Modelling Financial Data And Portfolio Optimization Problems - Uliege.be
Faculté d’Economie, de Gestion et de Sciences Sociales Ecole d’Administration des Affaires Modelling Financial Data and Portfolio Optimization Problems Dissertation présentée par Michaël Schyns pour l’obtention du grade de Docteur en Sciences de Gestion Membres du jury: Pr. A. Corhay (Université de Liège) Pr. Y. Crama (Université de Liège) Pr. W.G. Hallerbach (Erasmus University) Pr. G. Hübner (Université de Liège) Pr. A.W.J. Kolen (Maastricht University) Pr. M. Roubens (Université de Liège) Promoteur: Pr. Y. Crama Université de Liège Année Académique 2000-2001
Acknowledgment I would like to thank all the people who helped me in one way or another to complete my dissertation. Since the beginning, people have accepted to share their knowledge and time with me: Pr. Y. Crama has been a dynamic and motivating scientific supervisor. Despite his many occupations, his time has never mattered to answer my queries. Pr. A. Kolen, who initiated the second part of the thesis, has agreed to be a member of my thesis comittee. As such, we often met to coordinate the different parts of the work. Before and after jumping on board of my thesis comittee, Pr. G. Hübner managed to spare time to discuss with me about the financial part of my thesis each time I asked. Pr. M. Roubens has invited me several times to present my work and to discuss it with his team. Pr. A. Corhay’s door has always stayed open for me. Either for my thesis or for many other reasons, he has listened to me and has helped me throughout. At home, my wife and my family have endured my ups and downs and have never stopped supporting me. Last but not least, Pr. R. Moors who has been more than my boss. Even if he thinks that helping me was part of his job (‘I’m paid for it’ as he puts it), I do not dare estimating the number of hours he spent listening to me, talking to me, cheering me up,. I thank also Professor W.G. Hallerbach who, together with Professors A. Corhay, Y. Crama, G. Hübner, A. Kolen and M. Roubens, have accepted to be on my Jury. I should not forget to mention Alexandre and Axelle who pleasantly agreed to do without me and play with their mother and grandparents to let me finish my work. i
Contents 1 General introduction 1 PART ONE: Simulated Annealing for a generalized mean-variance model 3 2 Simulated Annealing for a generalized mean-variance model 4 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Portfolio selection issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.2 The optimization model . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.3 Solution approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Simulated annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Simulated annealing for portfolio selection . . . . . . . . . . . . . . . . . . . 12 2.4.1 Generalities: How to handle constraints . . . . . . . . . . . . . . . . 12 2.4.2 Budget and return constraints . . . . . . . . . . . . . . . . . . . . . . 15 2.4.3 Maximum number of assets constraint . . . . . . . . . . . . . . . . . 17 2.4.4 Floor, ceiling and turnover constraints . . . . . . . . . . . . . . . . . 18 2.4.5 Trading constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4.6 Summary: Neighbor selection . . . . . . . . . . . . . . . . . . . . . . 20 2.5 Cooling schedule, stopping criterion and intensification . . . . . . . . . . . . 21 2.5.1 Cooling schedule and stopping criterion . . . . . . . . . . . . . . . . . 21 2.5.2 Intensification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.6 Computational experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.6.1 Environment and data . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.6.2 The Markowitz mean-variance model . . . . . . . . . . . . . . . . . . 24 2.6.3 Floor, ceiling and turnover constraints . . . . . . . . . . . . . . . . . 25 2.6.4 Trading constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.6.5 Maximum number of securities . . . . . . . . . . . . . . . . . . . . . 30 ii
iii 2.6.6 Complete model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 PART TWO: Optimization of a portfolio of options under VaR constraints 35 3 Introduction to Part Two 36 4 Financial concepts 43 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2 Financial securities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2.1 Stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2.2 Portfolio of stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2.3 Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2.4 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.2.5 Risk-free investment . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.3 Continuous compounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.4 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.4.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.4.2 State-prices and arbitrage . . . . . . . . . . . . . . . . . . . . . . . . 48 4.5 Option pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.5.1 Classical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.5.2 Binomial trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.5.3 Black-Scholes formula . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.6 Risk-neutral valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.6.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.7 Complete market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5 Modelling the future 5.1 A multi-period scenario approach . . . . . . . . . . . . . . . . . . . . . . . . 56 56 5.1.1 One-period multinomial model . . . . . . . . . . . . . . . . . . . . . . 56 5.1.2 Two-period multinomial model . . . . . . . . . . . . . . . . . . . . . 57 5.1.3 Interesting properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.1.4 Binomial tree vs multinomial tree . . . . . . . . . . . . . . . . . . . . 59 5.2 Empirical data and implied parameters . . . . . . . . . . . . . . . . . . . . . 62 5.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.2.2 The smile effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
