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CHAPTER 12.
12.1. a) Markets are complete. Find the state prices from
1q3 1.1
1
qqq
8q15q10q5
3
321
321
3/1q
02424.0q
55151.0q
3
2
1
b) The put option has a price of 3q1. Risk neutral probabilities are derived from
3 2, 1,s whereq1.1 Ss
36667.0
02667.0
60667.0
3
2
1
c) Consumption at date 0 is 1. The pricing kernel is given by
3 2, 1,s where
p
q
m
s
S
s
11111.0m
06060.0m
83838.1m
3
2
1
12.2.
3 2, 1,s where
p
q
m
s
S
s
Program for agent 1
2
1
1
12
2
11
1
12
c,c cln
3
2
cln
3
1
qcqcq51q10max
2
1
1
1
The FOC is
0
c
1
3
2
q
0
c
1
3
1
q
2
1
2
1
1
1
and similarly for agent 2. This yields
2
2
2
1
1
2
1
1
2
2
2
1
1
2
1
1
cc,cc
c
1
c
1
,
c
1
c
1
Using the market clearing conditions we get
2
11
cc
2
5
cc
2
2
2
1
1
2
1
1
so that q1 = 2/15 and q2 = 4/33.
Now construct the risk neutral probabilities as follows:
21
2
2
21
1
1
qq
q
qq
q
which satisfy the required conditions to be probabilities.
Computation of the risk-free rate is as usual: q1 + q2 = 1/(1+rf).
The market value of endowments can be computed as follows
2
i2
1
i1
2
i2
1
i1
f
ieqeqee
r1
1
MV
.
