Using In Class Case Study as an Example (FYI only)
This page explains the one-step binomial option pricing model in a very basic way.
First, we use the same kind of example discussed in class. Then, below that, there is a
simple interactive calculator so you can change the numbers and see how the option value changes.
The binomial option pricing model is a mathematical method used to estimate the fair value of an option.
It works by modeling the possible future prices of the stock and then working backward to find the option value today.
Case Study Example:
A stock is currently trading at $40. At the end of one year, the stock can go to either
$50 or $30. The exercise price is $35, the risk-free rate is
8%, and the time to expiration is 1 year.
Given
Current stock price: S = $40
Future up-state price: Su = $50
Future down-state price: Sd = $30
Exercise price: K = $35
Risk-free rate: r = 8%
Time to expiration: t = 1 year
What we want
We want to estimate the value of a call option today using a one-step binomial model.
In a one-step model, the stock has only two possible prices at the end of the period:
one up-state and one down-state.
Binomial Tree
$40
/ \
$50 $30
Step-by-Step Explanation
Step 1: Determine the inputs
First, collect the numbers needed for the model:
The current stock price
The possible future stock prices
The exercise price
The risk-free rate
The time until expiration
In our example:
S = 40
Su = 50
Sd = 30
K = 35
r = 0.08
t = 1
Step 2: Calculate the up and down factors
The up factor tells us how much the stock rises in the up-state.
The down factor tells us how much the stock falls in the down-state.
u = Su / S = 50 / 40 = 1.25
d = Sd / S = 30 / 40 = 0.75
So the stock either goes up by a factor of 1.25 or down by a factor of 0.75.
Step 3: Calculate the risk-neutral probability
The risk-neutral probability is not the real-world probability.
It is the probability used inside the pricing model.
Basic classroom version:
p = (1 + r·t - d) / (u - d)
p = (1 + 0.08×1 - 0.75) / (1.25 - 0.75)
p = 0.33 / 0.50
p = 0.66
More accurate continuous-compounding version:
p = (e^(r·t) - d) / (u - d)
p = (e^(0.08×1) - 0.75) / (1.25 - 0.75)
p = (1.0833 - 0.75) / 0.50
p = 0.6666
So the risk-neutral probability of the stock going up is about 0.6666.
The probability of going down is:
Pd = 1 - Pu = 1 - 0.6666 = 0.3334
Step 4: Calculate the option value at the final nodes
At expiration, the value of a call option is:
Call value = max(S - K, 0)
So in the up-state:
Vu = max(50 - 35, 0) = 15
In the down-state:
Vd = max(30 - 35, 0) = 0
Therefore, at the end of the one-year period:
If stock goes to $50, call value = $15
If stock goes to $30, call value = $0
Step 5: Work backward to find the option value today
The option value today is the discounted expected value of the future option values:
Option value = [Pu × Vu + Pd × Vd] / (1 + r)^t
Plug in the numbers:
Option value = [0.6666 × 15 + 0.3334 × 0] / (1.08)^1
Option value = 9.999 / 1.08
Option value ≈ 9.26
Using the rounded classroom value Pu = 0.66, you get:
Option value = [0.66 × 15 + 0.34 × 0] / 1.08
Option value = 9.90 / 1.08
Option value ≈ 9.17
Conclusion: the value of the call option today is about
$9.17 to $9.26, depending on rounding and whether you use the basic classroom probability or the more accurate exponential version.
Interactive One-Step Binomial Calculator
This calculator is intentionally basic. It follows the same one-step logic as the class example above.
Up factor
—
u = Su / S
Down factor
—
d = Sd / S
Risk-neutral probability
—
probability of up-state
Up-state option value
—
value if stock goes up
Down-state option value
—
value if stock goes down
Option value today
—
discounted expected value
Detailed calculation
FYI only: this is a simple teaching calculator for the one-step binomial model.
It is meant to help explain the logic clearly, not to replace live market pricing.
Textbook Case Study Page
This separate page focuses only on the textbook-style riskless hedge explanation for the Western Cellular example.
What is on that page?
It shows the exact case-study logic:
build the hedge, make the portfolio riskless, discount at the risk-free rate,
and solve for the call price.