Mathematicalmethods Lecture Slides Week7 1 LagrangesMethod

AMATH 460: Mathematical Methodsfor Quantitative Finance

7.1 Lagranges Method

Kjell KonisActing Assistant Professor, Applied Mathematics

University of Washington

Kjell Konis (Copyright 2013) 7.1 Lagranges Method 1 / 29

Outline

1 Optimal Investment Portfolios

2 Relative Extrema of Functions of Several Variables

3 Lagranges Method

4 Example

5 Minimum Variance Portfolio


Outline



3 Lagranges Method

4 Example



Investment Portfolios

Portfolio of n assetsLet wi be the proportion of the portfolio invested in asset iHave constraint n

i=1wi = 1

Can take long and short positions = no constraints on individual wiLet i be the expected rate of return on asset iLet 2i be the risk of asset iLet ij be the correlation between assets i and jExpected rate of return and risk of the portfolio:

Expected Return =n

i=1wii

Risk =n

i=1w2i 2i + 2

1i

Investment Portfolios: Matrix Notation

Let w = (w1, . . . ,wn) and = (1, . . . , n)The expected rate of return can be written in matrix notation as

Return =n

i=1wii = wT

The risk can be written as

Risk = wTw

is the covariance matrix of the n assets

=

21 1212 1n1n2121 22 2n

... ... . . . ...n1n1 n2n2 2n


Optimal Investment Portfolios

Given , , and investor selected w , can computeportfolio returnportfolio risk

Two notions of optimality

For a target expected return, choose w to minimize portfolio riskFor a target level of risk, choose w to maximize expected return

Both notions are constrained optimization problems that can besolved using Lagrange multipliers


Optimal Investment Portfolios

Minimum variance optimization

n asset case

minimize: wTwsubject to: eTw = 1

Tw = P

2 asset caseminimize: 21w21 + 212w1w2 + 22w22

subject to: w1 + w2 = 11w1 + 2w2 = P

Maximum expected return optimization

n asset case

maximize: Twsubject to: eTw = 1

wTw = 2P

2 asset casemaximize: 1w1 + 2w2

subject to: w1 + w2 = 121w21 + 212w1w2 + 22w22 = 2P


Outline



3 Lagranges Method

4 Example



Relative Extrema of Single Variable Functions

A local minimum (maximum) of a function f is a point x0 where

f (x0) () f (x) x (x0 , x0 + )

for some > 0

A local extrema is a point that is a local minimum or maximum

If f is twice differentiable and f is continuousAny local extremum is a critical point of f : f (x0) = 0Can classify critical points using second derivative testf (x0) < 0 local maximumf (x0) > 0 local minimumf (x0) = 0 anything possible


Relative Extrema of Functions of n Variables

A local minimum (maximum) of a function f : Rn R is a pointx0 Rn where

f (x0) () f (x) x : x x0 <

Every local extremum is a critical point: Df (x0) = 0

If f is twice differentiable and has continuous second order partialderivatives

D2f (x0) is a symmetric matrix with real eigenvaluesSecond order conditions

All eigenvalues of D2f (x0) > 0 local minimumAll eigenvalues of D2f (x0) < 0 local maximumD2f (x0) has eigenvalues saddle pointD2f (x0) singular anything can happen


Finding Extrema: Functions of 2 Variables

Find the local extrema of f (x , y) = x2 + xy + y2

Df (x , y) =[2x + y x + 2y

]Df (0, 0) =

[0 0

] (0, 0) is a critical point

D2f (x , y) =[

2 11 2

]

Can use R to compute the eigenvalues> A eigen(A)$values[1] 3 1

Since both eigenvalues are greater than 0 = (0, 0) a local minimumKjell Konis (Copyright 2013) 7.1 Lagranges Method 11 / 29

Finding Extrema: Functions of 2 Variables (Take 2)

Find the local extrema of f (x , y) = x2 xy y2

Df (x , y) =[ 2x y x 2y] Df (0, 0) = [0 0]

(0, 0) is a critical point

D2f (x , y) =[2 11 2

]

Can use R to compute the eigenvalues> A eigen(A)$values[1] -1 -3

Since both eigenvalues are less than 0 = (0, 0) a local maximumKjell Konis (Copyright 2013) 7.1 Lagranges Method 12 / 29


Find the local extrema of f (x , y) = x2 + 3xy + y2

Df (x , y) =[2x + 3y 3x + 2y

]Df (0, 0) =

[0 0

] (0, 0) is a critical point

D2f (x , y) =[

2 33 2

]

Can use R to compute the eigenvalues> A eigen(A)$values[1] 5 -1

One positive and one negative eigenvalue = (0, 0) a saddle pointKjell Konis (Copyright 2013) 7.1 Lagranges Method 13 / 29


Find the local extrema of f (x , y) = 2xy (1 y2) 32First order condition

Df (x , y) =[

2y 2x + 3y

1 y2]

Df (0, 0) =[0 0

]= (0, 0) is a critical point

Second order condition

D2f (x , y) =

0 22 36y2

1y2

D2f (0, 0) = 0 2

2 3

Compute the eigenvalues of the Hessian at the critical point> eigen(matrix(c(0, 3, 3, 2), 2, 2))$values[1] 4.162278 -2.162278One positive and one negative eigenvalue = (0, 0) a saddle point


