Textbook-style one-step example using both the hedge method and the Pu / Pd method.
This page explains the Western Cellular call option example in a classroom-friendly way.
It shows two paths to the same answer:
(1) riskless hedge / replication and
(2) risk-neutral probabilities Pu and Pd.
Western Cellular stock is selling for $40 today. One year from now,
the stock will be either $50 or $30.
A call option allows the holder to buy one share at an exercise price of $35.
The risk-free rate is 8%.
Inputs
Current stock price: $40
Up-state stock price: $50
Down-state stock price: $30
Exercise price: $35
Risk-free rate: 8%
Time to expiration: 1 year
Question
What is the value of the call option today?
We will solve the same problem in two different but equivalent ways.
So the hedge portfolio is:
buy 0.75 share and sell 1 call.
Step 3: Show that the hedge portfolio is riskless
State
0.75 Share Value
Short Call Value
Portfolio Value
Up
0.75 × 50 = $37.50
−$15.00
$22.50
Down
0.75 × 30 = $22.50
−$0.00
$22.50
The ending portfolio value is the same in both states: $22.50.
That is why it is a riskless hedge portfolio.
Step 4: Discount the riskless payoff back to today
PV of hedge portfolio = 22.50 / 1.08 = 20.83
Today, the hedge portfolio costs:
0.75 × Stock price - Call price = PV of hedge portfolio
0.75 × 40 - C = 20.83
30 - C = 20.83
C = 9.17
Delta
0.75
Shares per call sold
Riskless ending value
$22.50
Same in both states
Present value of hedge
$20.83
Discounted at 8%
Call price today
$9.17
Hedge method answer
Method 2: Same Answer Using Pu and Pd
The one-step binomial model can also be written using risk-neutral probabilities.
This gives the same option value as the hedge approach when the same compounding convention is used.
Step 1: Up and down factors
u = Su / S = 50 / 40 = 1.25
d = Sd / S = 30 / 40 = 0.75
Step 2: Risk-neutral probabilities
Pu = (1 + r - d) / (u - d)
= (1.08 - 0.75) / (1.25 - 0.75)
= 0.66
Pd = 1 - Pu
= 1 - 0.66
= 0.34
Step 3: Option payoffs at expiration
Cu = max(50 - 35, 0) = 15
Cd = max(30 - 35, 0) = 0
Step 4: Price the call using the binomial equation
Important: Pu and Pd are risk-neutral probabilities, not real-world probabilities.
They are pricing weights implied by no-arbitrage.
Why Do Both Methods Give the Same Answer?
Hedge method
Builds a portfolio that has the same payoff in both states.
Once the payoff is certain, it must earn the risk-free rate.
Pu / Pd method
Uses risk-neutral probabilities to compute the discounted expected payoff directly.
Both methods are based on the same no-arbitrage logic. They are just two different ways
to express the same one-step binomial pricing model.
Method
Main Equation
Answer
Riskless hedge
0.75 × 40 - C = 22.50 / 1.08
$9.17
Pu / Pd method
C0 = [0.66 × 15 + 0.34 × 0] / 1.08
$9.17
Plain-English Explanation
We are looking for a combination of stock and option that removes uncertainty.
In this problem, buying 0.75 share and selling 1 call creates a portfolio
that will be worth $22.50 no matter what happens.
What we should notice
The stock goes up or down, but the hedge portfolio ends at the same value in either case.
Why that matters
If the payoff is certain, the portfolio must earn the risk-free rate.
What that lets us do
Discount the certain future value back to today, then solve for the option price.
Correct idea:
Buy 0.75 share and sell 1 call → riskless hedge
Also correct:
Use Pu and Pd in the one-step binomial equation
Wrong shortcut:
Buy 1 share and sell 1 call → not riskless in this problem