5.2 Binomial Option Pricing:
NPeriods

n this topic, we apply the riskneutral valuation technique to value a European call option that lasts for nperiods. The presentation is based upon the work of Cox, Ross, and Rubinstein (1979).
The tree for the stock in the twoperiod problem is shown in
Figure 5.1.
Figure 5.1
Binomial Tree:
2Period
In this case, we can write the call option value with respect
to riskneutral probabilities p as:
This twoperiod form can be rearranged into the general Nperiod
binomial option pricing model by summing over the 2n
possible paths that the underlying stock price can take.
To take a closer look at the derivation of how the Nperiod
model generalizes the twoperiod model, you can read topic 5.3, Binomial
Option Pricing Model: NPeriod
Derivation.
The nperiod
binomial option pricing model is expressed as:
Here, p
is the riskneutral probability of a realized uptick u,
and q = pu/r, which has the property that 0 < q < 1 because q
= u(r  d)/(r(u  d))
and u > r. We can
interpret q
as a "riskneutral terminal valueweighted probability." F is
the probability of at least m upticks occurring from n ticks computed relative
to riskneutral probabilities p, and
riskneutral valueweighted probabilities q.
This model includes two binomial probability distribution
terms as a function of n, the number
of periods, and m, which is the
smallest number of upticks required for the call option to finish
"inthemoney."
The first term is the present value of the expected revenue
from exercising an inthemoney option. This
term is not simply equal to S times
the probability of the call being inthemoney, because the expected revenue
depends on the terminal stock price associated with each path.
The critical information associated with a path and its terminal value is
the number of upticks realized plus the constants u
and r along the path. Therefore,
we can calculate the expected revenue from the current stock price weighted by
the riskneutral terminal valueweighted probability.
The second term is the (riskneutral) probability that the
call option will be inthemoney multiplied by the exercise price and discounted
by the riskfree rate. Therefore,
it is the present value of the expected exercise cost.
If you want to see the derivation of how the Nperiod
model generalizes the twoperiod model, see
topic 5.3, Binomial Option Pricing
Model: NPeriod
Derivation, otherwise you can skip to the limiting behavior of the binomial
option pricing model, as periods become smaller and smaller, in
topic 5.4, Binomial Option Pricing:
Limiting Results).