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Social Sciences
Finance
1973
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Black-Scholes Equation

∂V∂t+12σ2S2∂2V∂S2+rS∂V∂S−rV=0\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0

PDE for option pricing—hedging eliminates risk and determines fair derivative value.

By Fischer Black, Myron Scholes, Robert Merton

Social Sciences
Black-Scholes Equation
1973 · Fischer Black
Human Reviewed
84%

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Five doors into the universe behind this equation. Choose your path.

Story PortalWho discovered this?Discover how Fischer Black and others shaped this equation.Visual PortalWhat does it look like?See the equation come alive in the Visual Studio.Machine PortalWhere is it used?Explore machines powered by this equation — options trading.Math PortalWhat does it derive from?Trace the mathematical lineage from normal distribution.Future PortalWhere is it going?DeFi protocols
Why it matters: Created the derivatives market worth trillions; transformed finance into applied math.

Discoverers: Fischer Black, Myron Scholes, Robert Merton (1973)

What does it mean?

PDE for option pricing—hedging eliminates risk and determines fair derivative value.

Why should I care?

Created the derivatives market worth trillions; transformed finance into applied math.

Equation Compass

North — Prerequisites

West — History

East — Applications

South — Derivations

Variables & Units

SymbolNameUnitMeaning
VVOption value—Derivative price
SSStock price—Underlying asset price
σσVolatility—Price volatility
rrRisk-free rate—Interest rate

Worked Example

Call option price increases with volatility and time to expiry.

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Equation Universe

Black-Scholes Equation

∂V∂t+12σ2S2∂2V∂S2+rS∂V∂S−rV=0\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0

Real-world impact

Global derivatives markets

Options pricing equations move trillions in modern finance.

Photo: Unsplash — trading screens

PDE for option pricing—hedging eliminates risk and determines fair derivative value.

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