Dean CabanesOptions Pricing

Implied volatility

The volatility at which the Black-Scholes-Merton formula reproduces an observed option price, found by a safeguarded Newton solver.

What it does

Every input to the Black-Scholes-Merton formula is observable except volatility. Implied volatility runs the formula backward: given an option's market price, it finds the σ that reproduces it. The solver uses Newton's method, which follows the slope of price against volatility (vega), inside a bracket that is known to contain the answer. Whenever a Newton step would leave the bracket, or vega is too small to trust, it bisects instead.

Why it is used

Traders quote and compare options by implied volatility rather than price, because it strips out the effect of strike, expiry and rates. A plain Newton solver can overshoot and fail for deep in- or out-of-the-money options where vega is tiny; the bracket guarantees convergence.

Inputs

  • An observed European option price.
  • Spot, strike, time to expiry, risk-free rate, dividend yield, call or put.

Formulas

Find σ such that BS(σ) = observed price Newton step: σ ← σ − (BS(σ) − price) / vega(σ) Bracket: [10⁻⁶, 10]; tightened each step; bisect if Newton would leave it Start: σ₀ = √(2·|ln(F/K)| / T), clipped to [0.1, 2] (Manaster-Koehler; F = S·e^((r − q)·T)) No-arbitrage range (call): max(S·e^(−qT) − K·e^(−rT), 0) < price < S·e^(−qT)

Assumptions

  • The price is for a European option under Black-Scholes-Merton with the stated rate and dividend yield.
  • The price is a single figure (in practice, the midpoint of the bid and ask).
  • Convergence when the repriced option is within 10⁻¹⁰ of the target (relative to the price, for prices above 1).

How to read the results

An implied volatility above your own volatility estimate means the price embeds more expected movement (or a risk premium) than you assume. Entering the model price recovers the input volatility, which is the round-trip check the page runs by default.

Limitations

  • Prices of American options include an early-exercise premium, so inverting the European formula overstates their implied volatility, mostly for in-the-money puts.
  • Near the no-arbitrage bounds the problem is ill-conditioned: vega is tiny, and a one-cent change in price can move the implied volatility by several points. The solver still converges; the answer is just not informative.
  • One implied volatility per option. Fitting a smile or surface across strikes is out of scope.

Where it can fail

  • A price below the zero-volatility value or above the upper bound has no implied volatility; the engine says so instead of returning a number.
  • Volatility above 1,000% a year is treated as a sign of a bad input.

Validation on current data

Hull's implied-volatility example, and round trips (price at a known σ, then solve back) across strikes, maturities and volatilities, recomputed by the running engine.

References

  • Manaster, S. and Koehler, G. (1982). The calculation of implied variances from the Black-Scholes model: a note. Journal of Finance 37(1).
  • Hull, J. C. Options, Futures, and Other Derivatives. Pearson (implied volatility).

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