 What Is Really Required To Complete the Market?

445

(4)

14.4 Analytic Description of Jump-Diffusions

449

(6)

14.4.1 The Stock Price Dynamics

449

(6)

14.5 Hedging with Jump-Diffusion Processes

455

(15)

14.5.1 Hedging with a Bond and the Underlying Only

455

(2)

14.5.2 Hedging with a Bond, a Second Option and the Underlying

457

(3)

14.5.3 The Case of a Single Possible Jump Amplitude

460

(5)

14.5.4 Moving to a Continuum of Jump Amplitudes

465

(1)

14.5.5 Determining the Function g Using the Implied Approach

465

(5)

14.5.6 Comparison with the Stochastic-Volatility Case (Again)

470

(1)

14.6 The Pricing Formula for Log-Normal Amplitude Ratios

470

(2)

14.7 The Pricing Formula in the Finite-Amplitude-Ratio Case

472

(13)

14.7.1 The Structure of the Pricing Formula for Discrete Jump Amplitude Ratios

474

(1)

14.7.2 Matching the Moments

475

(1)

14.7.3 Numerical Results

476

(9)

14.8 The Link Between the Price Density and the Smile Shape

485

(9)

14.8.1 A Qualitative Explanation

491

(3)

14.9 Qualitative Features of Jump-Diffusion Smiles

494

(6)

14.9.1 The Smile as a Function of the Risk-Neutral Parameters

494

(5)

14.9.2 Comparison with Stochastic-Volatility Smiles

499

(1)

14.10 Jump-Diffusion Processes and Market Completeness Revisited

500

(2)

14.11 Portfolio Replication in Practice: The Jump-Diffusion Case

502

(9)

14.11.1 A Numerical Example

503

(1)

14.11.2 Results

504

(5)

14.11.3 Conclusions

509

(2)

15 Variance-Gamma

511

(18)

15.1 Who Can Make Best Use of the Variance-Gamma Approach?

511

(2)

15.2 The Variance-Gamma Process

513

(9)

15.2.1 Definition

513

(1)

15.2.2 Properties of the Gamma Process

514

(1)

15.2.3 Properties of the Variance-Gamma Process

514

(3)

15.2.4 Motivation for Variance-Gamma Modelling

517

(1)

15.2.5 Properties of the Stock Process

518

(1)

15.2.6 Option Pricing

519

(3)

15.3 Statistical Properties of the Price Distribution

522

(1)

15.3.1 The Real-World (Statistical) Distribution

522

(1)

15.3.2 The Risk-Neutral Distribution

522

(1)

15.4 Features of the Smile

523

(4)

15.5 Conclusions

527

(2)

16 Displaced Diffusions and Generalizations

529

(34)

16.1 Introduction

529

(1)

16.2 Gaining Intuition

530

(1)

16.2.1 First Formulation

530

(1)

16.2.2 Second Formulation

531

(1)

16.3 Evolving the Underlying with Displaced Diffusions

531

(1)

16.4 Option Prices with Displaced Diffusions

532

(1)

16.5 Matching At-The-Money Prices with Displaced Diffusions

533

(20)

16.5.1 A First Approximation

533

(1)

16.5.2 Numerical Results with the Simple Approximation

534

(1)

16.5.3 Refining the Approximation

534

(10)

16.5.4 Numerical Results with the Refined Approximation

544

(9)

16.6 The Smile Produced by Displaced Diffusions

553

(7)

16.6.1 How Quickly is the Normal-Diffusion Limit Approached?

553

(7)

16.7 Extension to Other Processes

560

(3)

17 No-Arbitrage Restrictions on the Dynamics of Smile Surfaces

563

(38)

17.1 A Worked-Out Example: Pricing Continuous Double Barriers

564

(7)

17.1.1 Money For Nothing: A Degenerate Hedging Strategy for a Call Option

564

(2)

17.1.2 Static Replication of a Continuous Double Barrier

566

(5)

17.2 Analysis of the Cost of Unwinding

571

(4)

17.3 The Trader's Dream

575

(6)

17.4 Plan of the Remainder of the Chapter

581

(1)

17.5 Conditions of No-Arbitrage for the Stochastic Evolution of Future Smile Surfaces

582

(3)

17.5.1 Description of the Market

582

(2)

17.5.2 The Building Blocks

584

(1)

