Stochastic Programming and Financial Analysis IE447...

Stochastic Programming and Financial AnalysisIE447

Midterm Review

Dr. Ted Ralphs

IE447 Midterm Review 1

Forming a Mathematical Programming Model

The general form of a mathematical programming model is:

min f(x)

s.t. g(x) ≤ 0

h(x) = 0

x ∈ X

where f : Rn → R , g : Rn → Rm, h : Rn → Rl, and X may be adiscrete set, such as Zn.

Notes:

• There is an important assumption here that all input data are known andfixed.

• Such a mathematical program is called deterministic.

• Is this realistic?

1


Categorizing Mathematical Programs

• Deterministic mathematical programs can be categorized along severalfundamental lines.

– Constrained vs. Unconstrained– Convex vs. Nonconvex– Linear vs. Nonlinear– Discrete vs. Continuous

• What is the importance of these categorizations?

– Knowing what category an instance is in can tell us something abouthow difficult it will be to solve.

– Different solvers are designed for different categories.

2


Unconstrained Optimization

• When M = ∅ and X = Rn, we have an unconstrained optimizationproblem.

• Unconstrained optimization problems will not generally arise directly fromapplications.

• They do, however, arise as subproblems when solving mathematicalprograms.

• In unconstrained optimization, it is important to distinguish between theconvex and nonconvex cases.

• Recall that in the convex case, optimizing globally is “easy.”

3


Linear Programs

• A linear program is one that can be written in a form in which thefunctions f and gi, i ∈ M are all linear and X = Rn.

• In general, a linear program is one that can be written as

minimize c>x

s.t. a>i x≥ bi ∀i ∈ M1

a>i x≤ bi ∀i ∈ M2

a>i x = bi ∀i ∈ M3

xj ≥ 0 ∀j ∈ N1

xj ≤ 0 ∀j ∈ N2

• Equivalently, a linear program can be written as

minimize c>x

s.t. Ax≥ b

• Generally speaking, linear programs are also “easy” to solve.

4


Nonlinear Programs

• A nonlinear program is any mathematical program that cannot beexpressed as a linear program.

• Usually, this terminology also assumes X = Rn.

• Note that by this definition, it is not always obvious whether a giveninstance is really nonlinear.

• In general, nonlinear programs are difficult to solve to global optimality.

5


Special Case: Convex Programs

• A convex program is a nonlinear program in which the objective functionf is convex and the feasible region is a convex set.

• In practice, convex programs are usually “easy” to solve.

6


Special Case: Quadratic Programs

• If all of the functions f and gi for i ∈ M are quadratic functions, thenwe have a quadratic program.

• Often, the term quadratic program refers specifically to a program of theform

minimize12x>Qx + c>x

s.t. Ax≥ b

• Because x>Qx = 12x>(Q + Q>)x, we can assume without loss of

generality that Q is symmetric.

• The objective function of the above program is then convex if and onlyif Q is positive semidefinite, i.e., y>Qy ≥ 0 for all y ∈ Rn.

• There are specialized methods for solving convex quadratic programsefficiently.

7


Special Case: Integer Programs

• When X = Zn, we have an integer program.

• When X = Zr × Rn−r, we have a mixed integer program.

• By convention, all functions are assumed to be linear in these casesunless otherwise specified.

• If some of the functions are nonlinear, then we have a mixed integernonlinear program.

• All mathematical program with integer variables are difficult to solve ingeneral.

8


Stochastic Optimization

• In the real world, little is known ahead of time with certainty.

• Most of the applications we look at in this class will involve some degreeof uncertainty.

• A risky investment is one whose return is not known ahead of time.

• a risk-free investment is one whose return is fixed.

• To make decisions involving risky investments, we need to incorporatesome degree of stochasticity into our models.

• This can be done in a variety of ways.

9


Probability

• Stochastic optimization involves various “random” phenomena.

• To describe these phenomena, we need a little probability theory.

• The symbol ω will denote the outcome of a random experiment.

• The set of all possible outcomes, called the sample space, will generallybe denoted Ω.

• Subsets of Ω are called events.

