# Mathematical Techniques in Finance: Tools for Incomplete Markets

Modern finance overlaps with many fields of mathematics, and for students this can represent considerable strain. Mathematical Techniques in Finance is an ideal textbook for Masters finance courses with a significant quantitative element while also being suitable for finance Ph.D. students. Developed for the highly acclaimed Master of Science in Finance

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## Overview

Modern finance overlaps with many fields of mathematics, and for students this can represent considerable strain. Mathematical Techniques in Finance is an ideal textbook for Masters finance courses with a significant quantitative element while also being suitable for finance Ph.D. students. Developed for the highly acclaimed Master of Science in Finance program at Imperial College London, it offers a carefully crafted blend of numerical applications and theoretical grounding in economics, finance, and mathematics. In the best engineering tradition, Aleš ?erný mixes tools from calculus, linear algebra, probability theory, numerical mathematics, and programming to analyze in an accessible way some of the most intriguing problems in financial economics. Eighty figures, over 70 worked examples, 25 simple ready-to-run computer programs, and several spreadsheets further enhance the learning experience. Each chapter is followed by a number of classroom-tested exercises with solutions available on the book's web site.

Applied mathematics is a craft that requires practice--this textbook provides plenty of opportunities to practice it and teaches cutting-edge finance into the bargain. Asset pricing is a common theme throughout the book; and readers can follow the development from discrete one-period models to continuous time stochastic processes. This textbook sets itself apart by the comprehensive treatment of pricing and risk measurement in incomplete markets, an area of current research that represents the future in risk management and investment performance evaluation.

## Product Details

ISBN-13:
9780691088068
Publisher:
Princeton University Press
Publication date:
11/10/2003
Pages:
352
Product dimensions:
6.20(w) x 9.40(h) x 1.10(d)

Preface xiii

CHAPTER 1: The Simplest Model of Financial Markets 1

1.1 One-Period Finite State Model 1

1.2 Securities and Their Pay-Offs 3

1.3 Securities as Vectors 3

1.4 Operations on Securities 4

1.5 The Matrix as a Collection of Securities 6

1.6 Transposition 6

1.7 Matrix Multiplication and Portfolios 8

1.8 Systems of Equations and Hedging 10

1.9 Linear Independence and Redundant Securities 12

1.10 The Structure of the Marketed Subspace 14

1.11 The Identity Matrix and Arrow -Debreu Securities 16

1.12 Matrix Inverse 17

1.13 Inverse Matrix and Replicating Portfolios 17

1.14 Complete Market Hedging Formula 19

1.15 Summary 20

1.16 Notes 21

1.17 Exercises 21

CHAPTER 2: Arbitrage and Pricing in the One-Period Model 25

2.1 Hedging with Redundant Securities and Incomplete Market 25

2.2 Finding the Best Approximate Hedge 29

2.3 Minimizing the Expected Squared Replication Error 32

2.4 Numerical Stability of Least Squares 34

2.5 Asset Prices, Returns and Portfolio Units 37

2.6 Arbitrage 38

2.7 No-Arbitrage Pricing 40

2.8 State Prices and the Arbitrage Theorem 42

2.9 State Prices and Asset Returns 45

2.10 Risk-Neutral Probabilities 45

2.11 State Prices and No-Arbitrage Pricing 46

2.12 Summary 48

2.13 Notes 49

2.14 Appendix: Least Squares with QR Decomposition 49

2.15 Exercises 52

CHAPTER 3: Risk and Return in the One-Period Model 55

3.1 Utility Functions 55

3.2 Expected Utility Maximization 59

3.3 Reporting Expected Utility in Terms of Money 60

3.4 Scale-Free Formulation of the Optimal Investment Problem with the HARA Utility 62

3.6 Reporting Investment Potential in Terms of Sharpe Ratios 70

3.7 The Importance of Arbitrage Adjustment 77

3.8 Portfolio Choice with Near-Arbitrage Opportunities 78

3.9 Generalization of the Sharpe Ratio 82

3.10 Summary 83

3.11 Notes 84

3.12 Exercises 85

CHAPTER 4: Numerical Techniques for Optimal Portfolio Selection in Incomplete Markets 87

