Optimization Methods in Finance Optimization methods play a central role in financial modeling This textbook is devoted to explaining how state-of-the-art optimization theory, algorithms, and software can be used to efficiently solve problems in computational finance It discusses some classical mean–variance portfolio optimization models as well as more modern developments such as models for optimal trade execution and dynamic portfolio allocation with transaction costs and taxes Chapters discussing the theory and efficient solution methods for the main classes of optimization problems alternate with chapters discussing their use in the modeling and solution of central problems in mathematical finance This book will be interesting and useful for students, academics, and practitioners with a background in mathematics, operations research, or financial engineering The second edition includes new examples and exercises as well as a more detailed discussion of mean–variance optimization, multi-period models, and additional material to highlight the relevance to finance Gérard Cornuéjols is a Professor of Operations Research at the Tepper School of Business, Carnegie Mellon University He is a member of the National Academy of Engineering and has received numerous prizes for his research contributions in integer programming and combinatorial optimization, including the Lanchester Prize, the Fulkerson Prize, the Dantzig Prize, and the von Neumann Theory Prize Javier Peña is a Professor of Operations Research at the Tepper School of Business, Carnegie Mellon University His research explores the myriad of challenges associated with large-scale optimization models and he has published numerous articles on optimization, machine learning, financial engineering, and computational game theory His research has been supported by grants from the National Science Foundation, including a prestigious CAREER award Reha Tütüncü is the Chief Risk Officer at SECOR Asset Management and an adjunct professor at Carnegie Mellon University He has previously held senior positions at Goldman Sachs Asset Management and AQR Capital Management focusing on quantitative portfolio construction, equity portfolio management, and risk management Optimization Methods in Finance Second Edition GÉRARD CORNUÉJOLS Carnegie Mellon University, Pennsylvania J AV I E R P E Ñ A Carnegie Mellon University, Pennsylvania REHA TÜTÜNCÜ SECOR Asset Management University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence www.cambridge.org Information on this title: www.cambridge.org/9781107056749 DOI: 10.1017/9781107297340 First edition © Gérard Cornuéjols and Reha Tütüncü 2007 Second edition © Gérard Cornjols, Javier Pa and Reha Tütüncü 2018 This publication is in copyright Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published 2007 Second edition 2018 Printed and bound in Great Britain by Clays Ltd, Elcograf S.p.A A catalogue record for this publication is available from the British Library ISBN 978-1-107-05674-9 Hardback Additional resources for this publication at www.cambridge.org/9781107056749 Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate Contents Preface Part I Introduction page xi 1 Overview of Optimization Models 1.1 Types of Optimization Models 1.2 Solution to Optimization Problems 1.3 Financial Optimization Models 1.4 Notes 10 Linear Programming: Theory and Algorithms 2.1 Linear Programming 2.2 Graphical Interpretation of a Two-Variable Example 2.3 Numerical Linear Programming Solvers 2.4 Sensitivity Analysis 2.5 *Duality 2.6 *Optimality Conditions 2.7 *Algorithms for Linear Programming 2.8 Notes 2.9 Exercises 11 11 15 16 17 20 23 24 30 31 Linear Programming Models: Asset–Liability Management 3.1 Dedication 3.2 Sensitivity Analysis 3.3 Immunization 3.4 Some Practical Details about Bonds 3.5 Other Cash Flow Problems 3.6 Exercises 3.7 Case Study 35 35 38 38 41 44 47 51 Linear Programming Models: