c
Mathematical Techniques in Finance, Solution of Exercise 9.2, °2003
Aleš Černý
1
9.2 (a) The coefficient of relative risk aversion of the log-utility is
−
V (ln V )00
V × V −2
=
= 1,
(ln V )0
V −1
whereas the square root utility has coefficient of relative risk aversion equal to
¡ ¢
√
3
V × 12 × − 12 × V − 2
1
V ( V )00
√
=−
= .
−
1 − 12
0
2
( V)
2V
Consequently a log-utility investor is more risk averse than a square root utility
investor.
(b) Let us find the unconditional objective and risk-neutral probabilities for
each path in the tree. Since the one-step objective probabilities are 12 each, the
uncondional probabilities will be
puu
pud
pdu
pdd
1
2
1
= pu × pd =
2
1
= pu × pd =
2
1
= pd × pd =
2
= pu × pu =
1
2
1
×
2
1
×
2
1
×
2
×
1
,
4
1
= ,
4
1
= ,
4
1
= .
4
=
The one-step risk-neutral probabilities are
qu
=
qd
=
Rf − Rd
1.005 − 0.98
5
=
=
Ru − Rd
1.04 − 0.98
12
7
,
12
therefore
25
,
144
quu
= qu × qu =
qud
= qdu = qu × qd =
qdd
= qd × qd =
35
,
144
49
.
144
Hence the unconditional change of measure is
m2 (uu) =
m2 (ud) =
m2 (du) =
m2 (dd) =
quu
puu
qud
pud
qdu
pdu
qdd
pdd
25 1
/
144 4
35 1
=
/
144 4
35 1
=
/
144 4
49 1
=
/
144 4
=
25
,
36
35
=
,
36
35
=
,
36
49
=
.
36
=
2
c
Mathematical Techniques in Finance, Solution of Exercise 9.2, °2003
Aleš Černý
(c) The budget constraint of our investor is
· ¸
V2
EQ
= V0 ,
0
Rf2
which means that the unique time t = 0 no-arbitrage value of the final period
wealth has to be V0 . With our specific numbers this is
25
144
35
49
× V2 (ud) + 144
× V2 (du) + 144
× V2 (dd)
= 1000.
2
1.005
Using the change of measure one can write this as follows:
·
¸
V2
m
EP
2 2 = V0 .
0
Rf
× V2 (uu) +
35
144
(1)
(d) We are maximizing:
EP
0 [ln V2 ] =
1
(ln V2 (uu) + ln V2 (ud) + ln V2 (du) + ln V2 (dd))
4
with respect to V2 (uu), V2 (ud), V2 (du), V2 (dd) and subject to the linear constraint (1). We can rewrite this as an unconstrained optimization using the
Lagrangean function
1
1
1
1
max L = ln V2 (uu) + ln V2 (ud) + ln V2 (du) + ln V2 (dd)−
V2
4
4
4
4
¶
µ 25
35
35
49
× V2 (uu) + 144
× V2 (ud) + 144
× V2 (du) + 144
× V2 (dd)
−
1000
.
− λ 144
1.0052
(e) The first order conditions yield
0 =
0 =
0 =
0 =
∂L
∂V2 (uu)
∂L
∂V2 (ud)
∂L
∂V2 (du)
∂L
∂V2 (dd)
1 1
4 V2 (uu)
1 1
=
4 V2 (ud)
1 1
=
4 V2 (du)
1 1
=
4 V2 (dd)
=
1
25
,
144 1.0052
35
1
,
−λ
144 1.0052
1
35
,
−λ
144 1.0052
49
1
.
−λ
144 1.0052
−λ
Solving for the wealth gives us
V2 (uu) =
V2 (ud) =
V2 (du) =
V2 (dd) =
36 × 1.0052
25λ
36 × 1.0052
35λ
36 × 1.0052
35λ
36 × 1.0052
.
49λ
(2)
(3)
(4)
(5)
3
c
Mathematical Techniques in Finance, Solution of Exercise 9.2, °2003
Aleš Černý
Note that the right hand side takes the form Rf2 / (m2 λ), which means the change
of measure determines the optimal level of wealth.
(f ) Substitute the optimal values of V2 from (2)-(5) back into te budget constraint (1) to obtain the value of λ
25
144
×
36×1.0052
25λ
+
35
144
×
36×1.0052
35λ
35
+ 144
×
1.0052
36×1.0052
35λ
+
49
144
×
36×1.0052
49λ
= 1000.
Log utility is a very special case when the left hand side simplifies enormously
and one ends up simply with
1
= 1000.
λ
(g) Now take this value of λ and substitute it back into (2)-(5) to obtain the
final expressions for optimal wealth
36
25
36
V2 (ud) = 1000 ×
35
36
V2 (du) = 1000 ×
35
36
V2 (dd) = 1000 ×
49
V2 (uu) = 1000 ×
× 1.0052 = 1454.44,
× 1.0052 = 1038.88,
× 1.0052 = 1038.88,
× 1.0052 = 742.06.
(h) This procedure will remind the reader of option pricing, except that the
terminal cash flow of the option is given, whereas here terminal wealth has been
determined optimally. The fastest way to perform the required calculations is
to find the value of self-financing portfolio by risk-neutral pricing, then compute
delta from (5.9) and finally evaluate the bank balance as the difference between
the portfolio value and the wealth invested in stock:
V1 (u) =
V1 (d) =
V0
=
5
7
1454.44 + 12
1038.88
qu V2 (uu) + qd V2 (ud)
= 12
= 1206.00
Rf
1.005
5
7
1038.88 + 12
742.06
qu V2 (uu) + qd V2 (ud)
= 12
= 861.43
Rf
1.005
5
7
1206.00 + 12
861.43
qu V1 (u) + qd V1 (d)
= 12
= 1000.00
Rf
1.005
Note that we must get V0 = 1000, otherwise there is an error in the calculation.
d
Now we can compute the number of stocks from the formula θ = SVuu −V
−Sd :
θ1 (u) =
θ1 (d) =
θ0
=
V2 (uu) − V2 (ud)
1454.44 − 1038.88
=
= 666.0,
S2 (uu) − S2 (ud)
10.816 − 10.192
V2 (du) − V2 (dd)
1038.88 − 742.06
=
= 504.6,
S2 (du) − S2 (dd)
10.192 − 9.604
V1 (u) − V1 (d)
1206.00 − 861.43
=
= 574.3.
S1 (u) − S1 (d)
10.4 − 9.8
4
c
Mathematical Techniques in Finance, Solution of Exercise 9.2, °2003
Aleš Černý
Finally, we will find the bank statement by calculating V −θS at every node.
The self-financing strategy is summarized in Figure 1.
stock price
optimal wealth
0.5
10.816
10.4
0.5
0.5
10.192
10
0.417
0.5
10.192
0.583
9.8
0.583
1038.9
0.417
1038.9
0.583
742.1
861.4
0.5
t=1
9.604
t=2
Bank account
t=0
t=1
t=2
Number of shares
1454.4
0
-5719.9
666.0
1038.9
-4742.9
0
574.3
1038.9
0
-4085.6
504.8
742.1
t=0
1454.4
1000.0
0.5
t=0
0.417
1206.0
t=1
t=2
0
t=0
t=1
t=2
Figure 1: Optimal investment strategy maximizing EP
0 [ln V2 ] with V0 = 1000.
(i) Because Vud = Vdu in this particular example we can express Vt as a
function of S and t. In general, even if stock returns are IID, optimal Vt cannot
be expressed as a function of St and t (this happens when Rf is time-dependent),
in which case Vt becomes an additional state variable.