By Igor V. Evstigneev, Thorsten Hens and Klaus Reiner Schenk-Hoppé
Contents
1 Portfolio Selection: Introductory Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Asset Prices and Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Investor’s Portfolio: Long and Short Positions . . . . . . . . . . . . . . . . . . . . . 4
1.3 Return on a Portfolio .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Mathematical Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Mean-Variance Portfolio Analysis: The Markowitz Model . . . . . . . . . . . . 11
2.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Optimization Problem: Formulation and Discussion . . . . . . . . . . . . . . . 13
2.3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Efficient Portfolios and Efficient Frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Solution to the Markowitz Optimization Problem . . . . . . . . . . . . . . . . . . . . . . 19
3.1 Statement of the Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Proof of the Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Properties of Efficient Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.1 Mean and Variance of the Return on an Efficient Portfolio . . . . . . . . 27
4.2 Description of the Efficient Frontier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 A Fund Separation Theorem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5 The Markowitz Model with a Risk-Free Asset . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.1 Data of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.2 Portfolio Optimization with a Risk-Free Asset . . . . . . . . . . . . . . . . . . . . . 36
5.3 Solution to the Portfolio Selection Problem . . . . . . . . . . . . . . . . . . . . . . . . 38
6 Efficient Portfolios in a Market with a Risk-Free Asset . . . . . . . . . . . . . . . . 43
6.1 Expectations and Variances of Portfolio Returns . . . . . . . . . . . . . . . . . . . 43
6.2 Efficient Frontier and the Capital Market Line . . . . . . . . . . . . . . . . . . . . . 44
6.3 Tangency Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.4 A Mutual Fund Theorem.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7 Capital Asset Pricing Model (CAPM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.1 A General Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.2 An Equilibrium Approach to the CAPM. . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7.3 The Sharpe-Lintner-Mossin Formula.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
8 CAPM Continued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
8.1 Security Market Line and the Pricing Formula . . . . . . . . . . . . . . . . . . . . . 61
8.2 CAPM as a Factor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
8.3 Applying Theory to Practice: Sharpe’s and Jensen’s Tests . . . . . . . . . 64
9 FactorModels and the Ross-Huberman APT . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
9.1 Single- and Multi-Factor Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
9.2 Exact Factor Pricing .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
9.3 Ross-Huberman APT: Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . 76
9.4 Formulation and Proof of the Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . 78
10 Problems and Exercises I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Part II Derivative Securities Pricing
11 Dynamic Securities Market Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
11.1 Multi-Period Model of an Asset Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
11.2 Basic Securities and Derivative Securities . . . . . . . . . . . . . . . . . . . . . . . . . . 108
11.3 No-Arbitrage Pricing: Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
11.4 The No-Arbitrage Hypothesis and Net Present Value . . . . . . . . . . . . . . 112
12 Risk-Neutral Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
12.1 Risk-Neutral Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
12.2 Fundamental Theorem of Asset Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
12.3 Asset Pricing in Complete Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
13 The Cox–Ross–Rubinstein Binomial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
13.1 The Structure of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
13.2 Completeness of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
13.3 Constructing a Risk-Neutral Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
13.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
14 American Derivative Securities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
14.1 The Notion of an American Derivative Security . . . . . . . . . . . . . . . . . . . . 137
14.2 Risk-Neutral Pricing of American Derivative Securities . . . . . . . . . . . 139
14.3 The Pricing Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
15 From Binomial Model to Black–Scholes Formula . . . . . . . . . . . . . . . . . . . . . . 145
15.1 Drift and Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
15.2 Modelling the Price Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
15.3 Binomial Approximation of the Price Process . . . . . . . . . . . . . . . . . . . . . . 147
15.4 Derivation of the Black–Scholes Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 150
16 Problems and Exercises II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Part III Growth and Equilibrium
17 Capital Growth Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
17.1 Growth-Optimal Investments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
17.2 Strategies in Terms of Investment Proportions.. . . . . . . . . . . . . . . . . . . . . 171
17.3 Results for Simple Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
18 Capital Growth Theory: Continued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
18.1 Log-Optimal Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
18.2 Growth-Optimal and Numeraire Strategies . . . . . . . . . . . . . . . . . . . . . . . . . 179
18.3 Growth-Optimality for General Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . 180
18.4 Volatility-Induced Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
19 General Equilibrium Analysis of Financial Markets . . . . . . . . . . . . . . . . . . . 187
19.1 Walrasian Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
19.2 On the Existence of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
19.3 Rational Expectations and Equilibrium Pricing.. . . . . . . . . . . . . . . . . . . . 192
19.4 Arbitrage and Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
20 Behavioral Equilibrium and Evolutionary Dynamics . . . . . . . . . . . . . . . . . . 197
20.1 A Behavioral Evolutionary Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
20.2 Survival Strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
20.3 Links to the Classical Theory .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
21 Problems and Exercises III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Mathematical Appendices
A Facts from Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
B Convexity and Optimization.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Preface
This textbook is a basic introduction to the key topics in mathematical finance and financial economics—two realms of ideas that substantially overlap but are often treated separately from each other. Our goal is to present the highlights in the field, with the emphasis on the financial and economic content of the models, concepts and results. The book provides a novel, unified treatment of the subject by deriving each topic from common fundamental principles and showing the interrelations between the key themes.
Although our presentation is fully rigorous, with some rare and clearly marked exceptions, we restrict ourselves to the use of only elementary mathematical concepts and techniques. No advanced mathematics (such as stochastic calculus) is used. The main source for the book, and a “proving ground” for testing our presentation of the material, are courses on mathematical finance, financial economics and risk management which we have delivered, over the last decade, to undergraduate and graduate students in economics and finance at the Universities of Manchester, Zurich and Leeds.
The textbook contains 18 chapters corresponding to 18 lectures in a course based upon it. There are three chapters with problems and exercises, most of which have been used in tutorials, take-home tests and examinations, with full and detailed answers. The problems and exercises contain not only numerical examples, but also theoretical questions that complement the material presented in the body of the textbook. Two mathematical appendices provide rigorous definitions of some of the mathematical notions and statements of general theorems used in the text.
The textbook covers the classical topics, such asmean-variance portfolio analysis (Markowitz, CAPM, factor models, the Ross-Huberman APT), derivative securities pricing, and general equilibrium models of asset markets (Arrow, Debreu and Radner). A less standard but very important topic, which to our knowledge has not previously been covered in elementary textbooks, is capital growth theory (Kelly, Breiman, Cover and others). Absolutely new material, reflecting research achievements of recent years, is an introduction to new dynamic equilibrium models of financial markets combining behavioral and evolutionary principles.
A characteristic feature of financial economics is that it has to focus on the analysis of random, unpredictable market situations. To model such situations our discipline created powerful theoretical tools based on probability and stochastic processes. However, the power of human mind is not unlimited, and it can never fully eliminate the influence of chance and fortune, personified by goddess Tyche, looking at us from the epigraph to this book.