Implied Volatility¶

One-line definition¶

The volatility that, plugged into Black-Scholes, reprices the model exactly to a given market option price — the market's own forward-looking volatility estimate, backed out rather than forecast.

Formula¶

No closed form — solved numerically. pm.options.implied_volatility_call and implied_volatility_put use bisection (scipy.optimize.brentq) to find the sigma where black_scholes_call_price(..., sigma, ...) == market_price, bracketed over a wide [1e-6, 5.0] volatility range (this always brackets a root because Black-Scholes price is strictly increasing in volatility — see the last row of the worked example below).

Why PMs care¶

Historical (realized) volatility describes what already happened; implied volatility (IV) is priced off what the market expects to happen over the option's remaining life, which is what actually determines an option's cost today. Comparing IV across strikes at the same expiry (the "volatility smile/skew") and across expiries (the "term structure") reveals the market's own view of tail risk and near-term event risk — information no historical-volatility number carries. The VIX index is exactly this idea aggregated across the S&P 500 option chain into one number: a market-implied volatility reading, not a statistical estimate from past returns.

Worked example¶

Take this repo's own Black-Scholes hand example forward: price a call at sigma = 20% (S=K=100, r=5%, T=1, q=0) to get market_price ≈ 10.4506, then solve backward for the implied volatility that reproduces that price. It recovers sigma = 0.20 to 8 decimal places (tests/test_options.py::test_implied_volatility_call_round_trips) — because Black-Scholes price is monotonic in volatility (verified directly in test_call_price_increases_with_volatility), the solver has exactly one root to find, which is what makes bisection both fast and reliable here.

Common mistakes¶

treating implied volatility as a forecast of what will actually happen — it's the market's current price of uncertainty, which itself moves with supply/demand for options, not only with a view on realized volatility
assuming one IV number describes an entire option chain — in reality IV varies by strike (skew/smile) and expiry (term structure); a single "the IV is X%" statement is always an approximation of a richer surface
solving for implied volatility on an option price that itself violates a no-arbitrage bound (e.g. below intrinsic value) — no real volatility reproduces it, and a numerical solver may fail to converge or return a meaningless answer instead of erroring cleanly

Limitations¶

This repo solves implied volatility only against the flat-volatility Black-Scholes formula, which is exactly why a smile exists in real markets: if a single volatility number correctly priced every strike, there would be nothing to plot. Building an actual volatility surface (smile/skew across strikes, term structure across expiries, and a consistent interpolation/extrapolation scheme between them) is materially more machinery than one root-find per quote and isn't implemented here.

Free resources¶

VIX Index — Cboe — how a market-wide implied-volatility measure is actually built and used