Black-Scholes Option Pricing
The Black-Scholes model prices European options assuming constant volatility and risk-free rate. Despite simplifications, it remains the benchmark for options pricing.
Inputs
S = current stock price
K = strike price
T = time to expiry (years)
r = risk-free rate (annual)
σ = volatility (annual, e.g. 0.20 = 20%)Black-Scholes Formula
d₁ = [ln(S/K) + (r + σ²/2)T] / (σ√T)
d₂ = d₁ - σ√T
Call = S×Φ(d₁) - K×e^(-rT)×Φ(d₂)
Put = K×e^(-rT)×Φ(-d₂) - S×Φ(-d₁)
Φ = standard normal CDFQuick Example
S=100, K=105, T=0.25, r=0.05, σ=0.20:
d₁ = [ln(100/105)+(0.05+0.02)×0.25]/(0.20×0.5)
= [-0.0488+0.0175]/0.10 = -0.313
d₂ = -0.313 - 0.10 = -0.413
Call ≈ $3.20, Put ≈ $7.57The Greeks
- Delta (Δ): option price change per $1 stock move
- Gamma (Γ): rate of change of Delta
- Theta (Θ): time decay — value lost per day
- Vega: sensitivity to volatility changes
- Rho: sensitivity to interest rate changes
Calculate option prices: Free Black-Scholes Calculator
Black-Scholes Quick-Reference: Option Pricing Inputs
| Parameter | Symbol | Effect on call price | Effect on put price |
|---|---|---|---|
| Stock price | S | ↑ price → ↑ call | ↑ price → ↓ put |
| Strike price | K | ↑ strike → ↓ call | ↑ strike → ↑ put |
| Time to expiry | T | ↑ T → ↑ call (theta) | ↑ T → ↑ put (theta) |
| Volatility | σ | ↑ σ → ↑ call (vega) | ↑ σ → ↑ put (vega) |
| Risk-free rate | r | ↑ r → ↑ call | ↑ r → ↓ put |
| Dividend yield | q | ↑ q → ↓ call | ↑ q → ↑ put |
How the Black-Scholes Model Works
The Black-Scholes model prices European options (exercised only at expiry) using: C = S·N(d₁) − K·e^(−rT)·N(d₂), where d₁ = [ln(S/K) + (r + σ²/2)T] / (σ√T) and d₂ = d₁ − σ√T. N(·) is the standard normal CDF. The model assumes: log-normal stock price, constant volatility, no dividends, continuous trading, no transaction costs, and a constant risk-free rate. The put price follows from put-call parity: P = K·e^(−rT)·N(−d₂) − S·N(−d₁).
Option Greeks describe price sensitivity: Delta (∂C/∂S = N(d₁)) measures directional exposure; Gamma (∂²C/∂S²) measures convexity; Theta (∂C/∂T) measures time decay (always negative for long options); Vega (∂C/∂σ) measures volatility sensitivity; Rho (∂C/∂r) measures interest-rate sensitivity. Market makers use Greeks to hedge option books dynamically (delta hedging).
Common Mistakes
- Using historical volatility instead of implied volatility: Black-Scholes takes future volatility as input, which is unknown. Implied volatility — backed out from market option prices — is often a better forward-looking estimate than historical σ.
- Applying Black-Scholes to American options: American options can be exercised early, so Black-Scholes underprices them (especially deep in-the-money puts or calls on high-dividend stocks). Use binomial trees or finite-difference methods instead.
- Ignoring volatility smile/skew: Real markets show implied volatility varying by strike and expiry (the volatility surface). Using a single flat σ in Black-Scholes misprices out-of-the-money options significantly.
Frequently Asked Questions
Implied volatility (IV) is the σ value that makes the Black-Scholes formula match the market option price. It represents the market's consensus forecast of future price volatility. The VIX index is derived from S&P 500 options IVs and is widely used as a "fear gauge." Options traders buy IV when they expect volatility to increase and sell IV (write options) when they expect it to decrease.
Collateralised Debt Obligations (CDOs) used Gaussian copula models assuming mortgage default correlations were low and normally distributed. In reality, during a housing crash, defaults became highly correlated — all at once. The Black-Scholes assumption of continuous log-normal returns also underestimates the frequency of extreme market moves (fat tails), leading to mispriced risk in derivatives portfolios across the financial system.
Put-call parity: C − P = S − K·e^(−rT). A portfolio of a long call and short put with the same strike/expiry is equivalent to holding the stock and borrowing the present value of the strike price. Violations create riskless arbitrage opportunities. Market makers use this relationship to price puts from calls (or vice versa) and to hedge positions efficiently.