10


Probability spaces

• Let A be a set of events.

• A probability measure (or distribution) P is a function that indicates theprobability that each event A ∈ A will occur.

• Probability measures must satisfy certain axioms and have the followingbasic properties.

– 0 ≤ P (A) ≤ 1– P (Ω) = 1, P (∅) = 0– P (A1 ∪A2) = P (A1) + P (A2) if A1 ∩A2 = ∅.

• The triple (Ω,A, P ) is called a probability space.

11


Random Variables

• A random variable ξ on a probability space (Ω,A, P ) is a functionξ : Ω → R such that ω|ξ(ω) ≤ x ∈ A for all finite x.

• ξ has a cumulative distribution given by Fξ(x) = P (ξ ≤ x).

• Discrete random variables are those that take on a finite number ofvalues ξk, k ∈ K

• Random variables have an associated probability density function.

• For a discrete random variable the density function f(ξk) ≡ P (ξ = ξk)

• A continuous random variables has density f with the property

P (a ≤ ξ ≤ b) =∫ b

a

f(ξ)dξ

=∫ b

a

dF (ξ)

= F (b)− F (a)

12


Expectation and Variance

• The Expected value of ξ is

– E(ξ) =∑

k∈K ξkf(ξk) (Discrete)– E(ξ) =

∫∞−∞ f(ξ)dξ =

∫∞−∞ dF (ξ). (Continuous)

• Variance of ξ is Var(ξ) = E(ξ − E(ξ)2).

13


Describing Uncertainty

• One of the challenges of dealing with uncertainty is how to describe it.

• In general, there are an infinite number of ways the future could turnout, but we must describe future possibilities succinctly if we hope todiscern anything from them.

• The scenario approach assumes that there are a finite number of possiblefuture outcomes of uncertainty.

• Each of these possible outcomes is called a scenario.

– Demand for a product is “low, medium, or high.”– Weather is “dry or wet.”– The market will go “up or down.”

• Even if this is not reality, such a discrete approximation is often “goodenough.”

• Using discrete approximations also results in discrete probability spaces,which are sometimes easier to deal with.

14


Linear Programming Models: Short Term Financing

• Short term financing models are to make provisions for a series of knowncash flows over a period of T months.

• For example, suppose the following sources of funds are available:

– Bank credit– Issue of zero-coupon bonds– Cash reserves in an interest-bearing account.

• How should a series of cash flows be provided for at minimum cost if nopayment obligations are to remain at the end of the period?

• Such questions can be answered using a linear programming model.

15


Linear Programming Models: Portfolio Dedication

Definition 1. Dedication or cash flow matching refers to the funding ofknown future liabilities through the purchase of a portfolio of risk-freenon-callable bonds.

• Suppose a pension fund faces liabilities totalling `j for years j = 1, ..., T .

• The fund wishes to dedicate these liabilities via a portfolio comprised ofn different types of bonds.

• Bond type i costs ci, matures in year ji, and yields a yearly couponpayment of di up to maturity.

• The principal paid out at maturity for bond i is pi.

• How should the fund invest in these bonds at minimum cost whilecovering all liabilities?

• This question can again be answered with a linear programming model.

16


Linear Programming Duality Theory

• Consider a linear program in standard form

min c>x

s.t. Ax = b

x≥ 0

• The dual is then

max p>b

s.t. p>A ≤ c>

17


From the Primal to the Dual

We can dualize general LPs as follows

PRIMAL minimize maximize DUAL

≥ bi ≥ 0constraints ≤ bi ≤ 0 variables

= bi free

≥ 0 ≤ cj

variables ≤ 0 ≥ cj constraintsfree = cj

18


Relationship of the Primal and the Dual

The following are the possible relationships between the primal and the dual:

Finite Optimum Unbounded Infeasible

Finite Optimum Possible Impossible Impossible

Unbounded Impossible Impossible Possible

Infeasible Impossible Possible Possible

19


Optimality Conditions

Let’s consider an LP in standard form. We have now shown that theoptimality conditions for (nondegenerate) x are

1. Ax = b (primal feasibility)

2. x ≥ 0 (primal feasibility)

3. xi = 0 if p>ai 6= ci (complementary slackness)

4. p>A ≤ c (dual feasibility)

• In standard form, the complementary slackness condition is simply x>c =0.

