Mathematical Techniques in Finance

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