Black-Scholes formula
Quantitative Finance glossary · Derivatives Pricing & Volatility Surfaces
Definition
Closed-form European call price under GBM with constant r, σ: C = S_0,N(d_1) - K,e^-rT,N(d_2), where d_1 = dfracln(S_0/K) + (r + 1/2σ^2)Tσ√(T) and d_2 = d_1 - σ√(T). Put follows from put-call parity. N(d_2) is the risk-neutral probability of finishing in-the-money; N(d_1) is the delta. The formula is a benchmark, not a belief — it is used to quote and risk-manage in implied-vol space, where its flat-vol assumption is visibly violated by the skew/surface.
Black-Scholes formula in plain English
The Black-Scholes formula is a famous equation that spits out a fair price for a "call option" — a contract giving you the right (not obligation) to buy a stock later at a locked-in price. It combines the sensible ingredients: today's stock price, that locked-in price, how long you have, interest rates, and how jumpy the stock is (its volatility). The bigger idea is that the price depends heavily on volatility — a wilder stock makes the option more valuable, because bigger swings raise the chance of a big payoff while your downside stays capped. In practice traders don't treat its number as gospel; they use it as a common language for quoting and comparing options, even though its "volatility is constant" assumption doesn't perfectly hold in the real world.
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