17.6 Deterministic Smile Surfaces

585

(8)

17.6.1 Equivalent Descriptions of a State of the World

585

(2)

17.6.2 Consequences of Deterministic Smile Surfaces

587

(1)

17.6.3 Kolmogorov-Compatible Deterministic Smile Surfaces

588

(1)

17.6.4 Conditions for the Uniqueness of Kolmogorov-Compatible Densities

589

(2)

17.6.5 Floating Smiles

591

(2)

17.7 Stochastic Smiles

593

(4)

17.7.1 Stochastic Floating Smiles

594

(1)

17.7.2 Introducing Equivalent Deterministic Smile Surfaces

595

(1)

17.7.3 Implications of the Existence of an Equivalent Deterministic Smile Surface

596

(1)

17.7.4 Extension to Displaced Diffusions

597

(1)

17.8 The Strength of the Assumptions

597

(1)

17.9 Limitations and Conclusions

598

(3)

III Interest Rates - Deterministic Volatilities

601

(100)

18 Mean Reversion in Interest-Rate Models

603

(22)

18.1 Introduction and Plan of the Chapter

603

(1)

18.2 Why Mean Reversion Matters in the Case of Interest-Rate Models

604

(4)

18.2.1 What Does This Mean for Forward-Rate Volatilities?

606

(2)

18.3 A Common Fallacy Regarding Mean Reversion

608

(2)

18.3.1 The Grain of Truth in the Fallacy

609

(1)

18.4 The BDT Mean-Reversion Paradox

610

(2)

18.5 The Unconditional Variance of the Short Rate in BDT - the Discrete Case

612

(4)

18.6 The Unconditional Variance of the Short Rate in BDT-the Continuous-Time Equivalent

616

(1)

18.7 Mean Reversion in Short-Rate Lattices: Recombining vs Bushy Trees

617

(3)

18.8 Extension to More General Interest-Rate Models

620

(2)

18.9 Appendix I: Evaluation of the Variance of the Logarithm of the Instantaneous Short Rate

622

(3)

19 Volatility and Correlation in the LIBOR Market Model

625

(14)

19.1 Introduction

625

(1)

19.2 Specifying the Forward-Rate Dynamics in the LIBOR Market Model

626

(5)

19.2.1 First Formulation: Each Forward Rate in Isolation

626

(2)

19.2.2 Second Formulation: The Covariance Matrix

628

(2)

19.2.3 Third Formulation: Separating the Correlation from the Volatility Term

630

(1)

19.3 Link with the Principal Component Analysis

631

(1)

19.4 Worked-Out Example 1: Caplets and a Two-Period Swaption

632

(3)

19.5 Worked-Out Example 2: Serial Options

635

(1)

19.6 Plan of the Work Ahead

636

(3)

20 Calibration Strategies for the LIBOR Market Model

639

(28)

20.1 Plan of the Chapter

639

(1)

20.2 The Setting

639

(4)

20.2.1 A Geometric Construction: The Two-Factor Case

640

(2)

20.2.2 Generalization to Many Factors

642

(1)

20.2.3 Re-Introducing the Covariance Matrix

642

(1)

20.3 Fitting an Exogenous Correlation Function

643

(3)

20.4 Numerical Results

646

(13)

20.4.1 Fitting the Correlation Surface with a Three-Factor Model

646

(4)

20.4.2 Fitting the Correlation Surface with a Four-Factor Model

650

(4)

20.4.3 Fitting Portions of the Target Correlation Matrix

654

(5)

20.5 Analytic Expressions to Link Swaption and Caplet Volatilities

659

(3)

20.5.1 What Are We Trying to Achieve?

659

(1)

20.5.2 The Set-Up

659

(3)

20.6 Optimal Calibration to Co-Terminal Swaptions

662

(5)

20.6.1 The Strategy

662

(5)

21 Specifying the Instantaneous Volatility of Forward Rates

667

(20)

21.1 Introduction and Motivation

667

(1)

21.2 The Link between Instantaneous Volatilities and the Future Term Structure of Volatilities

668

(3)

21.3 A Functional Form for the Instantaneous Volatility Function

671

(2)

21.3.1 Financial Justification for a Humped Volatility

672

(1)

21.4 Ensuring Correct Caplet Pricing

673

(4)

