This is a very clear and detailed breakdown of how to price a call option using the binomial model. Great for students new to finance or those who want to see every step explained with care.
🎯 Step 0: What Are We Doing?
We want to find out what a European call option is worth today. This option gives you the right to buy a stock in the future at a fixed price (called the strike price). We use a simple 1-step binomial tree to calculate the price.
📌 Step 1: Start with What’s Given
You are given these values:
• Stock price today: S = $40
• Up factor: u = 1.25 → If price goes up: 40 × 1.25 = $50
• Down factor: d = 0.75 → If price goes down: 40 × 0.75 = $30
• Strike price: K = $35
• Risk-free interest rate: r = 8% or 0.08
• Time to expiration: T = 1 year
📈 Step 2: What Happens at the End?
We figure out the option value at the end (maturity):
• If stock goes up to $50: Option =
max(50 - 35, 0) = $15
• If stock goes down to $30: Option =
max(30 - 35, 0) = $0
We only make money if the stock ends up above $35.
🧮 Step 3: Calculate Risk-Neutral Probability
We pretend we’re in a “risk-neutral” world (a math trick) to figure out expected value.
Formula:
Pu = (1 + r − d) / (u − d)
Plug in values:
(1.08 − 0.75) / (1.25 − 0.75) = 0.66
This means there's a 66% “neutral chance” the price goes up.
💰 Step 4: Calculate the Option Price Today
Now use the expected value formula:
Option = [Pu × Payoff(up) + Pd × Payoff(down)] / (1 + r)
• Up payoff = $15, Down payoff = $0
• Pu = 0.66 → Pd = 1 − 0.66 = 0.34
• Expected Value =
[0.66 × 15 + 0.34 × 0] / 1.08 = 9.9 / 1.08 = $9.17
✅ Final Answer: The option is worth $9.17 today.