Preface vii
1. The Theory of Interest 1
1.1 Simple Interest 1
1.2 Compound Interest 3
1.3 Continuously Compounded Interest 4
1.4 Present Value 5
1.5 Rate of Return 11
1.6 Exercises 12
2. Discrete Probability 15
2.1 Events and Probabilities 15
2.2 Addition Rule 17
2.3 Conditional Probability and Multiplication Rule 18
2.4 Random Variables and Probability Distributions 21
2.5 Binomial Random Variables 23
2.6 Expected Value 24
2.7 Variance and Standard Deviation 29
2.8 Exercises 32
3. Normal Random Variables and Probability 35
3.1 Continuous Random Variables 35
3.2 Expected Value of Continuous Random Variables 38
3.3 Variance and Standard Deviation 40
3.4 Normal Random Variables 42
3.5 Central Limit Theorem 49
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X l l An Undergraduate Introduction to Financial Mathematics
3.6 Lognormal Random Variables 51
3.7 Properties of Expected Value 55
3.8 Properties of Variance 58
3.9 Exercises 61
4. The Arbitrage Theorem 63
4.1 The Concept of Arbitrage 63
4.2 Duality Theorem of Linear Programming 64
4.2.1 Dual Problems 66
4.3 The Fundamental Theorem of Finance 72
4.4 Exercises 74
5. Random Walks and Brownian Motion 77
5.1 Intuitive Idea of a Random Walk 77
5.2 First Step Analysis 78
5.3 Intuitive Idea of a Stochastic Process 91
5.4 Stock Market Example 95
5.5 More About Stochastic Processes 97
5.6 Ito's Lemma 98
5.7 Exercises 101
6. Options 103
6.1 Properties of Options 104
6.2 Pricing an Option Using a Binary Model 107
6.3 Black-Scholes Partial Differential Equation 110
6.4 Boundary and Initial Conditions 112
6.5 Exercises 114
7. Solution of the Black-Scholes Equation 115
7.1 Fourier Transforms 115
7.2 Inverse Fourier Transforms 118
7.3 Changing Variables in the Black-Scholes PDE 119
7.4 Solving the Black-Scholes Equation 122
7.5 Exercises 127
8. Derivatives of Black-Scholes Option Prices 131
8.1 Theta 131
8.2 Delta 133
Contents xiii
8.3 Gamma 135
8.4 Vega 136
8.5 Rho 138
8.6 Relationships Between A, 9, and T 139
8.7 Exercises 141
9. Hedging 143
9.1 General Principles 143
9.2 Delta Hedging 145
9.3 Delta Neutral Portfolios 149
9.4 Gamma Neutral Portfolios 151
9.5 Exercises 153
10. Optimizing Portfolios 155
10.1 Covariance and Correlation 155
10.2 Optimal Portfolios 164
10.3 Utility Functions 165
10.4 Expected Utility 171
10.5 Portfolio Selection 173
10.6 Minimum Variance Analysis 177
10.7 Mean Variance Analysis 186
10.8 Exercises 191
Appendix A Sample Stock Market Data 195
Appendix B Solutions to Chapter Exercises 203
B.l The Theory of Interest 203
B.2 Discrete Probability 206
B.3 Normal Random Variables and Probability 212
B.4 The Arbitrage Theorem 225
B.5 Random Walks and Brownian Motion 231
B.6 Options 235
B.7 Solution of the Black-Scholes Equation 239
B.8 Derivatives of Black-Scholes Option Prices 245
B.9 Hedging 249
B.10 Optimizing Portfolios 255