21.5 Fitting the Instantaneous Volatility Function: Imposing Time Homogeneity of the Term Structure of Volatilities

677

(2)

21.6 Is a Time-Homogeneous Solution Always Possible?

679

(1)

21.7 Fitting the Instantaneous Volatility Function: The Information from the Swaption Market

680

(6)

21.8 Conclusions

686

(1)

22 Specifying the Instantaneous Correlation Among Forward Rates

687

(14)

22.1 Why Is Estimating Correlation So Difficult?

687

(1)

22.2 What Shape Should We Expect for the Correlation Surface'?

688

(1)

22.3 Features of the Simple Exponential Correlation Function

689

(2)

22.4 Features of the Modified Exponential Correlation Function

691

(3)

22.5 Features of the Square-Root Exponential Correlation Function

694

(3)

22.6 Further Comparisons of Correlation Models

697

(1)

22.7 Features of the Schonmakers-Coffey Approach

697

(1)

22.8 Does It Make a Difference (and When)?

698

(3)

IV Interest Rates - Smiles

701

(104)

23 How to Model Interest-Rate Smiles

703

(26)

23.1 What Do We Want to Capture? A Hierarchy of Smile-Producing Mechanisms

703

(1)

23.2 Are Log-Normal Co-Ordinates the Most Appropriate?

704

(2)

23.2.1 Defining Appropriate Co-ordinates

705

(1)

23.3 Description of the Market Data

706

(9)

23.4 Empirical Study I: Transforming the Log-Normal Co-ordinates

715

(3)

23.5 The Computational Experiments

718

(1)

23.6 The Computational Results

719

(2)

23.7 Empirical Study II: The Log-Linear Exponent

721

(4)

23.8 Combining the Theoretical and Experimental Results

725

(1)

23.9 Where Do We Go From Here?

725

(4)

24 (CEV) Processes in the Context of the LMM

729

(22)

24.1 Introduction and Financial Motivation

729

(1)

24.2 Analytical Characterization of CEV Processes

730

(2)

24.3 Financial Desirability of CEV Processes

732

(2)

24.4 Numerical Problems with CEV Processes

734

(1)

24.5 Approximate Numerical Solutions

735

(12)

24.5.1 Approximate Solutions: Mapping to Displaced Diffusions

735

(1)

24.5.2 Approximate Solutions: Transformation of Variables

735

(1)

24.5.3 Approximate Solutions: the Predictor-Corrector Method

736

(11)

24.6 Problems with the Predictor-Corrector Approximation for the LMM

747

(4)

25 Stochastic-Volatility Extensions of the LMM

751

(14)

25.1 Plan of the Chapter

751

(2)

25.2 What is the Dog and What is the Tail?

753

(1)

25.3 Displaced Diffusion vs CEV

754

(1)

25.4 The Approach

754

(2)

25.5 Implementing and Calibrating the Stochastic-Volatility LMM

756

(8)

25.5.1 Evolving the Forward Rates

759

(1)

25.5.2 Calibrating to Caplet Prices

759

(5)

25.6 Suggestions and Plan of the Work Ahead

764

(1)

26 The Dynamics of the Swaption Matrix

765

(18)

26.1 Plan of the Chapter

765

(1)

26.2 Assessing the Quality of a Model

766

(1)

26.3 The Empirical Analysis

767

(1)

26.3.1 Description of the Data

767

(1)

26.3.2 Results

768

(8)

26.4 Extracting the Model-Implied Principal Components

776

(5)

26.4.1 Results

778

(3)

26.5 Discussion, Conclusions and Suggestions for Future Work

781

(2)

27 Stochastic-Volatility Extension of the LMM: Two-Regime Instantaneous Volatility

783

(22)

27.1 The Relevance of the Proposed Approach

783

(1)

27.2 The Proposed Extension

783

(2)

27.3 An Aside: Some Simple Properties of Markov Chains

785

(3)

27.3.1 The Case of Two-State Markov Chains

787

(1)

27.4 Empirical Tests

788

(10)

27.4.1 Description of the Test Methodology

788

(2)

27.4.2 Results

790

(8)

27.5 How Important Is the Two-Regime Feature?

798

(3)

27.6 Conclusions

801

(4)

Bibliography

805

(8)

Index

813

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Volatility and Correlation The Perfect Hedger and the Fox

Summary

Author Biography

Table of Contents

Supplemental Materials