Financial Derivatives in Theory and Practice

by Hunt, Philip; Kennedy, Joanne

Edition: 1st

ISBN13: 9780470863589

ISBN10: 0470863587

Format: Hardcover

Pub. Date: 2004-07-02

Publisher(s): WILEY

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Summary

The term Financial Derivative is a very broad term which has come to mean any financial transaction whose value depends on the underlying value of the asset concerned. Sophisticated statistical modelling of derivatives enables practitioners in the banking industry to reduce financial risk and ultimately increase profits made from these transactions.The book originally published in March 2000 to widespread acclaim. This revised edition has been updated with minor corrections and new references, and now includes a chapter of exercises and solutions, enabling use as a course text. Comprehensive introduction to the theory and practice of financial derivatives. Discusses and elaborates on the theory of interest rate derivatives, an area of increasing interest. Divided into two self-contained parts – the first concentrating on the theory of stochastic calculus, and the second describes in detail the pricing of a number of different derivatives in practice. Written by well respected academics with experience in the banking industry.A valuable text for practitioners in research departments of all banking and finance sectors. Academic researchers and graduate students working in mathematical finance.

Author Biography

Philip Hunt is the author of Financial Derivatives in Theory and Practice, Revised Edition, published by Wiley.

Joanne Kennedy is the author of Financial Derivatives in Theory and Practice, Revised Edition, published by Wiley.

Preface to revised edition

Preface

xvii

Acknowledgements

xxi

Part I: Theory

(212)

Single-Period Option Pricing

(16)

Option pricing in a nutshell

(1)

The simplest setting

(1)

General one-period economy

(10)

Pricing

(1)

Conditions for no arbitrage: existence of Z

(2)

Completeness: uniqueness of Z

(3)

Probabilistic formulation

(3)

Units and numeraires

(1)

A two-period example

(4)

Brownian Motion

(12)

Introduction

(1)

Definition and existence

(1)

Basic properties of Brownian motion

(5)

Limit of a random walk

(2)

Deterministic transformations of Brownian motion

(1)

Some basic sample path properties

(2)

Strong Markov property

(5)

Reflection principle

(3)

Martingales

(32)

Definition and basic properties

(3)

Classes of martingales

(6)

Martingales bounded in L1

(1)

Uniformly integrable martingales

(3)

Square-integrable martingales

(2)

Stopping times and the optional sampling theorem

(8)

Stopping times

(4)

Optional sampling theorem

(4)

Variation, quadratic variation and integration

(7)

Total variation and Stieltjes integration

(2)

Quadratic variation

(4)

Quadratic covariation

(1)

Local martingales and semimartingales

(5)

The space cMloc

(3)

Semimartingales

(2)

Supermartingales and the Doob--Meyer decomposition

(2)

Stochastic Integration

(28)

Outline

(2)

Predictable processes

(2)

Stochastic integrals: the L2 theory

(7)

The simplest integral

(1)

The Hilbert space L2(M)

(1)

The L2 integral

(2)

Modes of convergence to H • M

(2)

Properties of the stochastic integral

(3)

Extensions via localization

(4)

Continuous local martingales as integrators

(1)

Semimartingales as integrators

(2)

The end of the road!

(1)

Stochastic calculus: Ito's formula

(10)

Integration by parts and Ito's formula

(2)

Differential notation

(2)

Multidimensional version of Ito's formula

(3)

Levy's theorem

(3)

Girsanov and Martingale Representation

(24)

Equivalent probability measures and the Randon--Nikodym derivative

(8)

Basic results and properties

(4)

Equivalent and locally equivalent measures on a filtered space

(2)

Novikov's condition

(2)

Girsanov's theorem

(6)

Girsanov's theorem for continuous semimartingales

(2)

Girsanov's theorem for Brownian motion

101

(4)

Martingale representation theorem

105

(10)

The space I2(M) and its orthogonal complement

106

(4)

Martingale measures and the martingale representation theorem

110

(1)

Extensions and the Brownian case

111

(4)

Stochastic Differential Equations

115

(26)

Introduction

115

(1)

Formal definition of an SDE

116

(1)

An aside on the canonical set-up

117

(2)

Weak and strong solutions

119

(6)

Weak solutions

119

(2)

Strong solutions

121

(3)

Tying together strong and weak

124

(1)

Establishing existence and uniqueness: Ito theory

125

(9)

Picard--Lindelof iteration and ODEs

126

(1)

A technical lemma

127

(3)

Existence and uniqueness for Lipschitz coefficients

130

(4)

Strong Markov property

134

(5)

Martingale representation revisited

139

(2)

Option Pricing in Continuous Time

141

(42)

Asset price processes and trading strategies

142

(4)

A model for asset prices

142

(2)

Self-financing trading strategies

144

(2)

Pricing European options

146

(5)

Option value as a solution to a PDE

147

(2)

Option pricing via an equivalent martingale measure

149

(2)

Continuous time theory

151

(25)

Information within the economy

152

(1)

Units, numeraires and martingale measures

153

(5)

Arbitrage and admissible strategies

158

(5)

Derivative pricing in an arbitrage-free economy

163

(1)

Completeness

164

(9)

Pricing kernels

173

(3)

Extensions

176

(7)

General payout schedules

176

(2)

Controlled derivative payouts

178

(1)

More general asset price processes

179

(1)

Infinite trading horizon

180

(3)

Dynamic Term Structure Models

183

(30)

Introduction

183

(1)

An economy of pure discount bonds

183

(4)

Modelling the term structure

187

(26)

Pure discount bond models

191

(1)

Pricing kernel approach

191

(1)

Numeraire models

192

(2)

Finite variation kernel models

194

(3)

Absolutely continuous (FVK) models

197

(1)

