CHAPTER 13.
13.1. a.
06944.ln25.1ln5.44.1ln25.96.833.ln5.2.1ln5.96.1EU 2
b.
The maximisation problem of the representative agent is
232322222121121211110232322222121121211110
232322222121
2
121211110
s.t.
))ln()ln()ln(())ln()ln(()ln(max
cqcqcqcqcqceqeqeqeqeqe
cccccc
(take consumption at date 0 as a numeraire, its price is 1;
qij is time 0 price of AD security that pays 1 unit of consumption at date i in state j )
The Lagrangian is given by
232322222121121211110
232322222121121211110
cqcqcqcqcqc
eqeqeqeqeqe
EUL
FOC’s are:
0
1
.
.
.
0
1
0
1
23
23
2
23
23
11
11
11
11
00
q
cc
L
q
cc
L
cc
L
A-D prices, risk neutral probabilities, and the pricing kernel can be derived easily from
the FOC’s. For example, at date t = 0 we have:
1111
0
11
11
11
0
11
11
1111 Mc
c
c
1m
U
MU
q
…
2323
0
23
23
23
0
2
23
23
2
2323 Mc
c
c
1m
U
MU
q
where mij is the pricing kernel. Risk neutral probabilities at date 0 are given by:
1211
12
12
1211
11
11 and qq
q
qq
qRNRN
state prices
1
0.16
0.4608
0.331776
0.576
0.4
risk neutral probabilities
1
0.1679656
0.48374093
0.34829347
0.590163934
0.409836066
pricing kernel
1
0.64
0.9216
1.327104
1.152
0.8
c.
valuation
1
0.2304
0.4608
0.2304
0.48
0.48
value
2.8816
2.8816
2.352
1.44
price
1.8816
1.152
space
1.63333333
1
0.8
0.694444444
At date one and two we have one value with payoffs (upper cell) and the value after the
cash flow arrived (lower cell).
option value
0.0608
0
0.152
f.
The price process is as in e. Now we need to solve for u, d, R, and risk neutral
probabilities.
4098.
8681.25.1
8681.02459.1
du
dR
q
02459.1r1R
8681.
8816.1
6333.1
d
25.1
152.1
44.1
8816.1
352.2
u
11
(Compare this value with q11 in b))
The value of the option is
option value
0.0608
0
0.152
g.
In part b we saw that pricing via A-D prices, risk-neutral probabilities, and pricing kernel are
essentially the same. These methods rely on the payoffs of the endowment stream. In contrast to