180 Mathematics for Finance
8.1.4 Cox–Ross–Rubinstein Formula .......................
The payoff for a call option with strike priceXsatisfiesf(x)=0forx≤X,
which reduces the number of terms in (8.4). The summation starts with the
leastmsuch that
S(0)(1 +u)m(1 +d)N−m>X.
Hence
CE(0) = (1 +r)−N∑N
k=m(
N
k)
pk∗(1−p∗)N−k(
S(0)(1 +u)k(1 +d)N−k−X)
.
This can be written as
CE(0) =x(1)S(0) +y(1),relating the option price to the initial replicating portfoliox(1),y(1), where
x(1) = (1 +r)−N∑N
k=m(
N
k)
pk∗(1−p∗)N−k(1 +u)k(1 +d)N−k,y(1) =−X(1 +r)−N∑N
k=m(
N
k)
pk∗(1−p∗)N−k.The expression forx(1) can be rewritten as
x(1) =
∑N
k=m(
N
k)(
p∗1+u
1+r)k(
(1−p∗)1+d
1+r)N−k
=∑N
k=m(
N
k)
qk
(1−q)N−k,where
q=p∗
1+u
1+r.
(Note thatp∗1+1+ur and (1−p∗)1+1+dradd up to one.) Similar formulae can be
derived for put options, either directly or using put-call parity.
These important results are summarised in the following theorem, in which
Φ(m, N, p) denotes the cumulative binomial distribution with N trials and
probabilitypof success in each trial,
Φ(m, N, p)=∑mk=0(
N
k)
pk(1−p)N−k.