- Option Pricing 177
Figure 8.1 Branchings in the two-step binomial treeFor each of the three subtrees in Figure 8.1 we can use the one-step repli-
cation procedure as described above. At time 2 the derivative security is rep-
resented by its payoff,
D(2) =f(S(2)),
which has three possible values. The derivative security priceD(1) has two
values
1
1+r[
p∗f(Suu)+(1−p∗)f(Sud)]
,
1
1+r[
p∗f(Sdu)+(1−p∗)f(Sdd)]
,
found by the one-step procedure applied to the two subtrees at nodes u and d.
This gives
D(1) =1
1+r[
p∗f(S(1)(1 +u)) + (1−p∗)f(S(1)(1 +d))]
=g(S(1)),where
g(x)=
1
1+r[
p∗f(x(1 +u)) + (1−p∗)f(x(1 +d))]
.
As a result,D(1) can be regarded as a derivative security expiring at time 1
with payoffg. (Though it cannot be exercised at time 1, the derivative security
can be sold forD(1) =g(S(1)).) This means that the one-step procedure can
be applied once again to the single subtree at the root of the tree. We have,
therefore,
D(0) =^1
1+r[
p∗g(S(0)(1 +u)) + (1−p∗)g(S(0)(1 +d))]
.
It follows that
D(0) =1
1+r[
p∗g(Su)+(1−p∗)g(Sd)]
=^1
(1 +r)^2[
p^2 ∗f(Suu)+2p∗(1−p∗)f(Sud)+(1−p∗)^2 f(Sdd)]
.
The last expression in square brackets is the risk-neutral expectation off(S(2)).
This proves the following result.