Optimization Methods in Finance Gerard Cornuejols Reha T ¨ ut ¨ unc ¨ u Carnegie Mellon University, Pittsburgh, PA 15213 USA January 2006 2 Foreword Optimization models play an increasingly important role in financial de- cisions. Many computational finance problems ranging from asset allocation to risk management, from option pricing to model calibration can be solved efficiently using modern optimization techniques. This course discusses sev- eral classes of optimization problems (including linear, quadratic, integer, dynamic, stochastic, conic, and robust programming) encountered in finan- cial models. For each problem class, after introducing the relevant theory (optimality conditions, duality, etc.) and efficient solution methods, we dis- cuss several problems of mathematical finance that can be modeled within this problem class. In addition to classical and well-known models such as Markowitz’ mean-variance optimization model we present some newer optimization models for a variety of financial problems. Acknowledgements This book has its origins in courses taught at Carnegie Mellon University in the Masters program in Computational Finance and in the MBA program at the Tepper School of Business (G´erard Cornu´ejols), and at the Tokyo In- stitute of Technology, Japan, and the University of Coimbra, Portugal (Reha T¨ut¨unc¨u). We thank the attendants of these courses for their feedback and for many stimulating discussions. We would also like to thank the colleagues who provided the initial impetus for this project, especially Michael Trick, John Hooker, Sanjay Srivastava, Rick Green, Yanjun Li, Lu´ıs Vicente and Masakazu Kojima. Various drafts of this book were experimented with in class by Javier Pe˜na, Fran¸cois Margot, Miroslav Karamanov and Kathie Cameron, and we thank them for their comments. Contents 1 Introduction 9 1.1 Optimization Problems . . . . . . . . . . . . . . . . . . . . . . 9 1.1.1 Linear and Nonlinear Programming . . . . . . . . . . 10 1.1.2 Quadratic Programming . . . . . . . . . . . . . . . . . 11 1.1.3 Conic Optimization . . . . . . . . . . . . . . . . . . . 12 1.1.4 Integer Programming . . . . . . . . . . . . . . . . . . 12 1.1.5 Dynamic Programming . . . . . . . . . . . . . . . . . 13 1.2 Optimization with Data Uncertainty . . . . . . . . . . . . . . 13 1.2.1 Stochastic Programming . . . . . . . . . . . . . . . . . 13 1.2.2 Robust Optimization . . . . . . . . . . . . . . . . . . . 14 1.3 Financial Mathematics . . . . . . . . . . . . . . . . . . . . . . 16 1.3.1 Portfolio Selection and Asset Allocation . . . . . . . . 16 1.3.2 Pricing and Hedging of Options . . . . . . . . . . . . . 18 1.3.3 Risk Management . . . . . . . . . . . . . . . . . . . . 19 1.3.4 Asset/Liability Management . . . . . . . . . . . . . . 20 2 Linear Programming: Theory and Algorithms 23 2.1 The Linear Programming Problem . . . . . . . . . . . . . . . 23 2.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . 28 2.4 The Simplex Method . . . . . . . . . . . . . . . . . . . . . . . 31 2.4.1 Basic Solutions . . . . . . . . . . . . . . . . . . . . . . 32 2.4.2 Simplex Iterations . . . . . . . . . . . . . . . . . . . . 35 2.4.3 The Tableau Form of the Simplex Method . . . . . . . 39 2.4.4 Graphical Interpretation . . . . . . . . . . . . . . . . . 42 2.4.5 The Dual Simplex Method . . . . . . . . . . . . . . . 43 2.4.6 Alternatives to the Simplex Method . . . . . . . . . . 45 3 LP Models: Asset/Liability Cash Flow Matching 47 3.1 Short Term Financing . . . . . . . . . . . . . . . . . . . . . . 47 3.1.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.1.2 Solving the Model with SOLVER . . . . . . . . . . . . 50 3.1.3 Interpreting the output of SOLVER . . . . . . . . . . 53 3.1.4 Modeling Languages . . . . . . . . . . . . . . . . . . . 54 3.1.5 Features of Linear Programs . . . . . . . . . . . . . . 55 3.2 Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.3 Sensitivity Analysis for Linear Programming . . . . . . . . . 