Definition
Option pricing is the mathematical determination of fair prices for financial options — contracts giving the right (but not the obligation) to buy or sell an asset at a specified price by a specified date. Modern option pricing relies on probability models of asset-price evolution (typically geometric Brownian motion) and the principle of no-arbitrage.
The Black-Scholes-Merton formula (1973) gave the first closed-form solution and won the 1997 Nobel Prize in Economics. It also turned finance into a heavily mathematical, probabilistic discipline.
Why it matters
How it works
Under Black-Scholes assumptions, the stock price follows a continuous random process with constant drift and volatility. The option's payoff at expiry depends on the stock price; its fair price today is the expected payoff under the risk-neutral measure, discounted at the risk-free rate. For a European call, this evaluates to the famous formula in terms of the cumulative normal distribution.
The deeper insight is that pricing rests on hedging: an option's price is what it would cost to dynamically replicate its payoff with a position in the stock and cash. Probability is the language; arbitrage-free replication is the mechanism.