📈 Black-Scholes Call Option Pricing – Beginner Friendly

🎯 What Are We Doing?
We are pricing a European call option using the Black-Scholes formula. This method gives us a fair price based on math, market conditions, and time.

📌 Step 1: Inputs Given
• Stock Price S = 21
• Strike Price X = 21
• Risk-Free Rate r = 5% = 0.05
• Time to Expiry t = 0.36 years (about 4.3 months)
• Volatility σ = 0.3 (30% annual standard deviation)

📈 Step 2: Formula Overview
C = S × N(d1) − X × e−rt × N(d2)
Where:
• d1 = [ln(S/X) + (r + σ²/2)t] / [σ × √t]
• d2 = d1 − σ × √t

🧮 Step 3: Calculate d1 and d2
1. ln(S/X) = ln(21/21) = 0
2. (r + σ²/2)t = (0.05 + 0.3² / 2) × 0.36 = (0.05 + 0.045) × 0.36 = 0.095 × 0.36 = 0.0342
3. σ × √t = 0.3 × √0.36 = 0.3 × 0.6 = 0.18
4. d1 = 0.0342 / 0.18 = 0.19
5. d2 = d1 − 0.18 = 0.19 − 0.18 = 0.01

📊 Step 4: Get N(d1) and N(d2)
We use the normal distribution table or a calculator:
• N(d1) = N(0.19) ≈ 0.5753
• N(d2) = N(0.01) ≈ 0.5040

💰 Step 5: Final Call Option Price
Plug into the formula:
C = 21 × 0.5753 − 21 × e−0.05×0.36 × 0.5040
• First part: 21 × 0.5753 = 12.0813
• Discount factor: e−0.05×0.36 = e−0.018 ≈ 0.9822
• Second part: 21 × 0.9822 × 0.5040 = 10.3943
• Call price = 12.0813 − 10.3943 = 1.687