روش‌های کمی برای مالی و سرمایه‌گذاری ۲۰۰۹
Quantitative Methods for Finance and Investments 2009

1. PREFACE

2. ACKNOWLEDGMENTS

3. C H A P T E R O N E INTRODUCTION AND OVERVIEW

1.1 THE IMPORTANCE OF MATHEMATICS IN FINANCE

1.2 MATHEMATICAL AND COMPUTER MODELING IN FINANCE

1.3 MONEY, SECURITIES, AND MARKETS

1.4 TIME VALUE, RISK, ARBITRAGE, AND PRICING

1.5 THE ORGANIZATION OF THIS BOOK

9. C H A P T E R T W O A REVIEW OF ELEMENTARY MATHEMATICS: FUNCTIONS AND OPERATIONS

2.1 INTRODUCTION

2.2 VARIABLES, EQUATIONS, AND INEQUALITIES

2.3 EXPONENTS

13. APPLICATION 2.1: INTEREST AND FUTURE VALUE (A more complete presentation of this material is provided in chapter 4)

2.4 THE ORDER OF ARITHMETIC OPERATIONS AND THE RULES OF ALGEBRA

15. APPLICATION 2.2: INITIAL DEPOSIT AMOUNTS (A more complete presentation of this material is provided in chapter 4)

2.5 THE NUMBER e

2.6 LOGARITHMS (Background reading: sections 2.2 and 2.5)

18. APPLICATION 2.3: THE TIME NEEDED TO DOUBLE YOUR MONEY (You may also wish to read related material in chapter 4)

2.7 SUBSCRIPTS

2.8 SUMMATIONS (Background reading: sections 2.3 and 2.7)

21. APPLICATION 2.4: MEAN VALUES (You may also wish to read related material in section 5.1)

2.9 DOUBLE SUMMATIONS (Background reading: section 2.8)

2.10 PRODUCTS (Background reading: section 2.8)

24. APPLICATION 2.5: GEOMETRIC MEANS (Background reading: application 2.4)

25. APPLICATION 2.6: THE TERM STRUCTURE OF INTEREST RATES (Background reading: application 2.5)

2.11 FACTORIAL PRODUCTS (Background reading: section 2.10)

27. APPLICATION 2.7: DERIVING THE NUMBER e (Background reading: section 2.5)

2.12 PERMUTATIONS AND COMBINATIONS (Background reading: section 2.11)

29. EXERCISES

30. APPENDIX 2.A AN INTRODUCTION TO THE EXCEL™ SPREADSHEET

31. C H A P T E R T H R E E A REVIEW OF ELEMENTARY MATHEMATICS: ALGEBRA AND SOLVING EQUATIONS

3.1 ALGEBRAIC MANIPULATIONS (Background reading: section 2.4)

33. APPLICATION 3.1: PURCHASE POWER PARITY

34. APPLICATION 3.2: FINDING BREAK-EVEN PRODUCTION LEVELS

35. APPLICATION 3.3: SOLVING FOR SPOT AND FORWARD INTEREST RATES (Background reading: sections 1.3 and 2.10, and application 2.6)