Arbitrage and Asset Pricing 4.1 Arbitrage Detection in the Foreign Exchange Market 4.2 The Fundamental Theorem of Asset Pricing 4.3 One-Period Binomial Pricing Model 53 53 55 56 vi Contents 4.4 4.5 4.6 4.7 Part II Static Arbitrage Bounds Tax Clientele Effects in Bond Portfolio Management Notes Exercises Single-Period Models 59 63 65 65 69 Quadratic Programming: Theory and Algorithms 5.1 Quadratic Programming 5.2 Numerical Quadratic Programming Solvers 5.3 Sensitivity Analysis 5.4 *Duality and Optimality Conditions 5.5 *Algorithms 5.6 Applications to Machine Learning 5.7 Exercises 71 71 74 75 76 81 84 87 Quadratic Programming Models: Mean–Variance Optimization 6.1 Portfolio Return 6.2 Markowitz Mean–Variance (Basic Model) 6.3 Analytical Solutions to Basic Mean–Variance Models 6.4 More General Mean–Variance Models 6.5 Portfolio Management Relative to a Benchmark 6.6 Estimation of Inputs to Mean–Variance Models 6.7 Performance Analysis 6.8 Notes 6.9 Exercises 6.10 Case Studies 90 90 91 95 99 103 106 112 115 115 121 Sensitivity of Mean–Variance Models to Input Estimation 7.1 Black–Litterman Model 7.2 Shrinkage Estimation 7.3 Resampled Efficiency 7.4 Robust Optimization 7.5 Other Diversification Approaches 7.6 Exercises 124 126 129 131 132 133 135 Mixed Integer Programming: Theory and Algorithms 8.1 Mixed Integer Programming 8.2 Numerical Mixed Integer Programming Solvers 8.3 Relaxations and Duality 8.4 Algorithms for Solving Mixed Integer Programs 8.5 Exercises 140 140 143 145 150 157 Contents vii Mixed Integer Programming Models: Portfolios with Combinatorial Constraints 9.1 Combinatorial Auctions 9.2 The Lockbox Problem 9.3 Constructing an Index Fund 9.4 Cardinality Constraints 9.5 Minimum Position Constraints 9.6 Risk-Parity Portfolios and Clustering 9.7 Exercises 9.8 Case Study 161 161 163 165 167 168 169 169 171 10 Stochastic Programming: Theory and Algorithms 10.1 Examples of Stochastic Optimization Models 10.2 Two-Stage Stochastic Optimization 10.3 Linear Two-Stage Stochastic Programming 10.4 Scenario Optimization 10.5 *The L-Shaped Method 10.6 Exercises 173 173 174 175 176 177 179 11 Stochastic Programming Models: Risk Measures 11.1 Risk Measures 11.2 A Key Property of CVaR 11.3 Portfolio Optimization with CVaR 11.4 Notes 11.5 Exercises 181 181 185 186 190 190 Part III Multi-Period Models 195 12 Multi-Period Models: Simple Examples 12.1 The Kelly Criterion 12.2 Dynamic Portfolio Optimization 12.3 Execution Costs 12.4 Exercises 197 197 198 201 209 13 Dynamic Programming: Theory and Algorithms 13.1 Some Examples 13.2 Model of a Sequential System (Deterministic Case) 13.3 Bellman’s Principle of Optimality 13.4 Linear–Quadratic Regulator 13.5 Sequential Decision Problem with Infinite Horizon 13.6 Linear–Quadratic Regulator with Infinite Horizon 13.7 Model of Sequential System (Stochastic Case) 13.8 Notes 13.9 Exercises 212 212 214 215 216 218 219 221 222 222 viii Contents 14 Dynamic Programming Models: Multi-Period Portfolio Optimization 14.1 Utility of Terminal Wealth 14.2 Optimal Consumption and Investment 14.3 Dynamic Trading with Predictable Returns and Transaction Costs 14.4 Dynamic Portfolio Optimization with Taxes 14.5 Exercises 225 225 227 228 230 234 15 Dynamic Programming Models: the Binomial Pricing Model 15.1 Binomial Lattice Model 15.2 Option Pricing 15.3 Option Pricing in Continuous Time 15.4 Specifying the Model Parameters 15.5 Exercises 238 238 238 244 245 246 16 Multi-Stage Stochastic Programming 16.1 Multi-Stage Stochastic Programming 16.2 Scenario Optimization 16.3 Scenario Generation 16.4 Exercises 248 248 250 255 259 17 Stochastic Programming Models: Asset–Liability Management 17.1 Asset–Liability Management 17.2 The Case of an Insurance Company 17.3 Option Pricing via Stochastic Programming 17.4 Synthetic Options 17.5 Exercises 262 262 263 265 270 273 Part IV Other Optimization Techniques 275 18 Conic Programming: Theory and Algorithms 18.1 Conic