Mathematical Techniques in Finance

Ales Cerny is professor of finance at the Cass Business School, City University London.

"Ales Černy'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, Černy has once again given us an up-to-date first-class textbook treatment of the subject."—Darrell Duffie, Stanford University

• 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 Ito Formulae 213 10.1 Continuous-Time Brownian Motion 213 10.2 Stochastic Integration and Ito Processes 218 10.3 Important Ito Processes 220 10.4 Function of a Stochastic Process: the Ito Formula 222 10.5 Applications of the Ito Formula 223 10.6 Multivariate Ito Formula 225 10.7 Ito Processes as Martingales 228 10.8 Appendix: Proof of the Ito 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