💹 Options Pricing — Black-Scholes & Monte Carlo
Price European options, explore the Greeks, and watch Monte Carlo GBM paths to expiry
Option Type
Presets
Parameters
Option Price
Greeks
About Options Pricing
Black-Scholes Model
The Black-Scholes formula (1973) gives a closed-form price for European options under assumptions of lognormal stock prices, constant volatility, and no dividends. The call price C = S·N(d₁) − K·e^(−rT)·N(d₂) depends on five inputs: current stock price S, strike K, risk-free rate r, volatility σ, and time to expiry T. Fischer Black and Myron Scholes (with Robert Merton) won the 1997 Nobel Prize in Economics for this formula.
The Greeks
The "Greeks" measure an option's sensitivity to input changes. Delta (Δ = ∂C/∂S) shows how much the option price changes per £1 move in the stock — it ranges from 0 to 1 for calls. Gamma (Γ = ∂²C/∂S²) measures the rate of delta change. Vega (∂C/∂σ) shows sensitivity to volatility — options become more valuable with higher volatility. Theta (∂C/∂T) is the daily time decay — options lose value as expiry approaches.
Monte Carlo Simulation
Monte Carlo simulation generates many random stock price paths using Geometric Brownian Motion: S(T) = S(0)·exp((r−σ²/2)T + σ·√T·Z), where Z is a standard normal variable. Averaging the discounted payoffs max(S(T)−K,0)·e^(−rT) over thousands of paths gives an unbiased estimate of the option price, and the error decreases as 1/√N. This approach extends naturally to path-dependent options (Asian, barrier) where no closed-form exists.