Black-Scholes-Merton (BSM) framework
Quantitative Finance glossary · Stochastic Calculus & Continuous Time Finance
Definition
Continuous-time arbitrage-free pricing framework: assume GBM dS = μ S,dt + σ S,dW, form a delta-hedged portfolio Pi = V - Δ,S, apply Itô, and require the hedged portfolio earn the risk-free rate. The result is the BSM PDE (partial V)/(partial t) + 1/2σ^2 S^2 (partial^2 V)/(partial S^2) + r S (partial V)/(partial S) - r V = 0, which by Feynman-Kac is equivalent to the risk-neutral expectation V = e^-rTmathbbE^mathbbQ[payoff]. The framework's assumptions (constant vol, no jumps, frictionless trading) are all violated in practice — its true contribution is the replication/hedging logic, not the price.
Black-Scholes-Merton (BSM) framework in plain English
The Black-Scholes-Merton framework is the deeper reasoning behind that famous formula, and its real genius is one idea: hedging. If you sell an option, you can offset your risk by continuously holding a carefully chosen amount of the underlying stock, so that no matter which way the stock wiggles moment to moment, your combined position barely moves. Since that hedged bundle is essentially risk-free, it can only fairly earn the risk-free interest rate — and forcing that to be true pins down the option's price. It's like balancing a seesaw: keep adjusting the weight on the other side and the whole thing stays level. The exact assumptions (smooth prices, no sudden jumps, no trading fees) aren't perfectly real, but the replicate-and-hedge logic is the lasting contribution.
Also called: BSM
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