If a call option is held on the stock at an exercise price of E then the payoff on the call is either

C_u=max(S_u-E,0)
or
C_d=max(S_d-E,0).
 

Let the risk-free interest be r and assume d<r<u.

Now consider a portfolis made up of one written call and h shares of the stock. That is to say, the owner of the portfolio owns h shares of the stock and then sells (writes) one call with an expiration date of one period. If the stock price goes up the portfolio has a value of

V_u = hS(1+u) - C_u
 

and if it goes down

V_d = hS(1+d) - C_d.
 

Suppose h is chosen so that the portfolio has the same price whether the stock price goes up or goes down. The value of h that achieves this condition is given by

hS(1+u) - C_u = hS(1+d) - C_d
or
h = (C_u-C_d)/(S_u-S_d)
= (max(S_u-E,0)-max(S_d-E,0))/(S_u-S_d).
 

Thus, given only S,E,u,and d, the ratio h can be determined. In particular, it does not depend upon the probability of a rise or fall.

The value of h that make the value of the portfolio independent of the stock price is called the hedge ratio. A portfolio that is perfectly hedged is a risk-free portfolio so its value should grow at the risk-free rate; i.e., r.

The current value of the hedged portfolio is the value of the stocks less the liability involved with having written the call. If C represents the value of owning the call then the liability involve with having written the call is -C. Therefore the value of the portfolio is (hS-C). After one period of growing at the risk-free rate its value will be (1+r)(hS-C), which is the same as (hS(1+u)-C_u)=(hS(1+d)-C_d). Solving for C gives

C = hS - (hS(1+u)-C_u)/(1+r)
= hS - hS(1+u)/(1+r) + C_u/(1+r)
= hS[1 -(1+u)/(1+r)] + C_u/(1+r)
= (hS(r-u) + C_u)/(1+r)
= [-hS(u-r) + C_u]/(1+r)
 

Noting that

h = (C_u-C_d)/(S(1+u)-S(1+d))
= (C_u-C_d)/S(u-d)
 

then

C = [(C_u-C_d)(r-u)/(u-d) + C_u]/(1+r)
= [C_u[(r-u)/(u-d) + 1] -C_d(r-u)/(u-d)]/(1+r)
C = [C_u(r-d)/(u-d) +C_d(u-r)/(u-d)]/(1+r)
 

If (r-d)/(u-d) is denoted as p then

1-p = [(u-d)-(r-d)]/(u-d)
= (u-r)/(u-d)
so
C = [pC_u + (1-p)C_d]/(1+r)
 

Thus the value of the call option is the discounted value of a weighted average of the expiration date value of the call.

Example:Let u=+0.1, d=-0.1, r= 0.05, S = 100, and E = 95.
Then S_u=110 and S_d=90
and consequently
C_u = 15 and C_d = 0.
 

Therefore

h=(15-0)/(110-90)=0.75
p = (0.05 - (-0.1))/(0.1 - (-0.1))
= 0.15/0.20 = 3/4
C = [(3/4)15 +(1/4)0]/(1.05) = 11.5/1.05 = $10.71.
 

Let us check this out by computing the value of the portfolio.

0.75 share of the $100 stock - $10.71 = $75.00 - $10.71 = $64.29.
 

If the price of the stock rises to $110 then the portfolio will be worth

(.75)(110) - 15 = 82.50 - 15.00 = $67.50.
 

If the price of the stock drops to $90 the portfolio will be worth

(.75)(90) = $67.50.
 

The one-period result may be used to determine the value of a call with two periods left before expiration. The two period results then gives the three period result and so on.

The results look the same as if one were computing the expected value of the expiration date payoff when the probability of stock price going up in one period is p and the probability of going down is (1-p).