Programming 18.2 Numerical Conic Programming Solvers 18.3 Duality and Optimality Conditions 18.4 Algorithms 18.5 Notes 18.6 Exercises 277 277 282 282 284 287 287 19 Robust Optimization 19.1 Uncertainty Sets 19.2 Different Flavors of Robustness 19.3 Techniques for Solving Robust Optimization Models 19.4 Some Robust Optimization Models in Finance 19.5 Notes 19.6 Exercises 289 289 290 294 297 302 302 Contents 20 Nonlinear Programming: Theory and Algorithms 20.1 Nonlinear Programming 20.2 Numerical Nonlinear Programming Solvers 20.3 Optimality Conditions 20.4 Algorithms 20.5 Estimating a Volatility Surface 20.6 Exercises ix 305 305 306 306 308 315 319 Appendices 321 Appendix 323 323 324 325 Basic Mathematical Facts A.1 Matrices and Vectors A.2 Convex Sets and Convex Functions A.3 Calculus of Variations: the Euler Equation References 327 Index 334 324 Basic Mathematical Facts Then ⎤ x1 ⎢ ⎥ ⎣ ⎦ = c x + · · · + cn x n ⎡ ··· cT x = c cn xn and ⎡ q11 ⎢ Qx = ⎣ q1n ··· ··· ⎤⎡ ⎤ ⎡ ⎤ q1n x1 q11 x1 + · · · + q1n xn ⎥ ⎥ ⎢ ⎥ = ⎢ ⎦ ⎦⎣ ⎦ ⎣ qnn x3 q1n x1 + · · · + qnn xn So ⎤ q11 x1 + · · · + q1n xn ⎥ ⎢ ⎦ ⎣ ⎡ xT Qx = x1 ··· xn q1n x1 + · · · + qnn xn n = = n qij xi xj i=1 j=1 q11 x21 + · · · + qnn x2n + 2q12 x1 x2 + 2q23 x2 x3 + · · · + 2qn−1,n xn−1 xn A symmetric matrix M ∈ Rn×n is positive semidefinite if xT Mx ≥ for all x ∈ Rn and it is positive definite if it satisfies the stronger condition xT Mx > for all non-zero x ∈ Rn A.2 Convex Sets and Convex Functions A set S ⊆ Rn is convex if for all x, y ∈ S the straight segment joining x and y is contained in S; that is, [x, y] := {λx + (1 − λ)y : λ ∈ [0, 1]} ⊆ S The following are types of convex sets that appear often in optimization models It is easy to verify that they are indeed convex sets Half-space: Given a non-zero a ∈ Rn and b ∈ R the half-space {x ∈ Rn : aT x ≤ b} is convex Intersections of convex sets: Given a collection Si ⊆ Rn , for i ∈ I, of convex sets, their intersection i∈I Si is a convex set A.3 Calculus of Variations: the Euler Equation 325 Affine images and preimages: Given a convex set S ⊆ Rn , matrices A ∈ Rm×n , B ∈ Rn×p and vectors a ∈ Rm , b ∈ Rn , the sets A(S) + a = {Ax + a : x ∈ S} ⊆ Rm and B−1 (S + b) := {v ∈ Rp : Bv − b ∈ S} ⊆ Rp are convex Suppose S ⊆ Rn is a convex set A function f : S → R is convex if, for all x, y ∈ S and λ ∈ [0, 1], f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y) A common way of dealing with the domain of a function is to consider extended valued functions; that is, functions defined on the whole space Rn and allowed to take the value ∞ The domain of an extended valued function f : Rn → R ∪ {∞} is the set dom(f ) := {x ∈ Rn : f (x) < ∞} An alternative and equivalent definition of convexity is the following An extended valued function f : Rn → R ∪ {∞} is convex if the set epigraph(f ) := {(x, t) ∈ Rn+1 : f (x) ≤ t} is convex Observe that if f is convex, then its domain is a convex set Furthermore, if f is convex then for all ∈ R the sublevel set {x ∈ Rn : f (x) ≤ } is a convex set The following relationship between differentiability and convexity is particularly useful to verify that functions are convex Theorem A.1 Suppose f : S → R is twice differentiable on the open set S ⊆ Rn and C ⊆ S is a convex set Then f is 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Technical University of Mă unchen Ye Y (1997) Interior-Point Algorithms: Theory and Analysis Wiley Zhao Y and W.T Ziemba (2001) A stochastic programming model using an endogenously determined worst case risk measure for dynamic asset allocation Mathematical Programming, 89(2):293–309 Index accrued interest, 42 active constraint, see binding constraint active return, 105 active risk, 105 active set, 313 active-set methods, 81 adaptive decision, adjoint, 285 Almgren–Chriss model, 201 alpha Jensen, 113 t-statistic, 113 alpha of a security, 104 APT, 111 arbitrage, 55 arbitrage pricing theory, see APT Armijo–Goldstein condition, 