58 3 4 CONTENTS 3.3.1 Short Term Financing . . . . . . . . . . . . . . . . . . 58 3.3.2 Dedication . . . . . . . . . . . . . . . . . . . . . . . . 63 3.4 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4 LP Models: Asset Pricing and Arbitrage 69 4.1 The Fundamental Theorem of Asset Pricing . . . . . . . . . . 69 4.1.1 Replication . . . . . . . . . . . . . . . . . . . . . . . . 71 4.1.2 Risk-Neutral Probabilities . . . . . . . . . . . . . . . . 72 4.1.3 The Fundamental Theorem of Asset Pricing . . . . . . 74 4.2 Arbitrage Detection Using Linear Programming . . . . . . . . 75 4.3 Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . 78 4.4 Case Study: Tax Clientele Effects in Bond Portfolio Manage- ment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5 Nonlinear Programming: Theory and Algorithms 85 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.3 Univariate Optimization . . . . . . . . . . . . . . . . . . . . . 88 5.3.1 Binary search . . . . . . . . . . . . . . . . . . . . . . . 88 5.3.2 Newton’s Method . . . . . . . . . . . . . . . . . . . . . 92 5.3.3 Approximate Line Search . . . . . . . . . . . . . . . . 95 5.4 Unconstrained Optimization . . . . . . . . . . . . . . . . . . . 97 5.4.1 Steepest Descent . . . . . . . . . . . . . . . . . . . . . 97 5.4.2 Newton’s Method . . . . . . . . . . . . . . . . . . . . . 101 5.5 Constrained Optimization . . . . . . . . . . . . . . . . . . . . 104 5.5.1 The generalized reduced gradient method . . . . . . . 107 5.5.2 Sequential Quadratic Programming . . . . . . . . . . . 112 5.6 Nonsmooth Optimization: Subgradient Methods . . . . . . . 113 6 NLP Models: Volatility Estimation 115 6.1 Volatility Estimation with GARCH Models . . . . . . . . . . 115 6.2 Estimating a Volatility Surface . . . . . . . . . . . . . . . . . 119 7 Quadratic Programming: Theory and Algorithms 125 7.1 The Quadratic Programming Problem . . . . . . . . . . . . . 125 7.2 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . 126 7.3 Interior-Point Methods . . . . . . . . . . . . . . . . . . . . . . 128 7.4 The Central Path . . . . . . . . . . . . . . . . . . . . . . . . . 131 7.5 Interior-Point Methods . . . . . . . . . . . . . . . . . . . . . . 132 7.5.1 Path-Following Algorithms . . . . . . . . . . . . . . . 132 7.5.2 Centered Newton directions . . . . . . . . . . . . . . . 133 7.5.3 Neighborhoods of the Central Path . . . . . . . . . . . 135 7.5.4 A Long-Step Path-Following Algorithm . . . . . . . . 138 7.5.5 Starting from an Infeasible Point . . . . . . . . . . . . 138 7.6 QP software . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 7.7 Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . 139 CONTENTS 5 8 QP Models: Portfolio Optimization 141 8.1 Mean-Variance Optimization . . . . . . . . . . . . . . . . . . 141 8.1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 143 8.1.2 Large-Scale Portfolio Optimization . . . . . . . . . . . 148 8.1.3 The Black-Litterman Model . . . . . . . . . . . . . . . 151 8.1.4 Mean-Absolute Deviation to Estimate Risk . . . . . . 155 8.2 Maximizing the Sharpe Ratio . . . . . . . . . . . . . . . . . . 158 8.3 Returns-Based Style Analysis . . . . . . . . . . . . . . . . . . 160 8.4 Recovering Risk-Neural Probabilities from Options Prices . . 162 8.5 Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . 166 8.6 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 9 Conic Optimization Tools 171 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 9.2 Second-order cone programming: . . . . . . . . . . . . . . . . 171 9.2.1 Ellipsoidal Uncertainty for Linear Constraints . . . . . 173 9.2.2 Conversion of quadratic constraints into second-order cone constraints . . . . . . . . . . . . . . . . . . . . . 175 9.3 Semidefinite programming: . . . . . . . . . . . . . . . . . . . 176 9.3.1 Ellipsoidal Uncertainty for Quadratic Constraints . . . 178 9.4 Algorithms and Software . . . . . . . . . . . . . . . . . . . . . 179 10 Conic Optimization Models in Finance 181 10.1 Tracking Error and Volatility Constraints . . . . . . . . . . . 