Short-rate models

197

(3)

Heath--Jarrow--Morton models

200

(6)

Flesaker--Hughston models

206

(7)

Part II: Practice

213

(46)

Modelling in Practice

215

(12)

Introduction

215

(1)

The real world is not a martingale measure

215

(3)

Modelling via infinitesimals

216

(1)

Modelling via macro information

217

(1)

Product-based modelling

218

(5)

A warning on dimension reduction

219

(2)

Limit cap valuation

221

(2)

Local versus global calibration

223

(4)

Basic Instruments and Terminology

227

(10)

Introduction

227

(1)

Deposits

227

(2)

Accrual factors and Libor

228

(1)

Forward rate agreements

229

(1)

Interest rate swaps

230

(2)

Zero coupon bonds

232

(1)

Discount factors and valuation

233

(4)

Discount factors

233

(1)

Deposit valuation

233

(1)

FRA valuation

234

(1)

Swap valuation

234

(3)

Pricing Standard Market Derivatives

237

(10)

Introduction

237

(1)

Forward rate agreements and swaps

237

(1)

Caps and floors

238

(4)

Valuation

240

(1)

Put-call parity

241

(1)

Vanilla swaptions

242

(2)

Digital options

244

(3)

Digital caps and floors

244

(1)

Digital swaptions

245

(2)

Futures Contracts

247

(12)

Introduction

247

(1)

Futures contract definition

247

(5)

Contract specification

248

(1)

Market risk without credit risk

249

(2)

Mathematical formulation

251

(1)

Characterizing the futures price process

252

(3)

Discrete resettlement

252

(1)

Continuous resettlement

253

(2)

Recovering the futures price process

255

(1)

Relationship between forwards and futures

256

(3)

Orientation: Pricing Exotic European Derivatives

259

(56)

Terminal Swap-Rate Models

263

(14)

Introduction

263

(1)

Terminal time modelling

263

(3)

Model requirements

263

(2)

Terminal swap-rate models

265

(1)

Example terminal swap-rate models

266

(3)

The exponential swap-rate model

266

(1)

The geometric swap-rate model

267

(1)

The linear swap-rate model

268

(1)

Arbitrage-free property of terminal swap-rate models

269

(4)

Existence of calibrating parameters

270

(1)

Extension of model to [0, ∞)

271

(2)

Arbitrage and the linear swap-rate model

273

(1)

Zero coupon swaptions

273

(4)

Convexity Corrections

277

(10)

Introduction

277

(1)

Valuation of `convexity-related' products

278

(4)

Affine decomposition of convexity products

278

(2)

Convexity corrections using the linear swap-rate model

280

(2)

Examples and extensions

282

(5)

Constant maturity swaps

283

(1)

Options on constant maturity swaps

284

(1)

Libor-in-arrears swaps

285

(2)

Implied Interest Rate Pricing Models

287

(16)

Introduction

287

(1)

Implying the functional form DTS

288

(4)

Numerical implementation

292

(1)

Irregular swaptions

293

(6)

Numerical comparison of exponential and implied swap-rate models

299

(4)

Multi-Currency Terminal Swap-Rate Models

303

(12)

Introduction

303

(1)

Model construction

304

(4)

Log-normal case

305

(2)

General case: volatility smiles

307

(1)

Examples

308

(7)

Spread options

308

(3)

Cross-currency swaptions

311

(4)

Orientation: Pricing Exotic American and Path-Dependent Derivatives

315

(102)

Short-Rate Models

319

(18)

Introduction

319

(1)

Well-known short-rate models

320

(5)

Vasicek--Hull--White model

320

(2)

Log-normal short-rate models

322

(1)

Cox--Ingersoll--Ross model

323

(1)

Multidimensional short-rate models

324

(1)

Parameter fitting within the Vasicek--Hull--White model

325

(4)

Derivation of φ, Ψ and B.T

326

(1)

Derivation of ξ, ζ and η

327

(1)

Derivation of μ, λ and A.T

328

(1)

Bermudan swaptions via Vasicek--Hull--White

329

(8)

Model calibration

330

(1)

Specifying the `tree'

330

(2)

Valuation through the tree

332

(1)

Evaluation of expected future value

332

(2)

Error analysis

334

(3)

Market Models

337

(14)

Introduction

337

(1)

Libor market models

338

(5)

Determining the drift

339

(2)

Existence of a consistent arbitrage-free term structure model

341

(2)

Example application

343

(1)

Regular swap-market models

343

(4)

Determining the drift

344

(2)

Existence of a consistent arbitrage-free term structure model

346

(1)

Example application

346

(1)

Reverse swap-market models

347

(4)

Determining the drift

348

(1)

Existence of a consistent arbitrage-free term structure model

349

(1)

Example application

350

(1)

Markov-Functional Modelling

351

(22)

Introduction

351

(1)

Markov-functional models

351

(3)

Fitting a one-dimensional Markov-functional model to swaption prices

354

(5)

Deriving the numeraire on a grid

355

(3)

Existence of a consistent arbitrage-free term structure model

358

(1)

Example models

359

(4)

Libor model

359

(2)

Swap model

361

(2)

Multidimensional Markov-functional models

363

(2)

Log-normally driven Markov-functional models

364

(1)

Relationship to market models

365

(2)

Mean reversion, forward volatilities and correlation

367

(3)

Mean reversion and correlation

367

(1)

Mean reversion and forward volatilities

368

(1)

Mean reversion within the Markov-functional Libor model

369

(1)

Some numerical results

370

(3)

Exercises and Solutions

373

(44)

Appendix 1 The Usual Conditions

417

(2)

Appendix 2 L2 Spaces

419

(2)

Appendix 3 Gaussian Calculations

421

(2)

References

423

(4)

Index

427

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