4.1 Sensitivity Analysis of Portfolio Decisions with the CRRA Utility 87

4.2 Newton's Algorithm for Optimal Investment with CRRA Utility 91

4.3 Optimal CRRA Investment Using Empirical Return Distribution 93

4.4 HARA Portfolio Optimizer 99

4.5 HARA Portfolio Optimization with Several Risky Assets 100

4.6 Quadratic Utility Maximization with Multiple Assets 104

4.7 Summary 106

4.8 Notes 107

4.9 Exercises 107

CHAPTER 5: Pricing in Dynamically Complete Markets 109

5.1 Options and Portfolio Insurance 109

5.2 Option Pricing 110

5.3 Dynamic Replicating Trading Strategy 113

5.4 Risk-Neutral Probabilities in a Multi-Period Model 121

5.5 The Law of Iterated Expectations 124

5.6 Summary 126

5.7 Notes 126

5.8 Exercises 126

CHAPTER 6: Towards Continuous Time 131

6.1 IID Returns, and the Term Structure of Volatility 131

6.2 Towards Brownian Motion 133

6.3 Towards a Poisson Jump Process 142

6.4 Central Limit Theorem and Infinitely Divisible Distributions 148

6.5 Summary 149

6.6 Notes 151

6.7 Exercises 151

CHAPTER 7: Fast Fourier Transform 153

7.1 Introduction to Complex Numbers and the Fourier Transform 153

7.2 Discrete Fourier Transform (DFT) 158

7.3 Fourier Transforms in Finance 159

7.4 Fast Pricing via the Fast Fourier Transform (FFT) 164

7.5 Further Applications of FFTs in Finance 167

7.6 Notes 171

7.7 Appendix 172

7.8 Exercises 174

CHAPTER 8: Information Management 175

8.1 Information: Too Much of a Good Thing? 175

8.2 Model-Independent Properties of Conditional Expectation 179

8.3 Summary 183

8.4 Notes 184

8.5 Appendix: Probability Space 184

8.6 Exercises 188

CHAPTER 9: Martingales and Change of Measure in Finance 193

9.1 Discounted Asset Prices Are Martingales 193

9.2 Dynamic Arbitrage Theorem 198

9.3 Change of Measure 199

9.4 Dynamic Optimal Portfolio Selection in a Complete Market 204

9.5 Summary 212

9.6 Notes 214

9.7 Exercises 214

CHAPTER 10: Brownian Motion and Itô Formulae 219

10.1 Continuous-Time Brownian Motion 219

10.2 Stochastic Integration and Itô Processes 224

10.3 Important Itô Processes 226

10.4 Function of a Stochastic Process: the Itô Formula 228

10.5 Applications of the Itô Formula 229

10.6 Multivariate Itô Formula 231

10.7 Itô Processes as Martingales 234

10.8 Appendix: Proof of the Itô Formula 235

10.9 Summary 235

10.10 Notes 236

10.11 Exercises 237

CHAPTER 11: Continuous-Time Finance 239

11.1 Summary of Useful Results 239

11.2 Risk-Neutral Pricing 240

11.3 The Girsanov Theorem 243

11.4 Risk-Neutral Pricing and Absence of Arbitrage 247

11.5 Automatic Generation of PDEs and the Feynman-Kac Formula 252

11.6 Overview of Numerical Methods 256

11.7 Summary 257

11.8 Notes 258

11.9 Appendix: Decomposition of Asset Returns into Uncorrelated Components 258

11.10 Exercises 261

CHAPTER 12: Dynamic Option Hedging and Pricing in Incomplete Markets 267

12.1 The Risk in Option Hedging Strategies 267

12.2 Incomplete Market Option Price Bounds 283

12.3 Towards Continuous Time 291

12.4 Derivation of Optimal Hedging Strategy 297

12.5 Summary 306

12.6 Notes 307

12.7 Appendix: Expected Squared Hedging Error in the Black -Scholes Model 307

12.8 Exercises 309

APPENDIX A: Calculus 313

A.1 Notation 313

A.2 Differentiation 316

A.3 Real Function of Several Real Variables 319

A.4 Power Series Approximations 321

A.5 Optimization 324

A.6 Integration 326

A.7 Exercises 332

APPENDIX B: Probability 337

B.1 Probability Space 337

B.2 Conditional Probability 337

B.3 Marginal and Joint Distribution 340

B.4 Stochastic Independence 341

B.5 Expectation Operator 343

B.6 Properties of Expectation 344

B.7 Mean and Variance 345

B.8 Covariance and Correlation 346

B.9 Continuous Random Variables 349

B.10 Normal Distribution 354

B.11 Quantiles 359

B.12 Relationships among Standard Statistical Distributions 360

B.13 Notes 361

B.14 Exercises 361

References 369

Index 373