181 10.2 Approximating Covariance Matrices . . . . . . . . . . . . . . 184 10.3 Recovering Risk-Neural Probabilities from Options Prices . . 187 10.4 Arbitrage Bounds for Forward Start Options . . . . . . . . . 189 10.4.1 A Semi-Static Hedge . . . . . . . . . . . . . . . . . . . 190 11 Integer Programming: Theory and Algorithms 195 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 11.2 Modeling Logical Conditions . . . . . . . . . . . . . . . . . . 196 11.3 Solving Mixed Integer Linear Programs . . . . . . . . . . . . 199 11.3.1 Linear Programming Relaxation . . . . . . . . . . . . 199 11.3.2 Branch and Bound . . . . . . . . . . . . . . . . . . . . 200 11.3.3 Cutting Planes . . . . . . . . . . . . . . . . . . . . . . 208 11.3.4 Branch and Cut . . . . . . . . . . . . . . . . . . . . . 212 12 IP Models: Constructing an Index Fund 215 12.1 Combinatorial Auctions . . . . . . . . . . . . . . . . . . . . . 215 12.2 The Lockbox Problem . . . . . . . . . . . . . . . . . . . . . . 216 12.3 Constructing an Index Fund . . . . . . . . . . . . . . . . . . . 219 12.3.1 A Large-Scale Deterministic Model . . . . . . . . . . . 220 12.3.2 A Linear Programming Model . . . . . . . . . . . . . 223 12.4 Portfolio Optimization with Minimum Transaction Levels . . 224 12.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 12.6 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 6 CONTENTS 13 Dynamic Programming Methods 227 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 13.1.1 Backward Recursion . . . . . . . . . . . . . . . . . . . 230 13.1.2 Forward Recursion . . . . . . . . . . . . . . . . . . . . 233 13.2 Abstraction of the Dynamic Programming Approach . . . . . 234 13.3 The Knapsack Problem. . . . . . . . . . . . . . . . . . . . . . 237 13.3.1 Dynamic Programming Formulation . . . . . . . . . . 237 13.3.2 An Alternative Formulation . . . . . . . . . . . . . . . 238 13.4 Stochastic Dynamic Programming . . . . . . . . . . . . . . . 239 14 DP Models: Option Pricing 241 14.1 A Model for American Options . . . . . . . . . . . . . . . . . 241 14.2 Binomial Lattice . . . . . . . . . . . . . . . . . . . . . . . . . 243 14.2.1 Specifying the parameters . . . . . . . . . . . . . . . . 244 14.2.2 Option Pricing . . . . . . . . . . . . . . . . . . . . . . 245 15 DP Models: Structuring Asset Backed Securities 249 15.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 15.2 Enumerating possible tranches . . . . . . . . . . . . . . . . . 253 15.3 A Dynamic Programming Approach . . . . . . . . . . . . . . 254 15.4 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 16 Stochastic Programming: Theory and Algorithms 257 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 16.2 Two Stage Problems with Recourse . . . . . . . . . . . . . . . 258 16.3 Multi Stage Problems . . . . . . . . . . . . . . . . . . . . . . 260 16.4 Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 262 16.5 Scenario Generation . . . . . . . . . . . . . . . . . . . . . . . 265 16.5.1 Autoregressive model . . . . . . . . . . . . . . . . . . 265 16.5.2 Constructing scenario trees . . . . . . . . . . . . . . . 267 17 SP Models: Value-at-Risk 273 17.1 Risk Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 273 17.2 Minimizing CVaR . . . . . . . . . . . . . . . . . . . . . . . . 276 17.3 Example: Bond Portfolio Optimization . . . . . . . . . . . . . 278 18 SP Models: Asset/Liability Management 281 18.1 Asset/Liability Management . . . . . . . . . . . . . . . . . . . 281 18.1.1 Corporate Debt Management . . . . . . . . . . . . . . 284 18.2 Synthetic Options . . . . . . . . . . . . . . . . . . . . . . . . 287 18.3 Case Study: Option Pricing with Transaction Costs . . . . . 290 18.3.1 The Standard Problem . . . . . . . . . . . . . . . . . . 291 18.3.2 Transaction Costs . . . . . . . . . . . . . . . . . . . . 292 19 Robust Optimization: Theory and Tools 295 19.1 Introduction to Robust Optimization . . . . . . . . . . . . . . 