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Mathematical Techniques in Finance: Tools for Incomplete Markets 2ed (ISE)

by Ales Cerny Princeton University Press
Pub Date:
07/2009
ISBN:
9780691141213
Format:
Pbk 412 pages
Price:
AU$56.99 NZ$60.86
Product Status: Not Our Publication - we no longer distribute
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Instructors
& Academics:
Originally published in 2003, Mathematical Techniques in Finance has become a standard textbook for master's-level finance courses containing a significant quantitative element while also being suitable for finance PhD students.

This fully revised second edition continues to offer a carefully crafted blend of numerical applications and theoretical grounding in economics, finance, and mathematics, and provides plenty of opportunities for students to practice applied mathematics and cutting-edge finance. Ales Cerný 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. The textbook is the perfect hands-on introduction to asset pricing, optimal portfolio selection, risk measurement, and investment evaluation.

The new edition includes the most recent research in the area of incomplete markets and unhedgeable risks, adds a chapter on finite difference methods, and thoroughly updates all bibliographic references. Eighty figures, over seventy examples, twenty-five simple ready-to-run computer programs, and several spreadsheets enhance the learning experience. All computer codes have been rewritten using MATLAB and online supplementary materials have been completely updated.

KEY FEATURES:
-A standard textbook for graduate finance courses
-Introduction to asset pricing, portfolio selection, risk measurement, and investment evaluation
-Detailed examples and MATLAB codes integrated throughout the text
-Exercises and summaries of main points conclude each chapter



Preface to the Second Edition xiii

From the Preface to the First Edition xix



Chapter 1: The Simplest Model of Financial Markets 1

1.1 One-Period Finite State Model 1

1.2 Securities and Their Payoffs 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 22



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 36

2.6 Arbitrage 38

2.7 No-Arbitrage Pricing 40

2.8 State Prices and the Arbitrage Theorem 41

2.9 State Prices and Asset Returns 44

2.10 Risk-Neutral Probabilities 45

2.11 State Prices and No-Arbitrage Pricing 46

2.12 Asset Pricing Duality 47

2.13 Summary 48

2.14 Notes 49

2.15 Appendix: Least Squares with QR Decomposition 49

2.16 Exercises 52



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

3.1 Utility Functions 56

3.2 Expected Utility Maximization 59

3.3 The Existence of Optimal Portfolios 61

3.4 Reporting Expected Utility in Terms of Money 62

3.5 Normalized Utility and Investment Potential 63

3.6 Quadratic Utility 67

3.7 The Sharpe Ratio 69

3.8 Arbitrage-Adjusted Sharpe Ratio 71

3.9 The Importance of Arbitrage Adjustment 75

3.10 Portfolio Choice with Near-Arbitrage Opportunities 77

3.11 Summary 79

3.12 Notes 81

3.13 Exercises 82



Chapter 4: Numerical Techniques for Optimal Portfolio Selection in Incomplete Markets 84

