The Black-Scholes equation is the differential-equation framework behind the classic continuous-time pricing model for options and other contingent claims.
The economic logic
The model starts from a no-arbitrage argument. If an option can be dynamically replicated by trading the underlying asset and a risk-free asset, then the option’s price must equal the cost of that replicating strategy.
That leads to the Black-Scholes partial differential equation:
dV/dt + 0.5*sigma^2*S^2*d2V/dS2 + r*S*dV/dS - rV = 0
where:
Vis the option value,Sis the underlying asset price,sigmais volatility,ris the risk-free rate.
For a standard European call with no dividends, the model produces the familiar closed-form pricing formula.
Assumptions behind it
The standard setup assumes:
- continuous trading,
- no arbitrage,
- constant volatility and interest rate,
- frictionless markets,
- lognormally evolving asset prices.
Those assumptions make the model tractable, but they are also the source of its limits in real markets.
Why economists still use it
Even when the assumptions are unrealistic, the Black-Scholes framework remains foundational because it teaches the key pricing idea: derivatives are valued from replication and hedging logic, not from raw expected returns alone.
It also provides a baseline language for implied volatility, delta hedging, and comparing market prices with model prices.