List of Figures xiii List of Tables xvi List of Examples xvii Foreword xix Preface to Volume I xxiii I.1 Basic Calculus for Finance 1 I.1.1 Introduction 1 I.1.2 Functions and Graphs, Equations and Roots 3 I.1.3 Differentiation and Integration 10 I.1.4 Analysis of Financial Returns 16 I.1.5 Functions of Several Variables 26 I.1.6 Taylor Expansion 31 I.1.7 Summary and Conclusions 35 I.2 Essential Linear Algebra for Finance 37 I.2.1 Introduction 37 I.2.2 Matrix Algebra and its Mathematical Applications 38 I.2.3 Eigenvectors and Eigenvalues 48 I.2.4 Applications to Linear Portfolios 55 I.2.5 Matrix Decomposition 61 I.2.6 Principal Component Analysis 64 I.2.7 Summary and Conclusions 70 I.3 Probability and Statistics 71 I.3.1 Introduction 71 I.3.2 Basic Concepts 72 I.3.3 Univariate Distributions 85 I.3.4 Multivariate Distributions 107 I.3.5 Introduction to Statistical Inference 118 I.3.6 Maximum Likelihood Estimation 130 I.3.7 Stochastic Processes in Discrete and Continuous Time 134 I.3.8 Summary and Conclusions 140 I.4 Introduction to Linear Regression 143 I.4.1 Introduction 143 I.4.2 Simple Linear Regression 144 I.4.3 Properties of OLS Estimators 155 I.4.4 Multivariate Linear Regression 158 I.4.5 Autocorrelation and Heteroscedasticity 175 I.4.6 Applications of Linear Regression in Finance 179 I.4.7 Summary and Conclusions 184 I.5 Numerical Methods in Finance 185 I.5.1 Introduction 185 I.5.2 Iteration 187 I.5.3 Interpolation and Extrapolation 193 I.5.4 Optimization 200 I.5.5 Finite Difference Approximations 206 I.5.6 Binomial Lattices 210 I.5.7 Monte Carlo Simulation 217 I.5.8 Summary and Conclusions 223 I.6 Introduction to Portfolio Theory 225 I.6.1 Introduction 225 I.6.2 Utility Theory 226 I.6.3 Portfolio Allocation 237 I.6.4 Theory of Asset Pricing 250 I.6.5 Risk Adjusted Performance Measures 256 I.6.6 Summary and Conclusions 266 References 269 Statistical Tables 273 Index 279

Written by leading market risk academic, Professor Carol Alexander, Quantitative Methods in Finance forms part one of the Market Risk Analysis four volume set. Starting from the basics, this book helps readers to take the first step towards becoming a properly qualified financial risk manager and asset manager, roles that are currently in huge demand. Accessible to intelligent readers with a moderate understanding of mathematics at high school level or to anyone with a university degree in mathematics, physics or engineering, no prior knowledge of finance is necessary. Instead the emphasis is on understanding ideas rather than on mathematical rigour, meaning that this book offers a fast-track introduction to financial analysis for readers with some quantitative background, highlighting those areas of mathematics that are particularly relevant to solving problems in financial risk management and asset management. Unique to this book is a focus on both continuous and discrete time finance so that Quantitative Methods in Finance is not only about the application of mathematics to finance; it also explains, in very pedagogical terms, how the continuous time and discrete time finance disciplines meet, providing a comprehensive, highly accessible guide which will provide readers with the tools to start applying their knowledge immediately. All together, the Market Risk Analysis four volume set illustrates virtually every concept or formula with a practical, numerical example or a longer, empirical case study. Across all four volumes there are approximately 300 numerical and empirical examples, 400 graphs and figures and 30 case studies many of which are contained in interactive Excel spreadsheets available from the accompanying CD-ROM . Empirical examples and case studies specific to this volume include: Principal component analysis of European equity indices;Calibration of Student t distribution by maximum likelihood;Orthogonal regression and estimation of equity factor models;Simulations of geometric Brownian motion, and of correlated Student t variables;Pricing European and American options with binomial trees, and European options with the Black-Scholes-Merton formula;Cubic spline fitting of yields curves and implied volatilities;Solution of Markowitz problem with no short sales and other constraints;Calculation of risk adjusted performance metrics including generalised Sharpe ratio, omega and kappa indices.