- Option Pricing 179
It follows that
D(0) =1
1+r[
p∗h(Su)+(1−p∗)h(Sd))]
=
1
(1 +r)^2[
p^2 ∗g(Suu)+2p∗(1−p∗)g(Sud)+(1−p∗)^2 g(Sdd))]
=
1
(1 +r)^3[
p^3 ∗f(Suuu)+3p^2 ∗(1−p∗)f(Suud)+3p∗(1−p∗)^2 f(Sudd)+(1−p∗)^3 f(Sddd))]
.
The emerging pattern is this: Each term in the square bracket is characterised
by the numberkof upward stock price movements. This number determines
the power ofp∗and the choice of the payoff value. The power of 1−p∗is the
number of downward price movements, equal to 3−kin the last expression,
andN−kin general, whereNis the number of steps. The coefficients in front
of each term give the number of scenarios (paths through the tree) that lead
to the corresponding payoff, equal to
(N
k)
=k!(NN−!k)!,thenumberofk-element
combinations out ofNelements. For example, there are three paths through
the 3-step tree leading to the node udd.
As a result, in theN-step model
D(0) =
1
(1 +r)N∑N
k=0(
N
k)
pk∗(1−p∗)N−kf(
S(0)(1 +u)k(1 +d)N−k)
. (8.4)
The expectation off(S(N)) under the risk-neutral probability can readily be
recognised in this formula. The result can be summarised as follows.
Theorem 8.4
The value of a European derivative security with payofff(S(N)) in theN-
step binomial model is the expectation of the discounted payoff under the
risk-neutral probability:
D(0) =E∗(
(1 +r)−Nf(S(N)))
.
Remark 8.1
There is no need to know the actual probabilitypto computeD(0). This re-
markable property of the option price is important in practice, as the value
ofpmay be difficult to estimate from market data. Instead, the formula for
D(0) featuresp∗, the risk-neutral probability, which may have nothing in com-
mon withp, but is easy to compute from (8.2).