Black-Scholes-Merton
An interactive exploration of option pricing, probability distributions, volatility, time value and the Greeks. Change the inputs and watch the mathematics change in real time.
From uncertainty to an option price
The Black-Scholes model connects probability distributions, logarithms, volatility, discounting and risk. Its central question is: what is a fair price for an option before it expires?
1. Model the stock
The future stock price is treated as a lognormally distributed random variable.
2. Define the payoff
A call benefits above the strike; a put benefits below it.
3. Price uncertainty
Risk-neutral pricing converts possible outcomes into a present value.
The assumptions
Volatility is assumed known and constant.
The continuously compounded rate is constant.
Continuously compounded returns are normally distributed.
Prices must be consistent with no-arbitrage relationships.
The theoretical model permits continuous adjustment.
The closed-form formula assumes exercise only at expiry.
The Black-Scholes equation
This partial differential equation describes how the option value changes through time and with the underlying asset.
European call
European put
Interactive option-pricing laboratory
These controls drive all the visualisations on the page.
Option value versus stock price
The nonlinear curve shows the value of the option before expiry. The strike and current stock price are marked.
Payoff at expiry
At expiry there is no remaining time value. The call payoff is max(S − K, 0); the put payoff is max(K − S, 0).
Watching time value disappear
As expiry approaches, option value converges toward the expiry payoff.
The effect of volatility
Higher uncertainty increases the value of both calls and puts because the holder's downside is limited while upside remains open.
The risk-neutral distribution
The model uses a lognormal distribution for the stock price at expiry. The distribution cannot extend below zero and has a right-hand tail.
The Greeks
The Greeks turn one option price into a map of its sensitivities.
Delta
Gamma
Vega
Theta
Rho
Three-dimensional option-value surface
Drag the surface to rotate it. The first surface shows option value against stock price and volatility; the second replaces volatility with time.
From Black-Scholes to Merton
The dividend-yield extension incorporates a continuously paid dividend yield q.
Implied volatility
We can run the model backwards: given a market option price, what volatility would make the Black-Scholes-Merton price equal that market price?
Where the model breaks down
Different strikes can imply different volatilities.
Market volatility varies through time.
Prices can move discontinuously after shocks.
Real markets have spreads, fees and funding costs.
Early exercise requires a different treatment.
Financial returns often have heavier tails than a simple normal model.
The wider risk journey
Normal and lognormal distributions
Volatility and correlation
Nonlinear payoffs and option pricing