CHAPTER 11.
11.1. a)
7424.0q
8224.0q
91.0q
r1
1
q
3
2
1
i
i
i
b) The matrix is the same at each date. The n-period A-D matrix is then
n
DA
. If we
n
in state 1
3
3
1
2
2
1
1
1
q7424.0DA
q8224.0DA
q91.030.033.028.0DA
11.2 The price of an A-D security is the (subjective) probability weighted MRS in the
corresponding state. It is determined by three considerations: the discount factor which is
imbedded in the MU of future consumption, the state probability and the relative
11.3 We determine a term structure for each initial state.
To–day’s state is 1:
0383.1r1 1193.1r1
0392.1r1 0800.1r1
0417.1
43.053.0
1
r1
3
1
3
3
1
2
1
2
2
1
1
1
11.4 We determine the price of A-D securities for each date, starting with bond 1 for date 1, q1
2
2
1
21 r1
1100
r1
100
q1100q100900
which gives
7309.0
1100
q100900
q1
2
. Similarly
11.5. Options and market completeness.
The put option has payoffs [ 1,1,1,0]. The payoff matrix is then
0111
1100
1110
1010
m
Of course, the fourth row gives the payoffs of the put option. We have to solve the system
1000
0100
0010
0001
mw
The matrix on the RHS is the A-D securities payoff matrix. The solution is
0111
0011
0110
1121
w
We could also have checked the determinant condition on matrix m, which states that for
11.6. a) An A-D security is an asset that pays out 1 unit of consumption in a particular state of
the world. The concept is very useful since if are able to extract A-D prices from traded
assets they enable us to price every complex security. This statement is valid even if no
b) Markets are not complete : Determinant of the payoff matrix = 0.
c) Note: # of assets # of states.
Completeness can be reached by adding a put on asset one with strike 12 (Det = 126).
3
2
2
1
1
0
0
0
An A-D security with puts:
long put on B (strike 8), two short puts on B (strike 7), long put on B (strike 6)
0
1
2
3
-2
0
0
1
2
+
0
0
0
1
=
0
1
0
0
11.7. If today’s state is state 1, to get $1.– for sure tomorrow using Arrow-Debreu prices, I need
to pay
q11 + q12; thus
1
1
0
0
0
0
0
1
0
q21 = 0.4
q22 = 0.4
11.8. The dividends (computed on the face value, 1000) are d1 =80, d2 =65. The ratio is d1/d2;
buying 1 unit of bond 1 and selling d1/d2 units of bond 2, we can build a 5-yr zero-coupon
11.9. a.
:r1
05263.r;
950
1000
)r1(;
)r1(
1000
950 11
1
:r2
0660.r;
880
1000
)r1(;
)r1(
1000
880 22
2
2
2
1
:r3
0863.r;
780
1000
)r1(;
)r1(
1000
780 33
3
3
3
1
b. We need the forward rates
21 f
and
12 f
.
(i)
3
3
2
211 )r1()f1)(r1(
1035.1
)05263.1(
0863.1
)f1( 2
1
3
21
3
312
2
2)r1()f1()r1(
1281.1
)0660.1(
)0863.1(
)f1( 2
3
12
So
21f
= .1035, and
12 f
= .1281.
The
2
21123T )f1(M1)f1)(M25.1(CF
2
)1035.1(M1)1281.1)(M25.1(
c. To lock in the
12 f
applied to the 1.25M, consider the following transactions :
We want to replicate
t=0
1
2
3
-1.25M
1.410M
Short : 1250 2 yr bond
Long : 1410 3 yr bond
Consider the corresponding cash flows :
t=0
1
2
3
Short : (1250)(880)= +1,100,000
-1,250,000
Long : -(1410)(780)= – 1,100,000
1,410,000
Total 0
-1,250,000
1,410,000
To lock in the
21 f
(compounded for two periods) applied to the 1M ; consider the
t=0
1
2
3
-1M
1.2177M
Short : 1000 1 yr bonds.
Consider the corresponding cash flows :
t=0
1
2
3
Short : (1000)(950)= 950,000
-1,000,000
Long : -(1217.7)(780)= – 950,000
+1,217,700
Total 0
-1,000,000
+1,217,700
The portfolio that will allow us to invest the indicated cash flows at the implied forward
rates is :
11.10 a.
The optimization problem of the first agent is
122111122111qqqq s.t. eecccMaxEU
.
1
2
21
22
11
12
21
22
11
12
2
111111 e
e
c
c
c
c
c
c
c
c
q
q
.
Using
21
1qq
and after some manipulation we get
21
1
2
21
2
1
1
1
1
ee
e
q
ee
e
q
.
b.
If
1
A-D prices are
21
1
2
21
2
1
ee
e
q
ee
e
q
.
The price of the risky asset is
212 2
2
1qqP
.
Now we insert A-D prices and since endowments are
212
211
2
2
1
QQe
QQe
the pricing formula
21
21
254
45
QQ
QQ
P
follows.