Black-Scholes-Merton Model (BSM)
The Black-Scholes-Merton model (1973) prices European options under the assumption of a lognormal stock price process, constant volatility, continuous trading, and a known risk-free rate. The formula derives from a dynamic hedging argument: a delta-neutral portfolio of stock and options must earn the risk-free rate. Call price: C = S·e^(−δT)·N(d₁) − K·e^(−rT)·N(d₂) where d₁ = [ln(S/K) + (r−δ+σ²/2)T]/(σ√T) and d₂ = d₁ − σ√T. N(·) is the standard normal CDF. BSM has well-known limitations (assumes constant vol, no jumps, European exercise only) but remains the market standard for quoting and risk management.
Related Terms
Delta (Δ)
The sensitivity of an option's price to a $1 change in the underlying spot price.
Gamma (Γ)
The rate of change of delta with respect to the spot price — the curvature of the option's value.
Vega (ν)
The sensitivity of an option's price to a 1% change in implied volatility.
Theta (Θ)
The rate at which an option loses value as time passes — time decay per calendar day.
Implied Volatility (IV)
The volatility that, when input into the BSM model, makes the model price equal to the market price of an option.
Binomial Option Pricing Model
A discrete-time model that prices options by working backward through a tree of possible stock price paths.
Risk-Neutral Probability
The artificial probability measure under which all assets earn the risk-free rate, used to price derivatives by expectation.