Affine images and preimages: Given a convex setS ⊆Rn, matricesA∈Rm×n, B∈Rn×p and vectorsa∈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.
SupposeS ⊆ Rn is a convex set. A function f : S →R is convex if, for all x,y∈S andλ∈[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 considerextended valuedfunctions; that is, functions defined on the whole spaceRn and allowed to take the value∞.Thedomainof an extended valued functionf :Rn →R∪{∞}
is the set
dom(f) :={x∈Rn:f(x)<∞}.
An alternative and equivalent definition of convexity is the following. An extended valued functionf :Rn→R∪ {∞}is convexif the set
epigraph(f) :={(x, t)∈Rn+1:f(x)≤t}
is convex. Observe that iff is convex, then its domain is a convex set. Further- more, iff is convex then for all∈Rthesublevelset {x∈Rn :f(x)≤} is a convex set.
The following relationship between differentiability and convexity is particu- larly useful to verify that functions are convex.
Theorem A.1 Suppose f : S → R is twice differentiable on the open set S⊆Rn andC⊆S is a convex set. Thenf is convex onC if and only if∇2f(x) is positive semidefinite for allx∈C.
As an immediate consequence of Theorem A.1 it follows that every affine functionf(x) =cTx+b is convex. It also follows that a quadratic function
f(x) =1
2xTQx+cTx+b is convex if and only ifQis positive semidefinite.
A.3 Calculus of Variations: the Euler Equation
The calculus of variations is the analog of calculus that works with function- als rather than functions. Functionals are often integrals of functions. Many problems in the calculus of variations arose from the need to find a function
that optimizes a given functional. The Euler equation for the minimization of a functional subject to boundary conditions is a kind of first-order optimality condition for the problem
minx
# T 0
L(t, x(t),x(t))dt, x(0) =˙ x0, x(T) =xT. The optimal solutionx∗(t) must satisfy the differential equation
Lx= d
dtLx˙. (A.1)
Equation (A.1) is called the Euler equation. For a derivation of this optimality condition as well as a detailed discussion on the interesting subject of calculus of variations, see Fleming and Rishel (1975).
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accrued interest, 42
active constraint,see binding constraint active return, 105
active risk, 105 active set, 313 active-set methods, 81 adaptive decision, 7 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, 3 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,seeCAPM 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, 3 consumption, 174 contingent claim, 55
Index 335
convex function, 325 convex optimization, 4 convex set, 324 cutting plane, 154
cutting-plane method, 150, 154 CVaR, 181, 184
decision variables, 3 descent direction, 309 deterministic model, 4 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,seefactor loading factor loading, 107
factor model, 106 factor portfolio, 120 Farkas’s lemma, 21, 34 feasibility cut, 178 feasible point, 3 feasible region, 3
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,seecone, second-order immunization, 39
immunized portfolio, 39
implementation shortfall,seetotal cost of trading
implied volatility, 273 index fund, 165 infeasible problem, 3 information ratio, 105
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, 5 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, 8 master problem, 177
matrix inverse, 323
positive semidefinite, 5, 280 mean absolute deviation,seeMAD 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, 4 mixed integer program, 140 mixed integer programming, 6 multi-period model, 4 multiple-factor risk model, 109 newsvendor problem, 173, 175, 177 Newton step, 29, 286
Newton’s method, 310 nonlinear program, 305 NP-hardness, 150
objective function, 3
one-fund separation theorem, 97 optimal decision rule, 216 optimal policy, 216 optimal solution, 3 optimal value, 3 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, 8 portfolio optimization
dynamic, 198
positive linear pricing rule, 55 primal problem, 21
principal,seeprincipal 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, 5
sequential, 313
quadratic programming model,seequadratic program
random sampling, 257 adjusted, 257 recourse, 7
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, 9 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,seehyperplane 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, 4 Stein paradox, 129 Stiemke’s theorem, 22, 34
stochastic and dynamic optimization, 4 stochastic discount factor, 56
stochastic model, 4
stochastic optimization, 6, 173, 174 two-stage with recourse, 6 with recourse, 174 stochastic program
linear two-stage, 175 stochastic programming, 248
multi-stage, 248
Index 337
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 tangent subspace, 308 term structure, 39
implied, 38
total cost of trading, 203 tracking error,seeactive risk tractability, 4
trade list, 202 trading strategy
execution, 202 trading trajectory, 202 treasury yield curve, 42 tree fitting, 257
two-fund separation theorem, 96 two-fund theorem, 115
unbounded problem, 4 uncertainty set, 289
ellipsoidal, 290 urgency, 206 utility, 173
logarithmic, 199 power, 199 quadratic, 93 value at risk,see VaR
conditional,seeCVaR value-to-go function, 215 VaR, 183
α, 183 volatility, 244 volatility smile, 316
volume-weighted average price (VWAP), 204
weak duality theorem, 21
winner selection problem,seecombinatorial auction problem