295 19.2 Uncertainty Sets . . . . . . . . . . . . . . . . . . . . . . . . . 296 19.3 Different Flavors of Robustness . . . . . . . . . . . . . . . . . 298 CONTENTS 7 19.3.1 Constraint Robustness . . . . . . . . . . . . . . . . . . 298 19.3.2 Objective Robustness . . . . . . . . . . . . . . . . . . 299 19.3.3 Relative Robustness . . . . . . . . . . . . . . . . . . . 301 19.3.4 Adjustable Robust Optimization . . . . . . . . . . . . 303 19.4 Tools and Strategies for Robust Optimization . . . . . . . . . 304 19.4.1 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . 305 19.4.2 Conic Optimization . . . . . . . . . . . . . . . . . . . 305 19.4.3 Saddle-Point Characterizations . . . . . . . . . . . . . 307 20 Robust Optimization Models in Finance 309 20.1 Robust Multi-Period Portfolio Selection . . . . . . . . . . . . 309 20.2 Robust Profit Opportunities in Risky Portfolios . . . . . . . . 313 20.3 Robust Portfolio Selection . . . . . . . . . . . . . . . . . . . . 315 20.4 Relative Robustness in Portfolio Selection . . . . . . . . . . . 317 20.5 Moment Bounds for Option Prices . . . . . . . . . . . . . . . 319 20.6 Additional Exercises . . . . . . . . . . . . . . . . . . . . . . . 320 A Convexity 323 B Cones 325 C A Probability Primer 327 D The Revised Simplex Method 331 8 CONTENTS Chapter 1 Introduction Optimization is a branch of applied mathematics that derives its importance both from the wide variety of its applications and from the availability of efficient algorithms. Mathematically, it refers to the minimization (or max- imization) of a given objective function of several decision variables that satisfy functional constraints. A typical optimization model addresses the allocation of scarce resources among possible alternative uses in order to maximize an objective function such as total profit. Decision variables, the objective function, and constraints are three es- sential elements of any optimization problem. Problems that lack constraints are called unconstrained optimization problems, while others are often re- ferred to as constrained optimization problems. Problems with no objective functions are called feasibility problems. Some problems may have multiple objective functions. These problems are often addressed by reducing them to a single-objective optimization problem or a sequence of such problems. If the decision variables in an optimization problem are restricted to integers, or to a discrete set of possibilities, we have an integer or discrete optimization problem. If there are no such restrictions on the variables, the problem is a continuous optimization problem. Of course, some problems may have a mixture of discrete and continuous variables. We continue with a list of problem classes that we will encounter in this book. 1.1 Optimization Problems We start with a generic description of an optimization problem. Given a function f (x) : IR n → IR and a set S ⊂ IR n , the problem of finding an x ∗ ∈ IR n that solves min x f(x) s.t. x ∈ S (1.1) is called an optimization problem. We refer to f as the objective function and to S as the feasible region. If S is empty, the problem is called infeasible. If it is possible to find a sequence x k ∈ S such that f(x k ) → −∞ as k → +∞, then the problem is unbounded. If the problem is neither infeasible nor 9 [...]... fundamental optimization problems is the linear optimization, or linear programming (LP) problem LP is the problem of optimizing a linear objective function subject to linear equality and inequality constraints A generic linear optimization problem has the following form: minx cT x aT x = bi , i ∈ E i aT x ≥ bi , i ∈ I, i (2.1) where E and I are the index sets for equality and inequality constraints, respectively... the Optimization Tree available from http://www-fp.mcs.anl.gov/otc/Guide/OptWeb/ 1.1.1 Linear and Nonlinear Programming One of the most common and easiest optimization problems is linear optimization or linear programming (LP) It is the problem of optimizing a linear objective function