3.2 THE QUADRATIC FORMULA (Background reading: section 3.1)

37. APPLICATION 3.4: FINDING BREAK-EVEN PRODUCTION LEVELS (Background reading: application 3.2)

38. APPLICATION 3.5: FINDING THE PERFECTLY HEDGED PORTFOLIO

3.3 SOLVING SYSTEMS OF EQUATIONS THAT CONTAIN MULTIPLE VARIABLES (Background reading: section 3.1)

40. APPLICATION 3.6: PRICING FACTORS

41. APPLICATION 3.7: EXTERNAL FINANCING NEEDS (Background reading: section 3.3)

3.4 GEOMETRIC EXPANSIONS (Background reading: section 3.1)

43. APPLICATION 3.8: MONEY MULTIPLIERS

3.5 FUNCTIONS AND GRAPHS (Background reading: section 2.3)

45. APPLICATION 3.9: UTILITY OF WEALTH

46. EXERCISES

47. APPENDIX 3.A SOLVING SYSTEMS OF EQUATIONS ON A SPREADSHEET (Background reading: section 3.3 and appendix 2.A)

48. C H A P T E R F O U R THE TIME VALUE OF MONEY

4.1 INTRODUCTION AND FUTURE VALUE

4.2 SIMPLE INTEREST (Background reading: sections 2.4, 2.7, and 4.1)

4.3 COMPOUND INTEREST (Background reading: sections 2.7, 3.1, and 4.2)

4.4 FRACTIONAL PERIOD COMPOUNDING OF INTEREST

53. APPLICATION 4.1: APY AND BANK ACCOUNT COMPARISONS

4.5 CONTINUOUS COMPOUNDING OF INTEREST (Background reading: sections 2.5 and 4.4)

4.6 ANNUITY FUTURE VALUES (Background reading: sections 2.8, 3.4, and 4.3)

56. APPLICATION 4.2: PLANNING FOR RETIREMENT (Background reading: sections 2.5, 3.1, and 4.5)

4.7 DISCOUNTING AND PRESENT VALUE (Background reading: section 4.3)

58. Deriving the present-value formula

4.8 THE PRESENT VALUE OF A SERIES OF CASH FLOWS (Background reading: sections 2.8 and 4.7)

4.9 ANNUITY PRESENT VALUES (Background reading: sections 3.4, 4.6, and 4.8)

61. Deriving the present-value annuity formula

62. APPLICATION 4.3: PLANNING FOR RETIREMENT, PART II (Background reading: application 4.2 and section 4.9)

63. APPLICATION 4.4: VALUING A BOND

4.10 AMORTIZATION (Background reading: section 4.9)

65. APPLICATION 4.5: DETERMINING THE MORTGAGE PAYMENT

4.11 PERPETUITY MODELS (Background reading: section 4.9)

4.12 SINGLE-STAGE GROWTH MODELS (Background reading: sections 4.9 and 4.11)

68. APPLICATION 4.6: STOCK VALUATION MODELS

4.13 MULTIPLE-STAGE GROWTH MODELS* (Background reading: section 4.12)

70. EXERCISES

71. APPENDIX 4.A TIME VALUE SPREADSHEET APPLICATIONS

72. C H A P T E R F I V E RETURN, RISK, AND CO-MOVEMENT

5.1 RETURN ON INVESTMENT (Background reading: section 2.8 and application 2.4)

74. APPLICATION 5.1: FUND PERFORMANCE (Background reading: sections 2.8 and 5.1 and application 2.4)

5.2 GEOMETRIC MEAN RETURN ON INVESTMENT (Background reading: sections 2.10 and 5.1 and application 2.5)

76. APPLICATION 5.2: FUND PERFORMANCE, PART II (Background reading: section 5.2 and application 5.1)

5.3 INTERNAL RATE OF RETURN (Background reading: sections 4.7, 4.8, and 4.9)

5.4 BOND YIELDS

5.5 AN INTRODUCTION TO RISK

5.6 EXPECTED RETURN (Background reading: sections 2.8 and 5.5)

5.7 VARIANCE AND STANDARD DEVIATION

5.8 HISTORICAL VARIANCE AND STANDARD DEVIATION

5.9 COVARIANCE

5.10 THE COEFFICIENT OF CORRELATION AND THE COEFFICIENT OF DETERMINATION

85. EXERCISES

86. APPENDIX 5.A RETURN AND RISK SPREADSHEET APPLICATIONS

87. C H A P T E R S I X ELEMENTARY PORTFOLIO MATHEMATICS

6.1 AN INTRODUCTION TO PORTFOLIO ANALYSIS (Background reading: sections 5.1 and 5.5)

6.2 PORTFOLIO RETURN (Background reading: sections 2.8, 5.5, and 6.1)

6.3 PORTFOLIO VARIANCE (Background reading: sections 2.9, 5.7–5.10, and 6.2)

6.4 DIVERSIFICATION AND EFFICIENCY

6.5 THE MARKET PORTFOLIO AND BETA

6.6 DERIVING THE PORTFOLIO VARIANCE EXPRESSION (Background reading: section 6.3)

94. EXERCISES

95. C H A P T E R S E V E N ELEMENTS OF MATRIX MATHEMATICS

7.1 AN INTRODUCTION TO MATRICES

97. APPLICATION 7.1: PORTFOLIO MATHEMATICS (Background reading: sections 2.9, 6.2, 6.3, and 7.1)

7.2 MATRIX ARITHMETIC (Background reading: sections 2.9 and 7.1)

99. APPLICATION 7.2: PORTFOLIO MATHEMATICS, PART II (Background reading: application 7.1 and section 7.2)

100. APPLICATION 7.3: PUT–CALL PARITY (Background reading: section 7.2)

7.3 INVERTING MATRICES (Background reading: sections 3.3 and 7.2)

7.4 SOLVING SYSTEMS OF LINEAR EQUATIONS (Background reading: sections 1.3, 3.3, and 7.3)

103. APPLICATION 7.4: EXTERNAL FUNDING REQUIREMENTS (Background reading: section 7.5 and application 3.7)

104. APPLICATION 7.5: COUPON BONDS AND DERIVING YIELD CURVES (Background reading: sections 4.8, 4.9, and 7.4 and applications 2.6 and 4.4)