iv 5.2.3 Risk free rate, index price and dividend yield . . . . . . . . . . . . . . 66 5.3 Probability density functions for index returns . . . . . . . . . . . . . . . . . 68 5.3.1 Subjective and risk-neutral probabilities . . . . . . . . . . . . . . . . 68 5.3.2 Normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.3.3 Theodossiou’s skewed distributions . . . . . . . . . . . . . . . . . . . 70 5.3.4 Fernandez and Steel’s skewed distributions . . . . . . . . . . . . . . . 71 5.3.5 Breeden, Litzenberger and Shimko’s implied distributions . . . . . . . 73 5.3.6 Rubinstein’s implied distribution . . . . . . . . . . . . . . . . . . . . 81 5.3.7 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.4 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.4.2 Monte-Carlo generator . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.4.3 A grid generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.4.4 Stratification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.4.5 Quality of the sample . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.4.6 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.5 Probability conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.5.2 From risk-neutral to consensus probabilities . . . . . . . . . . . . . . 90 5.5.3 From subjective to risk-neutral probabilities . . . . . . . . . . . . . . 93 5.5.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.6 Numerical results and conclusions . . . . . . . . . . . . . . . . . . . . . . . . 99 6 Modelling option prices 103 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.2.2 European and American options . . . . . . . . . . . . . . . . . . . . . 105 6.2.3 Strike price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.2.4 Expiration date . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6.2.5 Settlement price and date . . . . . . . . . . . . . . . . . . . . . . . . 107 6.2.6 Intrinsic, time and volatility values . . . . . . . . . . . . . . . . . . . 108 6.2.7 Ask and bid prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.2.8 Size of a contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.2.9 Market limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
v 6.2.10 Commissions and taxes . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.3 Option pricing models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.3.2 Arbitrage equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.3.3 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.3.4 First improvement: absolute relative objective function . . . . . . . . . . . . . . . . . . . 112 6.3.5 Second improvement: bid and ask prices . . . . . . . . . . . . . . . . 113 6.3.6 Third improvement: parity equations . . . . . . . . . . . . . . . . . . 116 6.3.7 Fourth improvement: state-prices . . . . . . . . . . . . . . . . . . . . 117 6.4 Option pricing optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.4.2 Optimization over one period . . . . . . . . . . . . . . . . . . . . . . 118 6.4.3 Optimization over two periods . . . . . . . . . . . . . . . . . . . . . . 118 6.4.4 Simulated Annealing algorithm . . . . . . . . . . . . . . . . . . . . . 120 6.4.5 Arbitrage and numerical instabilities . . . . . . . . . . . . . . . . . . 121 6.5 Target option prices and probabilities . . . . . . . . . . . . . . . . . . . . . . 121 6.5.1 First and second periods . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.5.2 Improved Black and Scholes formula for the target option prices . . . 122 6.5.3 Target option prices from state-prices . . . . . . . . . . . . . . . . . . 123 6.6 Pre and post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.6.1 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.6.2 Option cleaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.6.3 Selection of options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.7 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.7.2 Option cleaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.7.3 Mean deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.7.4 Smile effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.7.5 Density functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7 Modelling Value-at-Risk constraints 138 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 7.2 The portfolio optimization problem . . . . . . . . . . . . . . . . . . . . . . . 138 7.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
