The One-Period Binomial Model
First, let us define what we mean by a
one period world. An option has a defined life. Assume
the option's life is one unit of time, typically
expressed in days. This time period can be as
short or as long as necessary.
The model is called a binomial model.
It allows the stock price to either up or down, possibly at different rates. A
binomial probability distribution is a distribution in which there are two
outcomes or states. The probability of an up or down movement is governed by the binomial probability distribution. Because of this, the model is also called a two-state model.
Consider a world in
which there is a stock price at S on which call options are available. The
call has one period remaining before it expires. The beginning oldie period is today and is referred to as time 0. The
end of the period is referred to as time 1. When the call expires, the stock
can take one of two values: It can go up by a factor of u or
down by a factor of d. If it goes up,
the stock price will be Su.
If it goes up, the stock
price will be Sd.
Alternatively, suppose the stock price
is currently $10 and can go either up to $13 or up to $11. Thus, u = 1.30 and d = 1.1. The variable u and d, therefore,
are 1.0 plus the rate of return on the stock. When the call expires, the stock
will be either Su, =
10(1.3) = $13 or Sd 10(1.1) = $11.
Consider a call
option on the stock with an exercise price of X and a current price of C. When the option expires, it will be worth either Cu
or Cd. Because at expiration the call price is its intrinsic value:
Cu = Max[0, Su – X]
Cu =
Max[0, Sd – X]
The
following figure illustrates the paths of both the stock and the call price movements.
This diagram is simple but will become more complex when we introduce the two-period model.
The risk-free rate falls between the
rate of return if the stock goes up and the rate of return if the stock goes
down. Thus d < I+r < u. We shall assume that investors can borrow and lend at the risk free rate.
The formula for C is
developed by constructing a riskless portfolio of stock and options. A riskless
portfolio should earn the risk free rate. Given the stock's values and the
riskless return of the portfolio, the call's values can be inferred from the other
variables. This riskless portfolio is called
the hedge
portfolio and consists of h shares of stock and a single written call. The
model provides the hedge
ratio, h. The current value of the portfolio is the value of h shares minus the value of the short
call. We subtract the cell's value because the shares are assets and the short
call is a liability.
Thus, the portfolio
value is assets minus liabilities, or simply net worth. The current portfolio is
denoted as V, where V = hS
- C. At expiration, the portfolio will have value Vu, if the
stock goes up and Vd if
the stock goes down. Thus:
Vu = hSu-
Cu
Vd = hSd-
Cd
If the same outcome is achieved
regardless of what the stock price does, the position is riskless. We
can choose a value of h that will make this happen. We simply set Vu = Vd so that:
hSu - Cu
= hSd - Cd
Solving for h gives:
Moreover, if the
portfolio's current value grows at the risk free rate, its value at the option's expiration will be
(hS -
C)(1+r).
The two values of
the portfolio at expiration, Vu
and Vd, are equal, so we can choose either one. Choosing Vu and setting it equal to the original value of the portfolio compounded at the risk
free rate gives:
V(1
+ r) = Vu
(hS - C)(1+r) = hSu - Cu
Substituting the formula for h and
solving the equation for C gives the option pricing formula:
where p is defined as
The variables
affecting the call option price are the
current stock price, S, the exercise
price, X, the risk free rate, r, and the parameters, u and d, which define the possible future stock prices at expiration.
