84 Mathematics for Finance
Example 4.5
LetA(0) = 100,A(1) = 110,A(2) = 121 and suppose that stock prices can
follow four possible scenarios:
Scenario S(0) S(1) S(2)
ω 1 90 100 112
ω 2 90 100 106
ω 3 90 80 90
ω 4 90 80 80The tree of stock prices is shown in Figure 4.2. The risk-neutral probabilityP∗
is represented by the branching probabilitiesp∗,q∗,r∗at each node. Condition
Figure 4.2 Tree of stock prices in Example 4.5(4.3) forS ̃(n)=S(n)/A(n) can be written in the form of three equations, one
for each node of the tree,
100
110p∗+^80
110(1−p∗)=^90
100,
112
121
q∗+^106
121(1−q∗)=^100
110,
90
121
r∗+^80
121(1−r∗)=^80
110.
These can be solved to find
p∗=19
20
,q∗=2
3
,r∗=4
5
.
For each scenario (each path through the tree) the corresponding risk-neutral
probability can be computed as follows:
P∗(ω 1 )=p∗q∗=19
20
×
2
3
=
19
30
,
P∗(ω 2 )=p∗(1−q∗)=19
20
×
(
1 −
2
3
)
=
19
60
,
P∗(ω 3 )=(1−p∗)r∗=(
1 −
19
20
)
×
4
5
=
1
25
,
P∗(ω 4 )=(1−p∗)(1−r∗)=