309 asset allocation, 95 asset–liability management, 262 auction combinatorial, 161 autoregressive model, 256 backtracking, 309 Basel III, 15 basic feasible solution, 25 basis, 25 optimal, 25 Bellman’s optimality principle, 215, 216, 221 Benders decomposition, 255 Benders decomposition method, 177 bequest, 227 beta long–short threshold, 119 beta of a security, 104 bid, 161 bid–ask spread, 66 binary program, 140 binding constraint, binomial lattice, 238 binomial lattice model, 238 binomial pricing model, 56 Black–Litterman model, 126 Black–Scholes–Merton equation, 245 Black–Scholes–Merton option pricing formula, 316 blocking constraint, 82 bond clean price, 42 coupon rate, 42 dirty price, 42 maturity date, 42 term to maturity, 42 yield, 42 bond allocation, 12 bond portfolio dedicated, 35 boostrapping, 131 branch-and-bound method, 150, 151 branch-and-bound tree, 152 branch-and-cut method, 150 branching, 151 Brownian motion, 315 BSM formula, 316 bundle, 161 capital allocation line, 98 capital asset pricing model, see CAPM CAPM, 98, 108, 111, 112 captured value, 202 cash flow problems, 44 central path, 29, 83 clientele effects, 63 clustering, 142 combinatorial auction problem, 161 complementary slackness conditions, 24 conditional value at risk, see CVaR cone second-order, 277 symmetric, 287 conic program, 277 dual, 282 primal, 282 constraint turnover, 101 constraint set, consumption, 174 contingent claim, 55 Index convex function, 325 convex optimization, convex set, 324 cutting plane, 154 cutting-plane method, 150, 154 CVaR, 181, 184 decision variables, descent direction, 309 deterministic model, diffusion model, 316 dispersion measure, 181 diversification, 134 maximum, 135 drift, 244 dual cone, 282 dual problem, 21 efficient frontier, 93, 124 estimator inadmissable, 129 James–Stein shrinkage, 129 risk, 129 Euler equation, 206 event tree, 250 excess return, 102 factor exposure, see factor loading factor loading, 107 factor model, 106 factor portfolio, 120 Farkas’s lemma, 21, 34 feasibility cut, 178 feasible point, feasible region, feasible solution, see feasible point Fisher–Weil convexity, 40 Fisher–Weil dollar convexity, 40 Fisher–Weil dollar duration, 39 Frobenius inner product, 281 fund allocation linear programming model, 12 fundamental theorem of asset pricing, 56 geometric Brownian motion model, 244 Gomory mixed integer cut, 155 Gordan’s theorem, 22, 34 gradient descent method, 309 homogenization, 102 hyperplane separation theorem, 34 ice-cream cone, see cone, second-order immunization, 39 immunized portfolio, 39 implementation shortfall, see total cost of trading implied volatility, 273 index fund, 165 infeasible problem, information ratio, 105 335 insurance company ALM problem, 263 interior-point method, 28 infeasibility, 30 nonlinear programming, 313 quadratic program, 83 Jacobian, 310 Jensen’s alpha, 112 Kelly criterion, 197 L-shaped method, 177 Lagrange multiplier, 75 Lagrangian dual, 147 Lagrangian function, 22, 77 lasso regression, 87 line search, 309 line-search procedure, 30 linear independence constraint qualification, 307 linear optimization model, see linear programming model linear program, 11 linear programming, linear programming model, 11 non-degenerate, 19 standard form, 13 linear–quadratic regulator, 216 lockbox problem, 163 Lorenz cone, see cone, second-order loss function, 193 MAD, 181 market completeness, 59 Markowitz mean–variance, 91 Markowitz mean–variance model, master problem, 177 matrix inverse, 323 positive semidefinite, 5, 280 mean absolute deviation, see MAD mean–variance, 71, 296 stochastic optimization, 175 mean–variance model basic, 94 general, 100 mean–variance optimization model, 93 minimum position constraints, 168 mixed integer linear program, 140 mixed integer optimization, mixed integer program, 140 mixed integer programming, multi-period model, multiple-factor risk model, 109 newsvendor problem, 173, 175, 177 Newton step, 29, 286 