Options Pricing

Black-Scholes Option Price

Theoretical price of a European-style option based on stock price, strike, time, volatility, and risk-free rate.

01 / Formula

Formula

C = S₀ × N(d₁) − Ke^(−rT) × N(d₂) where d₁ = [ln(S₀/K) + (r + σ²/2)T] / (σ√T)

02 / Variables

Variables

C: Call option price
S₀: Current stock price
K: Strike price
r: Risk-free rate
σ: Volatility
T: Time to expiration (years)
N(x): Cumulative normal distribution

03 / Worked Example

Worked Example

Scenario: Stock at $100, strike $100, 1 year, 20% volatility, 5% rate ≈ $10.45 call value.

Steps:

Calculate d₁ using the full formula with ln(S/K), rate, volatility
Calculate d₂ = d₁ − σ√T
Look up N(d₁) and N(d₂) from normal distribution table
Apply formula: S₀ × N(d₁) − Ke^(−rT) × N(d₂)

04 / When to Use

When to Use This Formula

Estimate fair value for options
Compare market price to theoretical value
Evaluate whether options are mispriced

05 / Limitations

Limitations & Assumptions

Assumes European exercise (can only be exercised at expiration)
Assumes constant volatility (unrealistic)
Ignores dividends and transaction costs
Assumes continuous trading with no bid-ask spread

06 / Questions

Common Questions

Why is this model important if its assumptions are unrealistic?

Because it provides a structured framework and baseline. Deviations from Black-Scholes prices inform traders about market consensus on volatility and risk.

Can I use Black-Scholes for American options?

Not directly. American options (exercisable anytime) require more complex models that account for early exercise optionality.