Capital Asset Pricing
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Chapter 10
Capital Asset Pricing
10.1 An Optimization Problem
Consider an agent who has initial wealth
X
0
and wants to invest in the stock and money markets so
as to maximize
IE log X
n
:
Remark 10.1 Regardless of the portfolio used by the agent,
f
k
X
k
g
1
k=0
is a martingale under IP, so
IE
n
X
n
= X
0
BC
Here, (BC) stands for “Budget Constraint”.
Remark 10.2 If
is any random variable satisfying (BC), i.e.,
IE
n
= X
0
;
then there is a portfolio which starts with initial wealth
X
0
and produces
X
n
=
at time
n
.Tosee
this, just regard
as a simple European derivative security paying off at time
n
.Then
X
0
is its value
at time 0, and starting from this value, there is a hedging portfolio which produces
X
n
=
.
Remarks 10.1 and 10.2 show that the optimal
X
n
for the capital asset pricing problem can be
obtained by solving the following
Constrained Optimization Problem:
Find a random variable
which solves:
Maximize
IE log
Subject to
IE
n
= X
0
:
Equivalently, we wish to
Maximize
X
!2
log ! IP !
119
120
Subject to
X
! 2
n
! !IP ! , X
0
=0:
There are
2
n
sequences
!
in
.Callthem
!
1
;!
2
;::: ;!
2
n
. Adopt the notation
x
1
= !
1
;x
2
=!
2
; ::: ; x
2
n
= !
2
n
:
We can thus restate the problem as:
Maximize
2
n
X
k=1
log x
k
IP !
k
Subject to
2
n
X
k=1
n
!
k
x
k
IP !
k
, X
o
=0:
In order to solve this problem we use:
Theorem 1.30 (Lagrange Multiplier) If
x
1
;::: ;x
m
solve the problem
Maxmize
f x
1
;::: ;x
m
Subject to
g x
1
;::: ;x
m
=0;
then there is a number
such that
@
@x
k
f x
1
;::: ;x
m
=
@
@x
k
gx
1
;::: ;x
m
; k =1;::: ;m;
(1.1)
and
g x
1
;::: ;x
m
=0:
(1.2)
For our problem, (1.1) and (1.2) become
1
x
k
IP !
k
=
n
!
k
IP !
k
;k=1;::: ;2
n
; 1:1
0
2
n
X
k=1
n
!
k
x
k
IP !
k
=X
0
: 1:2
0
Equation (1.1’) implies
x
k
=
1
n
!
k
:
Plugging this into (1.2’) we get
1
2
n
X
k=1
IP !
k
=X
0
=
1
=X
0
:
CHAPTER 10. Capital Asset Pricing
121
Therefore,
x
k
=
X
0
n
!
k
;k=1;::: ;2
n
:
Thus we have shown that if
solves the problem
Maximize
IE log
Subject to
IE
n
=X
0
;
(1.3)
then
=
X
0
n
:
(1.4)
Theorem 1.31 If
is given by (1.4), then
solves the problem (1.3).
Proof: Fix
Z0
and define
f x = log x , xZ:
We maximize
f
over
x0
:
f
0
x=
1
x
,Z =0 x=
1
Z
;
f
00
x=,
1
x
2
0; 8x2 IR:
The function
f
is maximized at
x
=
1
Z
, i.e.,
log x , xZ f x
= log
1
Z
, 1; 8x0; 8Z0:
(1.5)
Let
be any random variable satisfying
IE
n
=X
0
and let
=
X
0
n
:
From (1.5) we have
log ,
n
X
0
log
X
0
n
, 1:
Taking expectations, we have
IE log ,
1
X
0
IE
n
IE log
, 1;
and so
IE log IE log
:
122
In summary, capital asset pricing works as follows: Consider an agent who has initial wealth
X
0
and wants to invest in the stock and money market so as to maximize
IE log X
n
:
The optimal
X
n
is
X
n
=
X
0
n
, i.e.,
n
X
n
= X
0
:
Since
f
k
X
k
g
n
k=0
is a martingale under IP, we have
k
X
k
= IE
n
X
n
jF
k
=X
0
;k=0;::: ;n;
so
X
k
=
X
0
k
;
and the optimal portfolio is given by
k
!
1
;::: ;!
k
=
X
0
k+1
!
1
;::: ;!
k
;H
,
X
0
k+1
!
1
;::: ;!
k
;T
S
k+1
!
1
;::: ;!
k
;H, S
k+1
!
1
;::: ;!
k
;T
:
