Financial Calculators
Black–Scholes Option Pricing Calculator
Calculate European call and put option prices and Greeks using the Black–Scholes–Merton model locally in your browser.
Price European options and compute their sensitivities.
The Black–Scholes–Merton model prices European options assuming log-normal price dynamics, constant volatility, and continuous trading. For a call: C = S e^(−qT) N(d₁) − K e^(−rT) N(d₂); for a put: P = K e^(−rT) N(−d₂) − S e^(−qT) N(−d₁), where d₁ = [ln(S/K) + (r − q + σ²/2)T] / (σ√T) and d₂ = d₁ − σ√T. The Greeks measure sensitivity: Delta (Δ) is the price change per $1 stock move; Gamma (Γ) is the Delta change per $1 move; Vega (ν) is the price change per 1% volatility change; Theta (Θ) is the daily time decay; Rho (ρ) is the price change per 1% rate change. Enter the stock price, strike price, time to expiry in years, risk-free rate and volatility as percentages, and an optional continuous dividend yield. For example, with S = $100, K = $100, T = 1 year, r = 5%, σ = 20%, q = 0%, the call price is $10.45 and the put price is $5.58. This model does not price American options, account for dividends paid discretely, or handle stochastic volatility. All calculations stay in your browser.
Frequently Asked Questions
Everything you need to know about this tool, how it works, and privacy.
What does the Black–Scholes model calculate?
The Black–Scholes–Merton model calculates the theoretical price of European call and put options assuming the underlying asset follows a log-normal distribution with constant volatility and continuous trading. It also computes the Greeks (Delta, Gamma, Vega, Theta, Rho) which measure the option's sensitivity to changes in the underlying price, volatility, time, and interest rates.
What are the limitations of the Black–Scholes model?
The model assumes constant volatility, log-normal price dynamics, continuous trading, no transaction costs, and a known constant risk-free rate. It does not price American options (which can be exercised early), account for discrete dividends, handle stochastic volatility, or model fat tails and volatility smiles observed in real markets. Use it as a theoretical benchmark, not a precise market price.
How does dividend yield affect option prices?
A continuous dividend yield q reduces the call price and increases the put price because the stock price is expected to drop by the dividend amount. The model adjusts the stock price drift by subtracting q. For discrete dividends, a more complex model or adjustment is needed; this calculator uses the continuous yield convention.
What do the Greeks mean?
Delta (Δ) measures the option price change per $1 change in the stock price. Gamma (Γ) measures the rate of change of Delta per $1 stock move. Vega (ν) measures the price change per 1% change in implied volatility. Theta (Θ) measures the price change as one calendar day passes; it is often negative for long options but can be positive. Rho (ρ) measures the price change per 1% change in the risk-free rate.
Is my data uploaded?
No. All calculations run locally in your browser. CodeASystem does not upload or store the values you enter.