vi 7.2.2 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 7.2.3 Time of creation and maturity of the options . . . . . . . . . . . . . . 142 7.2.4 The costs of transaction . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.2.5 Other option features . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 7.2.6 The guarantee constraint . . . . . . . . . . . . . . . . . . . . . . . . . 145 7.2.7 The Value-at-Risk constraints . . . . . . . . . . . . . . . . . . . . . . 148 7.3 The mathematical programming model . . . . . . . . . . . . . . . . . . . . . 149 7.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.3.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.3.3 Computation of the bid-ask spread and costs of transaction . . . . . . 152 7.3.4 The budget constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 152 7.3.5 The guarantee constraint: first approach . . . . . . . . . . . . . . . . 154 7.3.6 The VaR constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.3.7 The objective function . . . . . . . . . . . . . . . . . . . . . . . . . . 156 7.3.8 Mathematical programming model M1 . . . . . . . . . . . . . . . . . 157 7.3.9 Mathematical programming model M2 . . . . . . . . . . . . . . . . . 159 7.3.10 The guarantee constraint: improved approach . . . . . . . . . . . . . 159 8 Handling Value-at-Risk constraints 163 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 8.2 Structure of the portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 8.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 8.2.2 Theoretical structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 8.2.3 Empirical structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 8.3 Optimal VaR allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 8.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 8.3.2 Relaxation of the VaR constraint . . . . . . . . . . . . . . . . . . . . 168 8.4 Optimal VaR allocation vs. investment strategies . . . . . . . . . . . . . . . 169 8.4.1 Trading strategies involving options . . . . . . . . . . . . . . . . . . . 169 8.4.2 Selection of a strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 172 8.4.3 One-period strategy vs. two-period strategy . . . . . . . . . . . . . . 173 8.5 Optimal VaR allocation vs. Dybvig’s theorem . . . . . . . . . . . . . . . . . 174 8.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 8.5.2 Dybvig’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 8.5.3 Validity of Dybvig’s hypotheses . . . . . . . . . . . . . . . . . . . . . 176
vii 8.5.4 Handling Dybvig’s hypotheses . . . . . . . . . . . . . . . . . . . . . . 178 8.5.5 Index values and Dybvig’s theorem . . . . . . . . . . . . . . . . . . . 181 8.5.6 Strategies and Dybvig’s theorem . . . . . . . . . . . . . . . . . . . . . 185 8.5.7 One-period trees vs. two-period tree . . . . . . . . . . . . . . . . . . 187 8.5.8 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 9 Solving Value-at-Risk problems 188 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 9.2 The branch and bound method . . . . . . . . . . . . . . . . . . . . . . . . . 188 9.3 Improvements of the BB process and heuristics . . . . . . . . . . . . . . . . . 191 9.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 9.3.2 Investment strategies 9.3.3 Uses of Dybvig’s theorem 9.3.4 Rounding approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 . . . . . . . . . . . . . . . . . . . . . . . . . . 191 . . . . . . . . . . . . . . . . . . . . . . . . 195 9.4 Preselection of options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 9.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 9.4.2 Constructing the “universe” of options . . . . . . . . . . . . . . . . . 200 9.4.3 Advanced selection of options . . . . . . . . . . . . . . . . . . . . . . 201 10 Computational experiments 203 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 10.2 The financial problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 10.2.1 Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 10.2.2 Investor’s decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 10.3 Computer environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 10.3.1 The software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 10.3.2 CPlex parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 10.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 10.4.1 Number of scenarios and pdfs . . . . . . . . . . . . . . . . . . . . . . 208 10.4.2 Computation time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 10.4.3 Structure of optimal portfolios . . . . . . . . . . . . . . . . . . . . . . 212 10.4.4 Initial bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 10.4.5 Options and index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 10.4.6 One-period vs two-period model . . . . . . . . . . . . . . . . . . . . . 222 10.4.7 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
viii 10.4.8 Financial variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 10.4.9 Normal pdf vs. implied pdf . . . . . . . . . . . . . . . . . . . . . . . 224 10.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 11 Conclusions 230 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 11.2 Handling portfolio selection problems . . . . . . . . . . . . . . . . . . . . . . 230 11.3 Main contributions of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . 232 11.4 Future developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 Appendix 236 Appendix A : list of stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Appendix B : software parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 Bibliography 244
Chapter 1 General introduction This doctoral dissertation in management science, entitled “Modelling Financial Data and Portfolio Optimization Problems”, consists of two independent parts, whose unifying theme is the construction and solution of mathematical programming models motivated by portfolio selection problems. As such, this work is located at the interface of operations research and of finance. It draws heavily on techniques and theoretical results originating in both disciplines. The first part of the dissertation (Chapter 2) deals with an extension of Markowitz model and takes into account some of the side-constraints faced by a decision-maker when composing an investment portfolio, viz. lower and upper bounds on the quantities traded, and upper bounds on the number of assets included in the portfolio. We focus on the algorithmic difficulties raised by this model and we describe an original simulated annealing heuristic for its solution. The second (and largest) part of the thesis deals with a new multiperiod model for the optimization of a portfolio of options linked to a single index (Chapters 4-10). The objective of the model is to maximize the expected return of the portfolio under constraints limiting its value-at-risk. The model contains several interesting features, like the possibility to rebalance the portfolio with options introduced at the start of each period, explicit consideration of transaction costs, realistic pricing of options, consideration of advanced probability models to represent the future, etc. Some deep theoretical results from the financial literature are exploited in order to enrich the model and to extend its applicability. In particular, several available schemes for the generation of scenarios and for option pricing have been critically examined, and the most appropriate ones have been implemented. Furthermore, several optimization approaches (heuristic or exact procedures) have also been developed, implemented and tested. The models investigated in the dissertation bear on very different portfolio problems, draw 1
Chapter 1. General introduction 2 on separate streams of scientific literature, and are handled by distinct algorithmic techniques. Therefore, the corresponding parts of the dissertation are fully independent, and each part contains its own specific introduction and literature review.
PART ONE: Simulated Annealing for a generalized mean-variance model
Chapter 2 Simulated Annealing for a generalized mean-variance model 2.1 Introduction Markowitz’ mean-variance model of portfolio selection is one of the best known models in finance. In its basic form, this model requires to determine the composition of a portfolio of assets which minimizes risk while achieving a predetermined level of expected return. The pioneering role played by this model in the development of modern portfolio theory is unanimously recognized (see e.g. [11] for a brief historical account). From a practical point of view, however, the Markowitz model may often be considered too basic, as it ignores many of the constraints faced by real-world investors: trading limitations, size of the portfolio, etc. Including such constraints in the formulation results in a nonlinear mixed integer programming problem which is considerably more difficult to solve than the original model. Several researchers have attempted to attack this problem by a variety of techniques (decomposition, cutting planes, interior point methods, .), but there appears to be room for much improvement on this front. In particular, exact solution methods fail to solve large-scale instances of the problem. Therefore, in this chapter, we investigate the ability of the simulated annealing metaheuristic (SA) to deliver high-quality solutions for the mean-variance model enriched by additional constraints. The remainder of this chapter is organized in six sections. Section 2 introduces the portfolio selection model that we want to solve. Section 3 sums up the basic structure of simulated annealing algorithms. Section 2.4 contains a detailed description of our algorithm. Here, we make an attempt to underline the difficulties encountered when tailoring the SA metaheuristic to the problem at hand. Notice, in particular, that our model involves continuous 4
Chapter 2. Simulated Annealing for a generalized mean-variance model 5 as well as discrete variables, contrary to most applications of simulated annealing. Also, the constraints are of various types and cannot be handled in a uniform way. In Section 5, we discuss some details of the implementation. Section 6 reports on computational experiments carried out on a sample of 151 US stocks. Finally, the last section contains a summary of our work and some conclusions. 2.2 2.2.1 Portfolio selection issues Generalities In order to handle portfolio selection problems in a formal framework, three types of questions (at least) must be explicitly addressed: 1. data modelling, in particular the behavior of asset returns; 2. the choice of the optimization model, including: the nature of the objective function; the constraints faced by the investor; 3. the choice of the optimization technique. Although our work focuses mostly on the third step, we briefly discuss the whole approach since all the steps are interconnected to some extent. The first requirement is to understand the nature of the data and to be able to correctly represent them. Markowitz’ model (described in the next section) assumes for instance that the asset returns follow a multivariate normal distribution. In particular, the first two moments of the distribution suffice to describe completely the distribution of the asset returns and the characteristics of the different portfolios. Real markets often exhibit more intricacies, with distributions of returns depending on moments of higher-order (skewness, kurtosis, etc.), and distribution parameters varying over time. Analyzing and modelling such complex financial data is a whole subject in itself, which we do not tackle here explicitly. We rather adopt the classical assumptions of the mean-variance approach, where (pointwise estimates of) the expected returns and the variance-covariance matrix are supposed to provide a satisfactory description of the asset returns. Also, we do not address the origin of the numerical data. Note that some authors rely for instance on factorial models of the asset returns, and take advantage of the properties of such models to improve the efficiency of the optimization