4.1 Sensitivity Analysis of Portfolio Decisions with the CRRA Utility 84

4.2 Newton's Algorithm for Optimal Investment with CRRA Utility 88

4.3 Optimal CRRA Investment Using Empirical Return Distribution 90

4.4 HARA Portfolio Optimizer 94

4.5 HARA Portfolio Optimization with Several Risky Assets 96

4.6 Quadratic Utility Maximization with Multiple Assets 99

4.7 Summary 102

4.8 Notes 102

4.9 Exercises 102



Chapter 5: Pricing in Dynamically Complete Markets 104

5.1 Options and Portfolio Insurance 104

5.2 Option Pricing 105

5.3 Dynamic Replicating Trading Strategy 108

5.4 Risk-Neutral Probabilities in a Multi-Period Model 116

5.5 The Law of Iterated Expectations 119

5.6 Summary 121

5.7 Notes 121

5.8 Exercises 121



Chapter 6: Towards Continuous Time 125

6.1 IID Returns, and the Term Structure of Volatility 125

6.2 Towards Brownian Motion 127

6.3 Towards a Poisson Jump Process 136

6.4 Central Limit Theorem and Infinitely Divisible Distributions 142

6.5 Summary 143

6.6 Notes 145

6.7 Exercises 145



Chapter 7: Fast Fourier Transform 147

7.1 Introduction to Complex Numbers and the Fourier Transform 147

7.2 Discrete Fourier Transform (DFT) 152

7.3 Fourier Transforms in Finance 153

7.4 Fast Pricing via the Fast Fourier Transform (FFT) 158

7.5 Further Applications of FFTs in Finance 162

7.6 Notes 166

7.7 Appendix 167

7.8 Exercises 169



Chapter 8: Information Management 170

8.1 Information: Too Much of a Good Thing? 170

8.2 Model-Independent Properties of Conditional Expectation 174

8.3 Summary 178

8.4 Notes 179

8.5 Appendix: Probability Space 179

8.6 Exercises 183



Chapter 9: Martingales and Change of Measure in Finance 187

9.1 Discounted Asset Prices Are Martingales 187

9.2 Dynamic Arbitrage Theorem 192

9.3 Change of Measure 193

9.4 Dynamic Optimal Portfolio Selection in a Complete Market 198

9.5 Summary 206

9.6 Notes 208

9.7 Exercises 208



Chapter 10: Brownian Motion and Itˆo Formulae 213

10.1 Continuous-Time Brownian Motion 213

10.2 Stochastic Integration and Itˆo Processes 218

10.3 Important Itˆo Processes 220

10.4 Function of a Stochastic Process: the Itˆo Formula 222

10.5 Applications of the Itˆo Formula 223

10.6 Multivariate Itˆo Formula 225

10.7 Itˆo Processes as Martingales 228

10.8 Appendix: Proof of the Itˆo Formula 229

10.9 Summary 229

10.10 Notes 230

10.11 Exercises 231



Chapter 11: Continuous-Time Finance 233

11.1 Summary of Useful Results 233

11.2 Risk-Neutral Pricing 234

11.3 The Girsanov Theorem 237

11.4 Risk-Neutral Pricing and Absence of Arbitrage 241

11.5 Automatic Generation of PDEs and the Feynman-Kac Formula 246

11.6 Overview of Numerical Methods 250

11.7 Summary 251

11.8 Notes 252

11.9 Appendix: Decomposition of Asset Returns into Uncorrelated Components 252

11.10 Exercises 255



Chapter 12: Finite-Difference Methods 261

12.1 Interpretation of PDEs 261

12.2 The Explicit Method 263

12.3 Instability 264

12.4 Markov Chains and Local Consistency 266

12.5 Improving Convergence by Richardson's Extrapolation 268

12.6 Oscillatory Convergence Due to Grid Positioning 269

12.7 Fully Implicit Scheme 270

12.8 Crank-Nicolson Scheme 273

12.9 Summary 274

12.10 Notes 276

12.11 Appendix: Efficient Gaussian Elimination for Tridiagonal Matrices 276

12.12 Appendix: Richardson's Extrapolation 277

12.13 Exercises 277



Chapter 13: Dynamic Option Hedging and Pricing in Incomplete Markets 280

13.1 The Risk in Option Hedging Strategies 280

13.2 Incomplete Market Option Price Bounds 299

13.3 Towards Continuous Time 304

13.4 Derivation of Optimal Hedging Strategy 309

13.5 Summary 318

13.6 Notes 319

13.7 Appendix: Expected Squared Hedging Error in the Black-Scholes Model 320

13.8 Exercises 322



Appendix A Calculus 326

A.1 Notation 326

A.2 Differentiation 329

A.3 Real Function of Several Real Variables 332

A.4 Power Series Approximations 334

A.5 Optimization 336

A.6 Integration 338

A.7 Exercises 344



Appendix B Probability 348

B.1 Probability Space 348

B.2 Conditional Probability 348

B.3 Marginal and Joint Distribution 351

B.4 Stochastic Independence 352

B.5 Expectation Operator 354

B.6 Properties of Expectation 355

B.7 Mean and Variance 356

B.8 Covariance and Correlation 357

B.9 Continuous Random Variables 360

B.10 Normal Distribution 364

B.11 Quantiles 370

B.12 Relationships among Standard Statistical Distributions 371

B.13 Notes 372

B.14 Exercises 372



References 381

Index 385


Ales Cern and yacute;'s new edition of Mathematical Techniques in Finance is an excellent master's-level treatment of mathematical methods used in financial asset pricing. By updating the original edition with methods used in recent research, Cern and yacute; has once again given us an up-to-date first-class textbook treatment of the subject. Darrell Duffie, Stanford University
Ales Cerný is professor of finance at the Cass Business School, City University London.