Lecture 15
•
No Arbitrage Pricing
‣
Pricing of Forward Contracts
‣
Forward and Future Prices
‣
The Binomial Price Process
State Prices
For (v
1
,...,v
n
) choose the Arrow-Debreu Security for
state i,
i.e, v
i
=1, and v
j
=0 for j
!
i.
Let x
i
be the
resulting portfolio. The cost of the portfolio is
q
i
=px
i
=p
1
x
i
1
+...+p
n
x
i
n
.
We say that
q
i
is the state
price for state i
.
v
1
v
2
!
v
n
!
"
#
#
#
#
$
%
&
&
&
&
=
v
1,1
v
2,1
"
v
n
,1
v
1,2
v
2,2
"
v
n
,2
!
!
#
!
v
1,
n
v
2,
n
"
v
n
,
n
!
"
#
#
#
#
#
$
%
&
&
&
&
&
x
1
x
2
!
x
n
!
"
#
#
#
#
$
%
&
&
&
&
State Prices
For (v
1
,...,v
n
) choose the Arrow-Debreu Security for
state i,
i.e, v
i
=1, and v
j
=0 for j
!
i.
Let x
i
be the
resulting portfolio. The cost of the portfolio is
q
i
=px
i
=p
1
x
i
1
+...+p
n
x
i
n
.
We say that
q
i
is the state
price for state i
.
By construction the state prices have the property
that
p
i
=
q
1
,...,
q
n
(
)
v
i
,1
v
i
,2
!
v
i
,
n
!
"
#
#
#
#
#
$
%
&
&
&
&
&
=
q
i
v
i
,
s
s
=
1
n
'
.
State Prices
In matrix notation, state prices q
1
,...,q
n
must satisfy:
p
1
p
2
!
p
n
!
"
#
#
#
#
$
%
&
&
&
&
=
q
1
,...,
q
n
(
)
v
1,1
v
2,1
"
v
n
,1
v
1,2
v
2,2
"
v
n
,2
!
!
#
!
v
1,
n
v
2,
n
"
v
n
,
n
!
"
#
#
#
#
#
$
%
&
&
&
&
&
Risk Neutral Valuation
Let q
1
,...,q
l
be strictly positive state prices.
And
suppose that an asset generates payoffs v
1
,
....
,v
s
.
Then we already know that the price p of the asset
must satisfy
p
=
q
s
v
s
.
s
=
1
l
!
Risk Neutral Valuation
Let q
1
,...,q
l
be strictly positive state prices.
And
suppose that an asset generates payoffs v
1
,
....
,v
s
.
Then we already know that the price p of the asset
must satisfy
p
=
q
s
v
s
.
s
=
1
l
!
Payoff of riskless asset with price 1 (if exists)?
v=(1+r
0
,1+r
0
,..., 1+r
0
) .
Thus,
(1
+
r
0
)
q
s
=
1
s
=
1
l
!
q
s
=
1
1
+
r
0
s
=
1
l
!