Chapter 2. Simulated Annealing for a generalized mean-variance model 6 techniques (see e.g. [2, 58]). By contrast, the techniques that we develop here do not depend on any specific properties of the data, so that some changes of the model (especially of the objective function) can be performed while preserving our main conclusions. When building an optimization model of portfolio selection, a second requirement consists in identifying the objective of the investor and the constraints that he is facing. As far as the objective goes, the quality of the portfolio could be measured using a wide variety of utility functions. Following again Markowitz’ model, we assume here that the investor is risk averse and wants to minimize the variance of the investment portfolio subject to the expected level of final wealth. It should be noted, however, that this assumption does not play a crucial role in our algorithmic developments, and that the objective could be replaced by a more general utility function without much impact on the optimization techniques that we propose. As far as the constraints of the model go, we are especially interested in two types of complex constraints limiting the number of assets included in the portfolio (thus reflecting some behavioral or institutional restrictions faced by the investor), and the minimal quantities which can be traded when rebalancing an existing portfolio (thus reflecting individual or market restrictions). This topic is covered in more detail in Section 2.2.2. The final ingredient of a portfolio selection method is an algorithmic technique for the optimization of the chosen model. This is the main topic of the present chapter. In view of the complexity of our model (due, to a large extent, to the constraints mentioned in the previous paragraph), and to the large size of realistic problem instances, we have chosen to work with a simulated annealing metaheuristic. An in-depth study has been performed to optimize the speed and the quality of the algorithmic process, and to analyze the impact of various parameter choices. In the remainder of this section, we return in more detail to the description of the model, and we briefly survey previous work on this and related models. 2.2.2 The optimization model The Markowitz mean-variance model The problem of optimally selecting a portfolio among n assets was formulated by Markowitz in 1952 as a constrained quadratic minimization problem (see [50], [26], [48]). In this model, each asset is characterized by a return varying randomly with time. The risk of each asset is measured by the variance of its return. If each component xi of the n-vector x represents the proportion of an investor’s wealth allocated to asset i, then the total return of the portfolio
Chapter 2. Simulated Annealing for a generalized mean-variance model 7 is given by the scalar product of x by the vector of individual asset returns. Therefore, if R (R1 , . . . , Rn ) denotes the n-vector of expected returns of the assets and C the n n covariance matrix of the returns, we obtain the mean portfolio return by the expression n i 1 Ri xi and its level of risk by n i 1 n j 1 Cij xi xj . Markowitz assumes that the aim of the investor is to design a portfolio which minimizes risk while achieving a predetermined expected return, say Rexp. Mathematically, the problem can be formulated as follows for any value of Rexp: n n Cij xi xj min (2.1) i 1 j 1 n s.t. Ri xi Rexp i 1 n xi 1 i 1 xi 0 for i 1, . . . , n. The first constraint expresses the requirement placed on expected return. The second constraint, called budget constraint, requires that 100% of the budget be invested in the portfolio. The nonnegativity constraints express that no short sales are allowed. The set of optimal solutions of the Markowitz model, parametrized over all possible values of Rexp, constitutes the mean-variance frontier of the portfolio selection problem. It is usually displayed as a curve in the plane where the ordinate is the expected portfolio return and the abscissa is its standard deviation. If the goal is to draw the whole frontier, an alternative form of the model can also be used where the constraint defining the required expected return is removed and a new weighted term representing the portfolio return is included in the objective function. Hereafter, we shall use the initial fo
Modelling Financial Data and Portfolio Optimization Problems Dissertation pr esent ee par Membres du jury: Micha el Schyns Pr. A. Corhay (Universit edeLi ege) pour l'obtention du grade de Pr. Y. Crama (Universit edeLi ege) Docteur en Sciences de Gestion Pr. W.G. Hallerbach (Erasmus University) Pr. G. H ubner (Universit edeLi ege)
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