Contents
I Introduction to Finance and the Mathematics of Continuous-Time Models xv
1 Modern Finance 1
2 Introduction to Portfolio Selection and Capital Market Theory: Static
Analysis 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 One-Period Portfolio Selection . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Risk Measures for Securities and Portfolios in the One-Period Model 20
2.4 Spanning, Separation, and Mutual-Fund Theorems . . . . . . . . . . . 26
3 On the Mathematics and Economics Assumptions of Continuous-Time
Models 45
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Continuous-Sample-Path Processes with “No Rare Events” . . . . . . 52
3.3 Continuous-Sample-Path Processes with “Rare Events” . . . . . . . . 64
3.4 Discontinuous-Sample-Path Processes with “Rare Events” . . . . . . 68
II Optimum Consumption and Portfolio Selection in Continuous-Time Models 75
4 Lifetime Portfolio Selection Under Uncertainty: The Continuous-Time
Case 76
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2 Dynamics of the Model: The Budget Equation . . . . . . . . . . . . . 76
4.3 The Two-Asset Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4 Constant Relative Risk Aversion . . . . . . . . . . . . . . . . . . . . . 81
4.5 Dynamic Behavior and the Bequest Valuation Function . . . . . . . . 83
4.6 In?nite Time Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.7 Economic Interpretation of the Optimal Decision Rules for Portfolio
Selection and Consumption . . . . . . . . . . . . . . . . . . . . . . . . 86
4.8 Extension to Many Assets . . . . . . . . . . . . . . . . . . . . . . . . 90
4.9 Constant Absolute Risk Aversion . . . . . . . . . . . . . . . . . . . . 91
4.10 Other Extensions of the Model . . . . . . . . . . . . . . . . . . . . . . 92
5 Optimum Consumption and Portfolio Rules in a Continuous-timeModel 94
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2 A Digression on It ? o Processes . . . . . . . . . . . . . . . . . . . . . . 95
5.3 Asset-Price Dynamics and the Budget Equation . . . . . . . . . . . . 97
5.4 Optimal Portfolio and Consumption Rules: The Equations of Opti-
mality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.5 Log-Normality of Prices and the Continuous-Time Analog to Tobin-
Markowitz Mean-Variance Analysis . . . . . . . . . . . . . . . . . . . 103
5.6 Explicit Solutions for a Particular Class of Utility Functions . . . . . 107
5.7 Noncapital Gains Income: Wages . . . . . . . . . . . . . . . . . . . . 111
5.8 Poisson Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.9 Alternative Price Expectations to the Geometric Brownian Motion . . 117
5.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6 Further Developments in the Theory of Optimal Consumption and Port-
folio Selection 128
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.2 The Cox-Huang Alternative to Stochastic Dynamic Programming . . 130
6.2.1 The Growth-Optimum Portfolio Strategy . . . . . . . . . . . . 130
6.2.2 The Cox-Huang Solution of the Intertemporal Consumption-
Investment Problem . . . . . . . . . . . . . . . . . . . . . . . 133
6.2.3 The Relation Between the Cox-Huang and Dynamic Pro-
gramming Solutions . . . . . . . . . . . . . . . . . . . . . . . 139
6.3 Optimal Portfolio Rules when the Nonnegativity Constraint on Con-
sumption is Binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.4 Generalized Preferences and Their Impact on Optimal Portfolio De-
mands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
III Warrant and Option Pricing Theory 165
7 A Complete Model of Warrant Pricing that Maximizes Utility 166
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7.2 Cash-Stock Portfolio Analysis . . . . . . . . . . . . . . . . . . . . . . 166
7.3 Recapitulation of the 1965 Model . . . . . . . . . . . . . . . . . . . . 170
7.4 Determining Average Stock Yield . . . . . . . . . . . . . . . . . . . . 172
7.5 Determining Warrant Holdings and Prices . . . . . . . . . . . . . . . . 173
7.6 Digression: General Equilibrium Pricing . . . . . . . . . . . . . . . . 175
7.7 Utility-Maximizing Warrant Pricing: The Important “Incipient” Case 176
7.8 Explicit Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7.9 Warrants Never to be Converted . . . . . . . . . . . . . . . . . . . . . 180
7.10 Exact Solution to the Perpetual Warrant Case . . . . . . . . . . . . . . 182
7.11 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.12 Proof of the Superiority of Yield ofWarrants Over Yield of Common
Stock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
7.13 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
8 Theory of rational option pricing 196
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
8.2 Restrictions on Rational Option Pricing . . . . . . . . . . . . . . . . . 197
8.3 Effects of Dividends and Changing Exercise Price . . . . . . . . . . . 207
8.4 Restrictions on Rational Put Option Pricing . . . . . . . . . . . . . . . 213
8.5 Rational Option Pricing Along Black-Scholes Lines . . . . . . . . . . 216
8.6 An Alternative Derivation of the Black-Scholes Model . . . . . . . . 218
8.7 Extension of the Model to Include Dividend Payments And Exercise
Price Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
8.8 Valuing an American Put Option . . . . . . . . . . . . . . . . . . . . . 230
8.9 Valuing the “Down-and-Out” Call Option . . . . . . . . . . . . . . . . 232
8.10 Valuing a Callable Warrant . . . . . . . . . . . . . . . . . . . . . . . . 234
8.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
9 Option Pricing When Underlying Stock Returns are Discontinuous 239
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
9.2 The Stock Price and Option Price Dynamics . . . . . . . . . . . . . . 241
9.3 An Option Pricing Formula . . . . . . . . . . . . . . . . . . . . . . . . 246
9.4 A Possible Answer to an Empirical Puzzle . . . . . . . . . . . . . . . 251
10 Further Developments in Option Pricing Theory 256
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
10.2 Cox-Ross “Risk-Neutral” Pricing and the Binomial Option Pricing
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
10.3 Pricing Options on Futures Contracts . . . . . . . . . . . . . . . . . . 270
IV Contingent-Claims Analysis in the Theory of Corporate Finance and Finan-
cial Intermediation 277
11 A Dynamic General Equilibrium Model of the Asset Market and Its Ap-
plication to the Pricing of the Capital Structure of the Firm 278
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
11.2 A Partial-Equilibrium One-Period Model . . . . . . . . . . . . . . . . 278
11.3 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
11.4 A General Intertemporal Equilibrium Model of the Asset Market . . . 286
11.5 Model I: A Constant Interest Rate Assumption . . . . . . . . . . . . . 291
11.6 Model II: The “No Riskless Asset” Case . . . . . . . . . . . . . . . . 296
11.7 Model III: The General Model . . . . . . . . . . . . . . . . . . . . . . 297