Outline



3 Lagranges Method

4 Example



Lagranges Method

Problem:maximize: f (x1, x2, . . . , xn)

subject to: g1(x1, x2, . . . , xn) = 0g2(x1, x2, . . . , xn) = 0

...gm(x1, x2, . . . , xn) = 0

(1)

18th-century mathematician Joseph Louis Lagrange proposed thefollowing method for the solutionForm the function

F (x1, . . . , xn, 1, . . . , m) = f (x1, . . . , xn) +mi=1

igi(x1, x2, . . . , xn)

Optimal value for problem (1) occurs at one of the critical points of FKjell Konis (Copyright 2013) 7.1 Lagranges Method 16 / 29

Lagranges Method

Terminology:The function F (x1, . . . , xn, 1, . . . , m) is called the LagrangianThe column vector = (1, . . . , m) is called the Lagrangemultipliers vector

Necessary Condition:Let x = (x1, x2, . . . , xn)Let g(x) =

(g1(x), g2(x), . . . , gm(x)

)be a vector-valued function

of the constraintsThe gradient D

(g(x)

)must have full rank at any point where

the constraint g(x) = 0 is satisfied, that is

rank(Dg(x)

)= m x where g(x) = 0


Partial Derivatives of the Lagrangian

D F (x , ) has n + m variables, compute gradient in 2 parts

D F (x , ) =[DxF (x , ) DF (x , )

]Recall Lagrangian:

F (x , ) = f (x1, . . . , xn) +mi=1

igi(x1, x2, . . . , xn)

The partial derivatives are

Fxj

=fxj

+mi=1

igixj

Fi

= gi(x)

Gradient of f : Df (x) =[fx1

. . .fxn

]Kjell Konis (Copyright 2013) 7.1 Lagranges Method 18 / 29

Partial Derivatives of the Lagrangian

Gradient of g(x):

Dg(x) =

g1x1

g1x2

g1xn

g2x1

g2x2

g2xn

... ... . . . ...gmx1

gmx2

gmxn

Can express sum in second term in matrix notationmi=1

igixj

= T[Dg(x)

]j

It follows that

D F (x , ) =[Df (x) + TDg(x)

(g(x)

)T]Kjell Konis (Copyright 2013) 7.1 Lagranges Method 19 / 29

Outline



3 Lagranges Method

4 Example



Example

Want tomax/min: 4x2 2x3

subject to: 2x1 x2 x3 = 0x21 + x22 13 = 0

Start by writing down the Lagrangian

F (x , ) = f (x) + 1g1(x) + 2g2(x)

= 4x2 2x3 + 1(2x1 x2 x3) + 2(x21 + x22 13)

Check necessary condition:

Dg(x) =[

2 1 12x1 2x2 0

]


Derivatives of the LagrangianThe Lagrangian

F (x , ) = 4x2 2x3 + 1(2x1 x2 x3) + 2(x21 + x22 13)

Gradient of the Lagrangian

D F (x , ) =

21 + 22x1

4 1 + 22x22 1

2x1 x2 x3x21 + x22 13

T

Set D F (x , ) = 0 and solve for x and get 1 = 2 for free

21 + 22x1 set= 04 1 + 22x2 set= 0

2x1 x2 x3 set= 0x21 + x22 13 set= 0


Example (continued)

A little algebra gives

x1 =22

x2 =32

x3 =72

Also know that

x21 +x22 = 13 =( 22

)2+

(32

)2=

1322

= 13 = 2 = 1

The critical points are = (2,1), x = (2, 3,7), f (x) = 26 = (2, 1), x = (2,3, 7), f (x) = 26


Outline



3 Lagranges Method

4 Example



Minimum Variance Portfolio

Recall: minimum variance portfolio optimization

minimize: wTwsubject to: eTw = 1

Tw = PLagranges method setup

f (w) = wTw

g(w) =[g1(w)g2(w)

]=

[Tw P = 0eTw 1 = 0

]

First, check necessary condition

Dg(x) =[T

eT

]


Derivative of a Quadratic Form

Let A =[a bb c

]

Let f (x) = xTAx = ax21 + 2bx1x2 + cx22Then Df (x) =

[2ax1 + 2bx2 2bx1 + 2cx2

]= 2xTA

In general, let A be an n n symmetric matrixThe derivative (gradient) of the quadratic form f (x) = xTAx is

Df (x) = 2xTA



The Lagrangian

F (y , ) = wTw + 1[eTw 1]+ 2[Tw P]

Gradient of the Lagrangian

D F (w , ) =[Df (w) + T

(Dg(w)

) (g(w)

)T]=

[2wT + 1eT + 2T eTw 1 Tw P

]Find the critical point by solving the linear system2 e eT 0 0

T 0 0

w12

= 01P



Further reading:Second order conditions, e.g., Theorem 9.2 and Corollary 9.1 inPFME


http://computational-finance.uw.edu


Optimal Investment PortfoliosRelative Extrema of Functions of Several VariablesLagrange's MethodExampleMinimum Variance Portfolio

Mathematicalmethods Lecture Slides Week7 1 LagrangesMethod

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