Black–Scholes equation

The Black–Scholes equation is a partial differential equation for derivative prices under assumptions including continuous trading, constant volatility, and no arbitrage.

Black–Scholes formula: A closed-form price for European call and put options under the Black–Scholes model. It is the equation’s celebrated solution for European options.
European option: A financial contract that can be exercised only at its expiration date. The standard closed-form solution prices European options.
Option pricing: The valuation of contracts that give the right, but not the obligation, to trade an asset. The equation provides a foundational model for valuing options.
Local volatility model: An option-pricing model in which volatility varies with asset price and time. It extends the constant-volatility assumption to fit observed option prices.
Louis Bachelier: A French mathematician whose 1900 thesis modeled speculative prices using Brownian motion. His early stochastic treatment of markets anticipated later option-pricing mathematics.
Delta hedging: A strategy that offsets an option’s price sensitivity by holding a position in its underlying asset. Continuous delta hedging is the replication argument behind the equation.
No-arbitrage principle: The principle that equivalent financial positions should not permit a risk-free profit. The equation’s pricing logic rules out arbitrage through replication.
Implied volatility: The volatility value that makes an option-pricing model match an observed market price. Market option prices are often translated into volatility using the Black–Scholes formula.
Heston model: An option-pricing model in which volatility follows a stochastic process correlated with the asset price. It replaces fixed volatility with random, evolving volatility.
Fischer Black: An American economist and mathematician who co-developed the Black–Scholes option-pricing model. His work with Scholes produced the equation’s first published formulation.

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