Newton’s method, 310 nonlinear program, 305 NP-hardness, 150 336 Index objective function, one-fund separation theorem, 97 optimal decision rule, 216 optimal policy, 216 optimal solution, optimal value, optimality conditions, 24 optimality cut, 178, 180 optimization robust, 289 option American, 241 European, 239 option pricing, 238 par, see principal value pension fund, 263 performance analysis, 112 portfolio benchmark, 103 characteristic, 96 efficient, 93 equally weighted, 133 factor mimicking, 111 minimum risk, 95 risk-parity, 134 tangency, 97 value-weighted, 133 portfolio management, portfolio optimization dynamic, 198 positive linear pricing rule, 55 primal problem, 21 principal, see principal value principal value, 42 program semidefinite, 280 pruning a node, 152 pure integer linear program, 140 quadratic program, 71 dual, 76 primal, 76 standard form, 71 quadratic programming, sequential, 313 quadratic programming model, see quadratic program random sampling, 257 adjusted, 257 recourse, recourse problem, 177 reduced cost, 25 reduced gradient generalized, 312 regret, 292 relaxation, 141, 145 linear programming, 145 resampled efficiency, 131 residual vector, 83 reward-to-risk ratio, see Sharpe ratio Ricatti equation, 220 ridge regression, 86 risk contribution, 134 marginal, 134 risk management, risk measure, 6, 175, 181 coherent, 184 risk-neutral probability measure, 55 robust portfolio optimization, 299 robustness, 289 constraint, 290 objective, 291 relative, 292 sampling, 294 saddle-point problem, 292 scenario optimization, 176 scenario tree, 248 arbitrage-free , 258 scenario trees construction, 256 second-order program, 277 security selection, 95, 103 selection return, 114 semidefinite program standard form, 281 sensitivity, 18, 38, 75 separation theorem, see hyperplane sequential system, 214 shadow price, 17, 38 Sharpe ratio, 101, 116 shrinkage estimators, 129 shrinkage factor, 129 shrinkage procedure, 108 shrinkage target, 129 signal, 111 simplex method, 25 dual, 25, 27 single-factor risk model, 107 slack variable, 14 Slater condition, 284 static model, Stein paradox, 129 Stiemke’s theorem, 22, 34 stochastic and dynamic optimization, stochastic discount factor, 56 stochastic model, stochastic optimization, 6, 173, 174 two-stage with recourse, with recourse, 174 stochastic program linear two-stage, 175 stochastic programming, 248 multi-stage, 248 Index stochastic sequential decision problem, 221 stochastic sequential system, 221 strong duality theorem, 21 style analysis, 113 subdifferential, 311 subgradient, 311 subgradient method, 311 support vector machine, 85 surplus variable, 14 synthetic option, 270 synthetic option strategy, 270 two-fund separation theorem, 96 two-fund theorem, 115 tangent subspace, 308 term structure, 39 implied, 38 total cost of trading, 203 tracking error, see active risk tractability, trade list, 202 trading strategy execution, 202 trading trajectory, 202 treasury yield curve, 42 tree fitting, 257 value at risk, see VaR conditional, see CVaR value-to-go function, 215 VaR, 183 α, 183 volatility, 244 volatility smile, 316 volume-weighted average price (VWAP), 204 337 unbounded problem, uncertainty set, 289 ellipsoidal, 290 urgency, 206 utility, 173 logarithmic, 199 power, 199 quadratic, 93 weak duality theorem, 21 winner selection problem, see combinatorial auction problem .. .Optimization Methods in Finance Optimization methods play a central role in financial modeling This textbook is devoted to explaining how state-of-the-art optimization theory,... binding constraints at x are the equality constraints and those inequality constraints that hold with equality at x The term active constraint is also often used in lieu of “binding constraint”... Mixed Integer Programming Models: Portfolios with Combinatorial Constraints 9.1 Combinatorial Auctions 9.2 The Lockbox Problem 9.3 Constructing an Index Fund 9.4 Cardinality Constraints 9.5 Minimum