• This condition is always satisfied during the simplex algorithm, since thereduced costs of the basic variables are zero.

20


Dual Variables and Marginal Costs

• Consider an LP in standard form with a nondegenerate, optimal basicfeasible solution x∗ and optimal basis B.

• Suppose we wish to perturb the right hand side slightly by replacing bwith b + d.

• As long as d is “small enough,” we have B−1(b + d) > 0 and B is stillan optimal basis.

• The optimal cost of the perturbed problem is

c>BB−1(b + d) = p>(b + d)

• This means that the optimal cost changes by p>d.

• Hence, we can interpret the optimal dual prices as the marginal cost ofchanging the right hand side of the ith equation.

21


Economic Interpretation

• The dual prices, or shadow prices allow us to put a value on “resources”(broadly construed).

• Alternatively, they allow us to consider the sensitivity of the optimalsolution value to changes in the input.

• Consider the bond portfolio problem from Lecture 3.

• By examining the dual variable for the each constraint, we can determinethe value of an extra unit of the corresponding “resource”.

• We can then determine the maximum amount we would be willing topay to have a unit of that resource.

• Note that the reduced costs can be thought of as the shadow pricesassociated with the nonnegativity constraints.

22


Local Sensitivity Analysis

• For changes in the right-hand side,

– Recompute the values of the basic variables, B−1b.– Re-solve using dual simplex if necessary.

• For a changes in the cost vector,

– Recompute the reduced costs.– Re-solve using primal simplex.

• For changes in a nonbasic column Aj

– Recompute the reduced cost, cj − cBB−1Aj.– Recompute the column in the tableau, B−1Aj.

• For all of these changes, we can compute ranges within which the currentbasis remains optimal.

23


Optimality Conditions for Nonlinear Programs

• The KKT conditions provide a set of necessary conditions for optimalityin the case of a nonlinear program.

• These conditions apply only when the gradients of the binding constraintsare linearly independent.

• In this case, we get that x∗ locally optimal ⇒ there exists u ∈ Rm suchthat

5f(x∗) +∑

ui 5 gi(x∗) = 0

uigi(x∗) = 0 ∀i ∈ [1,m]

u≥ 0

24


Remarks on the KKT conditions

• As in the LP case, we have PF, DF and CS conditions.

• x∗ is a KKT point if the KKT conditions are satisfied at x∗.

• For a linear program, the KKT conditions are simply the standardoptimality conditions for LP.

• Furthermore, x∗ is a KKT point if and only if x∗ is the solution to thefirst-order LP approximation to the NLP

minf(x∗) +5f(x∗)>(x− x∗) | gi(x∗) +5gi(x∗)>(x− x∗) ≤ 0, i ∈ [1,m]

• The KKT conditions are sufficient for convex programs.

• The KKT conditions are necessary and sufficient for convex programswith all linear constraints.

25


Optimality Conditions for Quadratic Programs

• Consider the quadratic program

min12x>Qx + c>x

s.t. Ax = b

x ≥ 0

• KKT conditions are that there exists a solution to the system

F (x, y, s) =

Ax− bA>y −Qx + s− c

s>x

=

000

, (x, s) ≥ 0. (1)

• If Q is positive semi-definite, then we have a convex program and theabove conditions are necessary and sufficient.

26


Market Model

• A market consists of a set of n risky assets, denoted generally byS1, . . . , Sn, and a risk-free asset S0 and a probability space describingpossible future states of the market.

• The price of investment i at time 0 is known and denoted by Si0.

• The price of investment i at time t is a random variable denoted Sit.

• We will frequently wish to describe the state of prices at a future timet = 1 in terms of scenarios.

• We then let Ω = ∪mj=1Ωj be a partition of Ω into m events such that

the prices at time 1 are constants Si1(Ωj) for all ω ∈ Ωj, i = 0, . . . , n.

