Black-Scholes Formula
Reference for the Black-Scholes formula for European options.
Covers d1, d2, N(d) calculations, Greeks (delta, gamma, theta, vega), and model assumptions.
The Formula
Call Price: C = S × N(d₁) - K × e⁻ʳᵗ × N(d₂)
Put Price: P = K × e⁻ʳᵗ × N(-d₂) - S × N(-d₁)
d₁ = [ln(S/K) + (r + σ²/2) × t] / (σ × √t)
d₂ = d₁ - σ × √t
Put Price: P = K × e⁻ʳᵗ × N(-d₂) - S × N(-d₁)
d₁ = [ln(S/K) + (r + σ²/2) × t] / (σ × √t)
d₂ = d₁ - σ × √t
The Black-Scholes model, published in 1973 by Fischer Black and Myron Scholes, calculates the theoretical price of European-style options. It assumes constant volatility and no dividends.
Variables
| Symbol | Meaning |
|---|---|
| C, P | Call or put option price |
| S | Current stock price |
| K | Strike price of the option |
| t | Time to expiration (in years) |
| r | Risk-free interest rate (annual) |
| σ | Volatility of the stock (annual standard deviation) |
| N(x) | Cumulative standard normal distribution function |
| e | Euler's number (≈ 2.71828) |
| ln | Natural logarithm |
Example 1
Stock at $100, strike $105, 6 months to expiry, r = 5%, σ = 20%
d₁ = [ln(100/105) + (0.05 + 0.04/2) × 0.5] / (0.20 × √0.5)
= [-0.04879 + 0.035] / 0.1414 = -0.0975
d₂ = -0.0975 - 0.1414 = -0.2389
N(d₁) ≈ 0.4612, N(d₂) ≈ 0.4056
C = 100 × 0.4612 - 105 × e⁻⁰·⁰²⁵ × 0.4056 ≈ $4.63
When to Use It
Use the Black-Scholes formula when:
- Pricing European call and put options
- Calculating implied volatility from market prices
- Understanding how time, volatility, and price affect option values
- Building risk management and hedging strategies
Key Notes
- Call option price: C = S·N(d₁) − K·e^(−rT)·N(d₂): S is current stock price, K is strike price, r is risk-free rate, T is time to expiry, N() is the cumulative normal distribution, and d₁, d₂ depend on all five inputs plus volatility σ.
- Implied volatility: The model is often used in reverse — given an observed market option price, solve for σ. This "implied volatility" is the market's consensus forecast of future volatility. The VIX index is derived from implied volatilities of S&P 500 options.
- Key assumptions and their failures: Black-Scholes assumes constant volatility, continuous trading, no dividends, and log-normal price returns. Real markets have "fat tails" (crash events more common than predicted) and a "volatility smile" (implied σ varies with strike price).
- The Greeks quantify sensitivity: Delta (Δ) = price sensitivity to S; Gamma (Γ) = rate of change of Delta; Theta (Θ) = time decay per day; Vega (ν) = sensitivity to volatility; Rho (ρ) = sensitivity to interest rate. Traders hedge each Greek separately.
- Applications: Black-Scholes is used to price European equity options, convertible bonds, employee stock options, and structured financial products. It earned its creators the 1997 Nobel Memorial Prize in Economic Sciences.