5.3 Binomial
Option Pricing: NPeriod Derivation

n the twoperiod case, we can write the call option value
with respect to riskneutral probabilities
p
as:
This twoperiod form can be rearranged into a form that
allows the general structure to be deduced.
Written in this form, the general formula for nperiods
is
That is, we are summing over the 2n possible paths that the
underlying stock price can take. Note,
however, that because u, d, and r
are constants, the option pricing problem is not pathdependent.
In this case, a path can be defined in terms of some number
of realized upticks and some number of realized downticks that sum to n.
With this observation, let m be
the smallest number of upticks for which a path is inthemoney.
We can ignore any path with less than m upticks because the call option
will finish outofthemoney.
Thus, if we let m
be the smallest integer such that Sumdnm
> X, then we can simplify the general formula by eliminating all stock
paths with fewer than m upticks:
Since we know that the option will finish inthemoney on all
these paths, we no longer need to take the max, and so we get:
This expression still has the simple form of the twoperiod
model. The second term is the
(riskneutral) probability that the call option will be inthemoney multiplied
by the exercise price, all discounted by the riskfree rate.
Therefore, it is the discounted value of the expected payment.
The first term is the discounted value of the expected
revenue. Unlike the expected
payment, the expected revenue is not equal to S times the probability of the
call being inthemoney, because the revenue depends on the stock price (while
the payment does not). Rather, we
calculate the expected revenue, as done in the formula.
Using the definition of p,
this can be rewritten as
Let q = pu/r.
Then, using the binomial distribution notation, the present
value of the call option equals
Next, we consider the limiting behavior of this pricing model
as n gets larger.
This is described in topic 5.4, Binomial
Option Pricing: Limiting Results,
and formally derived in the technical appendix to this chapter.