• Since S0 is risk-free, we have

S01(Ω1) = · · · = S0

1(Ωm) = RS00 (2)

27


Arbitrage

• Arbitrage is getting something for nothing. It is the fabled “free lunch.”

• More formally, there are two types of arbitrage

Definition 2. Type A arbitrage is a trading strategy that has positiveinitial cash flow and nonnegative payoff under all future scenarios.

Definition 3. Type B arbitrage is a trading strategy that costs nothinginitially, has nonnegative payoff under all future scenarios and has astrictly positive expected payoff.

• Obviously finding and exploiting arbitrage opportunities can be verylucrative.

• Because market forces are quick to adjust, arbitrage opportunities do notusually exist for long.

28


Risk-Neutral Probability Measures

The existence of arbitrage is intrinsically linked to the existence of a risk-neutral probability measure.

Definition 4. A risk-neutral probability measure (RNPM) for the market(S0, . . . , Sn) is a vector p = (p1, . . . , pm) > 0 such that

Si0 = R−1

m∑

j=1

pjSi1(Ωj), (i = 0, . . . , n).

Note that because of (2), the constraint corresponding to i = 0 is equivalentto

m∑

j=1

pj = 1,

which, together with p ≥ 0, means that p must be a probability measureon Ω.

29


Fundamental Theorem of Asset Pricing

Theorem 1. (Fundamental Theorem of Asset Pricing) An RNPM for themarket (S0, . . . , Sn) exists if and only if the market is arbitrage-free.

Idea of Proof: Consider the following LP,

(P) minx

n∑

i=0

xiSi0

s.t.n∑

i=0

xiSi1(Ωj) ≥ 0, (j = 1, . . . , m).

Note that since x = 0 is a feasible solution, the optimal objective valuemust be nonpositive.

30


Asset Pricing Using the Risk Neutral Probabilities

• The Fundamental Theorem of Asset Pricing introduces a very generalnotion of risk-neutral probabilities.

• As in the simple case of two scenarios and two underlying assets, we canuse the risk-neutral probabilities to price assets whose prices are linearfunctions of the prices of known assets.

• The prices are again simply the discounted expected value of the assetwith respect to the risk neutral probabilities.

• This is the same as adding an extra (linearly dependent) row to thearbitrage detection LP.

• If an asset to be added is linearly dependent on existing assets, but itsprice is not not equal to the same combination of the prices of the otherassets, this makes the dual infeasible, i.e., introduces arbitrage.

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Assets with Piecewise Linear Payoffs

• Consider a portfolio

Sx :=n∑

i=1

xiSi

of assets Si, i = 1, . . . , n whose payoffs Si1 are piecewise linear functions

of a single underlying asset S01 (not necessarily risk-free!).

• When the payoff functions of each security is a piecewise linear functionof the underlying security, so is the payoff function of Sx.

Sx1 (ω) = Ψx(S0

1(ω)) ∀ω ∈ Ω,

where

Ψx(s) =n∑

i=1

xiΨi(s)

has breakpoints among the set Kij : j = 1, . . . , ki; i = 1, . . . , n.

• Let 0 < K1 < · · · < Km be these breakpoints listed in ascending order,and let K0 := 0.

32


Detecting Arbitrage

• The following optimization problem is designed to identify arbitrageopportunities in this case, if they exist.

(P) minx

n∑

i=1

xiSi0

s.t. Ψx(s) ≥ 0 ∀ s ∈ [0,∞).

• The problem is to find a minimum cost portfolio with nonnegative payofffor all realizations of S0

1.

• If there exists such a portfolio with negative cost, then an arbitrageopportunity of type A exists, as before.

• By utilizing the fact that a piecewise linear function defined on [0,∞) isnonnegative if and only if it has a nonnegative value at each breakpointand its last piece has nonnegative slope, we can rewrite this as a linearprogram.

33


Another Theorem on Asset Pricing

In analogy to the first fundamental theorem of asset pricing, LP duality canbe used to prove the following result.

Theorem 2. There is no arbitrage of type A if and only if the optimalobjective value of the following LP is zero.

