The Mathematics of Financial Derivatives: A

Finance is one of the fastest growing areas in the modern banking and corporate world. This, together with the sophistication of modern financial products, provides a rapidly growing impetus for new mathematical models and modern mathematical methods; the area is an expanding source for novel and relevant 'real-world' mathematics. In this book the authors describe the modelling of financial derivative products from an applied mathematician's viewpoint, from modelling through analysis to elementary computation. A unified approach to modelling derivative products as partial differential equations is presented, using numerical solutions where appropriate. Some mathematics is assumed, but clear explanations are provided for material beyond elementary calculus, probability, and algebra. Over 140 exercises are included. This volume will become the standard introduction to this exciting new field for advanced undergraduate students.

Preface

ix

Part One: Basic Option Theory

1

(132)

1 An Introduction to Options and Markets

3

(15)

1.1 Introduction

3

(1)

1.2 What is an Option?

4

(3)

1.3 Reading the Financial Press

7

(4)

1.4 What are Options For?

11

(2)

1.5 Other Types of Option

13

(1)

1.6 Forward and Futures Contracts

14

(1)

1.7 Interest Rates and Present Value

15

(3)

2 Asset Price Random Walks

18

(15)

2.1 Introduction

18

(1)

2.2 A Simple Model for Asset Prices

19

(6)

2.3 Ito's Lemma

25

(5)

2.4 The Elimination of Randomness

30

(3)

3 The Black-Scholes Model

33

(25)

3.1 Introduction

33

(1)

3.2 Arbitrage

33

(2)

3.3 Option Values, Payoffs and Strategies

35

(5)

3.4 Put-call Parity

40

(1)

3.5 The Black-Scholes Analysis

41

(3)

3.6 The Black-Scholes Equation

44

(2)

3.7 Boundary and Final Conditions

46

(2)

3.8 The Black-Scholes Formulae

48

(3)

3.9 Hedging in Practice

51

(1)

3.10 Implied Volatility

52

(6)

4 Partial Differential Equations

58

(13)

4.1 Introduction

58

(1)

4.2 The Diffusion Equation

59

(7)

4.3 Initial and Boundary Conditions

66

(2)

4.4 Forward versus Backward

68

(3)

5 The Black-Scholes Formulae

71

(19)

5.1 Introduction

71

(1)

5.2 Similarity Solutions

71

(4)

5.3 An Initial Value Problem

75

(1)

5.4 The Formulae Derived

76

(5)

5.5 Binary Options

81

(2)

5.6 Risk Neutrality

83

(7)

6 Variations on the Black-Scholes Model

90

(16)

6.1 Introduction

90

(1)

6.2 Options on Dividend-paying Assets

90

(8)

6.3 Forward and Futures Contracts

98

(2)

6.4 Options on Futures

100

(1)

6.5 Time-dependent Parameters

101

(5)

7 American Options

106

(27)

7.1 Introduction

106

(2)

7.2 The Obstacle Problem

108

(1)

7.3 American Options as Free Boundary Problems

109

(1)

7.4 The American Put

110

(4)

7.5 Other American Options

114

(1)

7.6 Linear Complementarity Problems

115

(6)

7.7 The American Call with Dividends

121

(12)

Part Two: Numerical Methods

133

(62)

8 Finite-difference Methods

135

(30)

8.1 Introduction

135

(1)

8.2 Finite-difference Approximations

136

(2)

8.3 The Finite-difference Mesh

138

(1)

8.4 The Explicit Finite-difference Method

139

(5)

8.5 Implicit Finite-difference Methods

144

(1)

8.6 The Fully-implicit Method

144

(11)

8.7 The Crank-Nicolson Method

155

(10)

9 Methods for American Options

165

(15)

9.1 Introduction

165

(2)

9.2 Finite-difference Formulation

167

(1)

9.3 The Constrained Matrix Problem

168

(1)

9.4 Projected SOR

169

(3)

9.5 The Time-stepping Algorithm

172

(2)

9.6 Numerical Examples

174

(2)

9.7 Convergence of the Method

176

(4)

10 Binomial Methods

180

(15)

10.1 Introduction

180

(3)

10.2 The Discrete Random Walk

183

(4)

10.3 Valuing the Option

187

(1)

10.4 European Options

187

(2)

10.5 American Options

189

(2)

10.6 Dividend Yields

191

(4)

Part Three: Further Option Theory

195

(68)

11 Exotic and Path-dependent Options

197

(9)

11.1 Introduction

197

(2)

11.2 Compound Options: Options on Options

199

(2)

11.3 Chooser Options

201

(1)

11.4 Barrier Options

201

(1)

11.5 Asian Options

202

(1)

11.6 Lookback Options

203

(3)

12 Barrier Options

206

(7)

12.1 Introduction

206

(1)

12.2 Knock-outs

207

(2)

12.3 Knock-ins

209

(4)

13 A Unifying Framework for Path-dependent Options

213

(9)

13.1 Introduction

213

(1)

13.2 Time Integrals of the Random Walk

214

(3)

13.3 Discrete Sampling

217

(5)

14 Asian Options

222

(14)

14.1 Introduction

222

(1)

14.2 Continuously Sampled Averages

223

(2)

14.3 Similarity Reductions

225

(1)

14.4 The Average Strike Option

226

(4)

14.5 Average Rate Options

230

(3)

14.6 Discretely Sampled Averages

233

(3)

15 Lookback Options

236

(16)

15.1 Introduction

236

(1)

15.2 Continuous Sampling of the Maximum

237

(6)

15.3 Discrete Sampling of the Maximum

243

(1)

15.4 Similarity Reductions

244

(2)

15.5 Some Numerical Examples

246

(2)

15.6 Two `Perpetual Options'

248

(4)

16 Options with Transaction Costs

252

(11)

16.1 Introduction

252

(1)

16.2 Discrete Hedging

252

(5)

16.3 Portfolios of Options

257

(6)

Part Four: Interest Rate Derivative Products

263

(32)

17 Interest Rate Derivatives

265

(21)

17.1 Introduction

265

(1)

17.2 Basics of Bond Pricing

265

(3)

17.3 The Yield Curve

268

(2)

17.4 Stochastic Interest Rates

270

(1)

17.5 The Bond Pricing Equation

270

(3)

17.6 Solutions of the Bond Pricing Equation

273

(7)

17.7 The Extended Vasicek Model of Hull & White

280

(1)

17.8 Bond Options

281

(1)

17.9 Other Interest Rate Products

282

(4)

18 Convertible Bonds

286

(9)

18.1 Introduction

286

(1)

18.2 Convertible Bonds

286

(4)

18.3 Convertible Bonds with Random Interest Rate

290

(5)

Hints to Selected Exercises

295

(13)

Bibliography

308

(4)

Index

312

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The Mathematics of Financial Derivatives: A Student Introduction

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