Preface
Acknowledgments
1 Introduction and Overview 1
1.1 The importance of mathematics in finance 1
1.2 Mathematical and computer modeling in finance 2
1.3 Money, securities, and markets 3
1.4 Time value, risk, arbitrage, and pricing 5
1.5 The organization of this book 6
2 A Review of Elementary Mathematics: Functions and Operations 7
2.1 Introduction 7
2.2 Variables, equations, and inequalities 7
2.3 Exponents 8
Application 2.1: Interest and future value 9
2.4 The order of arithmetic operations and the rules of algebra 10
Application 2.2: Initial deposit amounts 11
2.5 The number e 11
2.6 Logarithms 12
Application 2.3: The time needed to double your money 13
2.7 Subscripts 14
2.8 Summations 14
Application 2.4: Mean values 15
2.9 Double summations 16
2.10 Products 17
Application 2.5: Geometric means 17
Application 2.6: The term structure of interest rates 18
2.11 Factorial products 19
Application 2.7: Deriving the number e 19
2.12 Permutations and combinations 20
Exercises 21
Appendix 2.A An introduction to the Excel™ spreadsheet 23
3 A Review of Elementary Mathematics: Algebra and Solving Equations 25
3.1 Algebraic manipulations 25
Application 3.1: Purchase power parity 27
Application 3.2: Finding break-even production levels 28
Application 3.3: Solving for spot and forward interest rates 29
3.2 The quadratic formula 29
Application 3.4: Finding break-even production levels 30
Application 3.5: Finding the perfectly hedged portfolio 31
3.3 Solving systems of equations that contain multiple variables 32
Application 3.6: Pricing factors 35
Application 3.7: External financing needs 35
3.4 Geometric expansions 38
Application 3.8: Money multipliers 40
3.5 Functions and graphs 41
Application 3.9: Utility of wealth 43
Exercises 44
Appendix 3.A Solving systems of equations on a spreadsheet 48
4 The Time Value of Money 51
4.1 Introduction and future value 51
4.2 Simple interest 51
4.3 Compound interest 52
4.4 Fractional period compounding of interest 53
Application 4.1: APY and bank account comparisons 55
4.5 Continuous compounding of interest 56
4.6 Annuity future values 57
Application 4.2: Planning for retirement 59
4.7 Discounting and present value 60
4.8 The present value of a series of cash flows 61
4.9 Annuity present values 62
Application 4.3: Planning for Retirement, Part Ii 64
Application 4.4: Valuing a bond 64
4.10 Amortization 65
Application 4.5: Determining the mortgage payment 66
4.11 Perpetuity models 67
4.12 Single-stage growth models 68
Application 4.6: Stock valuation models 70
4.13 Multiple-stage growth models 72
Exercises 73
Appendix 4.A Time value spreadsheet applications 77
5 Return, Risk, and Co-movement 79
5.1 Return on investment 79
Application 5.1: Fund performance 81
5.2 Geometric mean return on investment 82
Application 5.2: Fund Performance, Part Ii 83
5.3 Internal rate of return 84
5.4 Bond yields 87
5.5 An introduction to risk 88
5.6 Expected return 88
5.7 Variance and standard deviation 89
5.8 Historical variance and standard deviation 91
5.9 Covariance 93
5.10 The coefficient of correlation and the coefficient of determination 94
Exercises 95
Appendix 5.A Return and risk spreadsheet applications 99
6 Elementary Portfolio Mathematics 103
6.1 An introduction to portfolio analysis 103
6.2 Portfolio return 103
6.3 Portfolio variance 104
6.4 Diversification and efficiency 106
6.5 The market portfolio and beta 110
6.6 Deriving the portfolio variance expression 111
Exercises 113
7 Elements of Matrix Mathematics 115
7.1 An introduction to matrices 115
Application 7.1: Portfolio mathematics 116
7.2 Matrix arithmetic 117
Application 7.2: Portfolio Mathematics, Part Ii 120
Application 7.3: Put–call parity 121
7.3 Inverting matrices 123
7.4 Solving systems of equations 125
Application 7.4: External funding requirements 126
Application 7.5: Coupon bonds and deriving yield curves 127
Application 7.6: Arbitrage with riskless bonds 130
Application 7.7: Fixed income portfolio dedication 131
Application 7.8: Binomial option pricing 132
7.5 Spanning the state space 133
Application 7.9: Using options to span the state space 136
Exercises 137
Appendix 7.A Matrix mathematics on a spreadsheet 142
8 Differential Calculus 145
8.1 Functions and limits 145
Application 8.1: The natural log 146
8.2 Slopes, derivatives, maxima, and minima 147
8.3 Derivatives of polynomials 149
Application 8.2: Marginal utility 151
Application 8.3: Duration and immunization 153
Application 8.4: Portfolio risk and diversification 156
8.4 Partial and total derivatives 157
8.5 The chain rule, product rule, and quotient rule 158
Application 8.5: Plotting the Capital Market Line 159
8.6 Logarithmic and exponential functions 165
8.7 Taylor series expansions 166
Application 8.6: Convexity and immunization 167
Exercises 172
Appendix 8.A Derivatives of polynomials 176
Appendix 8.B A table of rules for finding derivatives 177
Appendix 8.C Portfolio risk minimization on a spreadsheet 178
9 Integral Calculus 180
9.1 Antidifferentiation and the indefinite integral 180
9.2 Riemann sums 181
9.3 Definite integrals and areas 185
Application 9.1: Cumulative densities 186
Application 9.2: Expected value and variance 188
Application 9.3: Valuing continuous dividend payments 189
Application 9.4: Expected option values 191
9.4 Differential equations 191
Application 9.5: Security returns in continuous time 193
Application 9.6: Annuities and growing annuities 194
Exercises 195
Appendix 9.A Rules for finding integrals 198
Appendix 9.B Riemann sums on a spreadsheet 199
10 Elements of Options Mathematics 203
10.1 An introduction to stock options 203
10.2 Binomial option pricing: one time period 205
10.3 Binomial option pricing: multiple time periods 207
10.4 The Black–Scholes option pricing model 210
10.5 Puts and valuation 212
10.6 Black–Scholes model sensitivities 213
10.7 Estimating implied volatilities 215
Exercises 219
References 222
Appendix A Solutions to Exercises 224
Appendix B The z-Table 266
Appendix C Notation 267
Appendix D Glossary 270
Index 274