(P′) minx

n∑

i=1

Si0xi

s.t.n∑

i=1

Ψi(0)xi ≥ 0,

n∑

i=1

Ψi(Kx` )xi ≥ 0, (` = 1, . . . , m),

n∑

i=1

(Ψi(Km + 1)−Ψi(Km)

)xi ≥ 0.

Furthermore, if there is no arbitrage of type A, then there is no arbitrage oftype B if and only if the dual of (P′) has a strictly feasible solution.

34


The Portfolio Optimization Problem

Decision variables: xi, proportion of wealth invested in asset i.

Constraints:

• the entire wealth is assumed invested,∑

i xi = 1,

• if short-selling of asset i is not allowed, xi ≥ 0,

• bounds on exposure to groups of assets,∑

i∈G xi ≤ b, . . .

Objective function: The investor wants to maximize expected return whileminimizing risk. What to do?

• Let R = [R1 . . . Rn]> be the random vector of asset returns andµ = E[R] the vector of their expectations.

• Then the random return of the portfolio y is

∑i yiS

i1 −

∑i yiS

i0∑

i yiSi0

=∑

i

yiSi0∑

i yiSi0

· Si1 − Si

0

Si0

= R>x.

35


The Konno & Yamazaki Model

• The expected portfolio return is

E[R>x] =∑

i

xiE[Ri] = µ>x.

• How do we measure risk?

• Konno & Yamazaki proposed an LP model for portfolio optimization bymeasuring risk based on the `1 norm.

`(x) := E[|(R− µ)>x|] ,

• In other words, we consider the mean absolute deviation of the portfolioreturn from its mean.

36


The Konno & Yamazaki Model

• We assume the self-financing constraint∑

i xi = 1 is taken into accountamong the constraints Bx = b

• The target return constraint rmin ≤ µ>x can be modelled among theconstraints Ax ≥ a.

• Subject to these constraints, we now want to minimize the risk,

(KY) minx

`(x)

s.t. Ax ≥ a,

Bx = b.

37


A Linear Portfolio Optimization Model

• The main motivation for using `(x) as a risk measure instead of thevariance of the portfolio return

σ2(x) := σ2(R>x) = E[(

(R− µ)>x)2

]

is that the resulting model is linear rather than quadratic.

• This allows us to handle a much larger number of assets.

• If R ∼ N(µ,Q) is a multivariate normal random vector with covariancematrix Q, then σ2(x) = x>Qx and

`(x) =1√

2πσ2(x)

∫ +∞

−∞|ϑ| exp

(− ϑ2

2σ2(x)

)dϑ =

√2σ2(x)

π.

• Thus, minimizing σ2(x) is the same as minimizing `(x) in this case.

38


Analyzing Tradeoffs

• In the case of the above portfolio optimization problem, there is anobvious tradeoff to be analyzed between risk and return.

• The general framework of biobjective programming can be used toanalyze such tradeoffs.

• A biobjective or bicriterion mathematical program is an optimizationproblem of the form

vmin f(x)subject to x ∈ X,

where

• f : Rn → R2 is the (bicriterion) objective function, and

• X is the feasible region, usually defined to be

x ∈ Zp × Rn−p | gi(x) ≥ 0, i = 1, . . . , m

for functions gi : Rn → R, i = 1, . . . , m.

• The vmin operator indicates that we are interested in generating theefficient solutions (defined next).

39


The Parametric Simplex Method

The parametric simplex method is a variant of the simplex method that canbe used to analyze bicriteria LPs. We assume the objective vector is of theform c + θd and we want to know the set of optimal solutions for all valuesof θ.

• Determine an initial feasible basis.

• Determine the interval [θ1, θ2] for which this basis is optimal.

• Determine a variable j whose reduced cost is nonpositive for θ ≥ θ2.

• If the corresponding column has no positive entries, then the problem isunbounded for θ > θ2.

• Otherwise, rotate column j into the basis.

• Determine a new interval [θ2, θ3] in which the current basis is optimal.

• Iterate to find all breakpoint ≥ θ1.