subject to linear equality and inequality constraints This corresponds to the case where the functions f and gi in. .. Assume as above that U is the uncertainty set that contains all possible values of the uncertain parameters p Then, an objective robust solution is obtained by solving: minx∈S maxp∈U f (x, p) (1.14) Note that objective robustness is a special case of constraint robustness Indeed, by introducing a new variable t (to be minimized) into (1.13) and imposing the constraint f (x, p) ≤ t, we get an equivalent... period The constraints indicate that the surplus left after liability 1.3 FINANCIAL MATHEMATICS 21 Lt is covered will be invested as follows: xi,t invested in asset class i In this formulation, xi,0 are the fixed, and possibly nonzero initial positions in the different asset classes 22 CHAPTER 1 INTRODUCTION Chapter 2 Linear Programming: Theory and Algorithms 2.1 The Linear Programming Problem One of... constraints and minimizes the objective value among all feasible vectors Similar definitions apply to nonlinear programming problems The best known and most successful methods for solving LPs are the simplex method and interior-point methods NLPs can be solved using gradient search techniques as well as approaches based on Newton’s method such as interior-point and sequential quadratic programming methods. .. OPTIMIZATION WITH DATA UNCERTAINTY 13 When there are both continuous variables and integer constrained variables, the problem is called a mixed integer linear program (MILP): minx cT x Ax ≥ b x ≥ 0 xj ∈ I for j = 1, , p N (1.7) where A, b, c are given data and the integer p (with 1 ≤ p < n) is also part of the input 1.1.5 Dynamic Programming Dynamic programming refers to a computational method involving... be incorporated into the model simply by removing the nonnegativity constraint on the corresponding variables If regulations or investor preferences limit the amount of investment in a subset of the securities, the model can be augmented with a linear constraint to reflect such a limit In principle, any linear constraint can be added to the model without making it significantly harder to solve Asset allocation... uncertain parameters 1.2 OPTIMIZATION WITH DATA UNCERTAINTY 15 There are different definitions and interpretations of robustness and the resulting models differ accordingly One important concept is constraint robustness, often called model robustness in the literature This refers to solutions that remain feasible for all possible values of the uncertain inputs This type of solution is required in several... explained by some probability distribution Robust optimization is used when one wants a solution that behaves well in all possible realizations of the uncertain data These two alternative approaches are not problem classes (as in LP, QP, etc.) but rather modeling techniques for addressing data uncertainty 1.2.1 Stochastic Programming The term stochastic programming refers to an optimization problem in. .. (1.13) 16 CHAPTER 1 INTRODUCTION The constraint robust formulation of the resulting problem is equivalent to (1.14) Constraint robustness and objective robustness are concepts that arise in conservative decision making and are not always appropriate for optimization problems with data uncertainty 1.3 Financial Mathematics Modern finance has become increasingly technical, requiring the use of sophisticated . of optimization problems (including linear, quadratic, integer, dynamic, stochastic, conic, and robust programming) encountered in finan- cial models. For each problem class, after introducing. constraint robustness. Indeed, by introducing a new variable t (to be minimized) into (1.13) and imposing the constraint f(x, p) ≤ t, we get an equivalent problem to (1.13). 16 CHAPTER 1. INTRODUCTION The. and interior-point methods. NLPs can be solved using gradient search techniques as well as approaches based on Newton’s method such as interior-point and sequential quadratic programming methods. 1.1.2