Implied volatility
Quantitative Finance glossary · Derivatives Pricing & Volatility Surfaces
Definition
The volatility input σ_imp that, plugged into the Black-Scholes formula, returns the observed market price: C^BS(S, K, T, r, σ_imp) = C^mkt. Solved numerically because BSM has no closed-form inverse in σ (Newton-Raphson on Vega, or Jäckel's 'Let's Be Rational' algorithm for tail-precision). The IV surface across (K, T) is the market's nonparametric encoding of jump risk, stochastic vol, and skew — the workhorse interpolation/extrapolation object on every options desk.
Implied volatility in plain English
Everyone quotes options at market prices, but it's more useful to talk about the 'volatility' baked into that price — how wildly the market thinks the stock might swing. Implied volatility is the volatility number you'd have to feed into the Black-Scholes formula to make it spit out the price you actually see; it's the price re-expressed in the language of expected wobbliness. You can't solve for it with algebra, so computers zero in on it numerically. What's powerful is that plotting implied volatility across different strike prices and expiration dates gives a 'surface' that quietly encodes the market's fears — crash risk, sudden jumps, uncertainty about volatility itself — making it the central object traders read and interpolate on every options desk.
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