• Repeat the process to find breakpoints ≤ θ1.

40


General Markowitz Portfolio Model

• We next consider a portfolio optimization model that uses variance asthe risk measure.

• A general Markovitz portfolio optimization problem would thus take thefollowing form,

(M) minx

x>Qx

s.t. µ>x ≥ r,

Ax ≥ a,

Bx = b,

• We’ve singled out the inequality constraint µ>x ≥ r because it containsthe extra parameter r.

• Note that since the covariance matrix Q is positive semidefinite, (M) isa convex QP.

41


• Since Q Â 0 (positive definite), we have σmin > 0, where

σ2min := min

xx>Qx

s.t. Ax ≥ a

Bx = b.

• Note that here the constraint µ>x ≥ r has been dropped.

• Let

(R) r(σ) = maxx

µ>x

s.t. Ax ≥ a

Bx = b

x>Qx ≤ σ2,

and note that for σ ≥ σmin the function r(σ) is well-defined, as (R) hasfeasible solutions.

42


Efficient Portfolios

Note that µ>x ≤ r(√

x>Qx) for all feasible x, and that it can never makesense to hold a portfolio x for which

µ>x < r(√

x>Qx)

,

since the portfolio x∗ obtained from solving problem (R) with σ2 = x>Qxwould yield the more desirable expected return

µ>x∗ = r(√

x>Qx)

.

Definition 5. Portfolios that satisfy the relation

µ>x = r(√

x>Qx)

are called efficient. The curve σ 7→ r(σ), defined for σ ≥ σmin, is calledthe efficient frontier.

43


The Market Price of Risk

• We now consider the situation where the universe of investable assetscontains

– One risk-free asset S0 with return rf and– n risky assets S1, . . . , Sn with random return vector R and E[R] = µ.

• We write

x =[x0

x

], R =

[rf

R

], µ =

[rf

µ

].

• The covariance matrix of R then has the block structure

Q =[0 00 Q

],

where Q Â 0 is the covariance matrix of R.

44


Markowitz Model with Risk-free asset

We consider the Markowitz problem

(P) minx

x>Qx

s.t. Ax ≥ a

Bx = b,

µ>x ≥ r,

where the constraint matrices are of the form

A =[0 ea A

],

B = [b B],

a =[

0a

]

where e = [1 . . . 1], and B has first row [1 e].

45


The Efficient Frontier

Theorem 3. Under the above assumptions there exists a portfolio xm ∈Rn on the efficient frontier of (M) such that the efficient frontier of (P) isgiven by the ray x(θ) : θ ≤ 1, where

x(θ) =(

θe0

(1− θ)xm

).

• In other words, for any σ2 ≥ 0 there exists a θ ≤ 1 such that the portfoliox(θ) achieves the maximum return µ>x(θ) = r(σ) at the risk level σ2.

• Note that this means that the relative proportions of wealth allocationamong the risky assets alone is the same for all investors, no matter howrisk-averse they are.

46


The Maximum Sharpe Ratio Problem

• In the proof of theorem 3, the portfolio xm is chosen as an optimalsolution of the maximum Sharpe ratio problem

(SR) maxx

µ>x− rf√x>Qx

s.t. Ax ≥ a

Bx = b

• Thus, another implicit assumption on the problem data A, a, B, b is thatthe feasible set x ∈ Rn : Ax ≥ a, Bx = b is nonempty and that a(finite) optimal solution of (SR) exists.

47


Reformulating

Therefore, (SR) can be solved by solve the convex QP,

(HSR) min(y,τ)

y>Qy

s.t. (µ− rfe)>y = 1

Ay ≥ τa

By = τb

τ ≥ 0,

and converting an optimal solution (y∗, τ∗) of (HSR) into an optimalsolution of (SR) by setting x∗ = τ−1y∗. Note that τ cannot be zero in anyfeasible solution.

48


rm−r0/sm

sm

R0

Rm

port

folio

ret

urn

captia

l marke

t line

portfolio risk (std dev.)

efficient fro

ntier

market portfolio

49

Stochastic Programming and Financial Analysis IE447...

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