105. APPLICATION 7.6: ARBITRAGE WITH RISKLESS BONDS (Background reading: application 7.5)

106. APPLICATION 7.7: FIXED INCOME PORTFOLIO DEDICATION (Background reading: application 7.6)

107. APPLICATION 7.8: BINOMIAL OPTION PRICING (Background reading: section 7.4 and application 7.3)

7.5 SPANNING THE STATE SPACE (Background reading: sections 7.3 and 7.4)

109. APPLICATION 7.9: USING OPTIONS TO SPAN THE STATE SPACE (Background reading: section 7.5 and application 7.8)

110. EXERCISES

111. APPENDIX 7.A MATRIX MATHEMATICS ON A SPREADSHEET (Background reading: sections 7.3, 7.4, and 7.5 and appendix 3.A)

112. C H A P T E R E I G H T DIFFERENTIAL CALCULUS

8.1 FUNCTIONS AND LIMITS (Background reading: sections 2.2 and 3.5)

114. APPLICATION 8.1: THE NATURAL LOG (Background reading: sections 2.5, 2.11, 4.4, 4.5, and 8.1 and application 2.7)

8.2 SLOPES, DERIVATIVES, MAXIMA, AND MINIMA (Background reading: section 8.1)

8.3 DERIVATIVES OF POLYNOMIALS (Background reading: section 8.2)

117. APPLICATION 8.2: MARGINAL UTILITY (Background reading: section 8.3 and application 3.9)

118. APPLICATION 8.3: DURATION AND IMMUNIZATION (Background reading: sections 4.8, 4.9, 5.4, and 8.3 and application 4.4)

119. APPLICATION 8.4: PORTFOLIO RISK AND DIVERSIFICATION (Background reading: sections 6.1, 6.4, 6.5 and 8.2)

8.4 PARTIAL AND TOTAL DERIVATIVES (Background reading: section 8.3)

8.5 THE CHAIN RULE, PRODUCT RULE, AND QUOTIENT RULE (Background reading: sections 8.3 and 8.4)

122. APPLICATION 8.5: PLOTTING THE CAPITAL MARKET LINE (Background reading: sections 6.4, 7.4, and 8.5)

8.6 LOGARITHMIC AND EXPONENTIAL FUNCTIONS (Background reading: sections 2.5, 2.6, 8.1, and 8.2)

8.7 TAYLOR SERIES EXPANSIONS (Background reading: sections 2.11 and 8.3)

125. APPLICATION 8.6: CONVEXITY AND IMMUNIZATION (Background reading: section 8.7 and application 8.4)

8.8 THE METHOD OF LAGRANGE MULTIPLIERS (Background reading: sections 7.4 and 8.4)

127. APPLICATION 8.7: OPTIMAL PORTFOLIO SELECTION (Background reading: sections 6.3, 6.4, and 8.7)

128. EXERCISES

129. APPENDIX 8.A DERIVATIVES OF POLYNOMIALS (Background reading: sections 2.12 and 8.3)

130. APPENDIX 8.C PORTFOLIO RISK MINIMIZATION ON A SPREADSHEET (Background reading: section 8.8, application 8.8, and appendix 7.C)

131. C H A P T E R N I N E INTEGRAL CALCULUS

9.1 ANTIDIFFERENTIATION AND THE INDEFINITE INTEGRAL (Background reading: section 8.3)

9.2 RIEMANN SUMS (Background reading: section 9.1)

9.3 DEFINITE INTEGRALS AND AREAS (Background reading: sections 9.1 and 9.2)

135. APPLICATION 9.1: CUMULATIVE DENSITIES (Background reading: sections 2.4 and 9.3)

136. APPLICATION 9.2: EXPECTED VALUE AND VARIANCE (Background reading: sections 5.6, 5.7, and 9.2 and application 9.1)

137. APPLICATION 9.3: VALUING CONTINUOUS DIVIDEND PAYMENTS (Background reading: sections 4.5, 4.8, 9.2, and 9.3)

138. APPLICATION 9.4: EXPECTED OPTION VALUES (Background reading: section 9.3 and application 7.8)

9.4 DIFFERENTIAL EQUATIONS

140. APPLICATION 9.5: SECURITY RETURNS IN CONTINUOUS TIME (Background reading: section 9.4 and application 9.3)

141. APPLICATION 9.6: ANNUITIES AND GROWING ANNUITIES (Background reading: section 9.4 and application 9.5)

142. EXERCISES

143. APPENDIX 9.A RULES FOR FINDING INTEGRALS

144. APPENDIX 9.B RIEMANN SUMS ON A SPREADSHEET (Background reading: section 9.2)

145. C H A P T E R T E N ELEMENTS OF OPTIONS MATHEMATICS

10.1 AN INTRODUCTION TO STOCK OPTIONS

10.2 BINOMIAL OPTION PRICING: ONE TIME PERIOD (Background reading: section 10.1 and application 7.8)

10.3 BINOMIAL OPTION PRICING: MULTIPLE TIME PERIODS (Background reading: section 10.2)

10.4 THE BLACK–SCHOLES OPTION PRICING MODEL (Background reading: section 10.3)

10.5 PUTS AND VALUATION (Background reading: sections 10.1 and 10.4 and application 7.9)

10.6 BLACK–SCHOLES MODEL SENSITIVITIES (Background reading: sections 8.4, 8.5, and 10.4)

10.7 ESTIMATING IMPLIED VOLATILITIES (Background reading: sections 8.6 and 10.6)

153. EXERCISES

154. REFERENCES

155. A P P E N D I X A SOLUTIONS TO EXERCISES

156. A P P E N D I X B THE Z-TABLE

157. A P P E N D I X C NOTATION

158. A P P E N D I X D GLOSSARY

159. INDEX