Implied Volatility: Inversion, Quote Bounds, and Model Risk

Treat implied volatility as a model-specific price inversion, validate bounds and solver convergence, and distinguish quote-side IV, surface coordinates, forecasts, and executable P/L.

For educational purposes only; not investment advice. Investing may result in loss.

Direct answer

Implied volatility (IV) is a model coordinate: for selected price P_target and fixed inputs theta, it is a nonnegative root sigma_star of V_model(sigma_star;theta)=P_target. It is inferred, not directly observed. Changing the quote side, timestamp, rate, dividend, borrow, exercise model or numerical convention can change the answer.

Each strike and expiry can have its own IV, forming a surface. IV is annualized and normally reported as a decimal or percentage; one volatility point means Delta_sigma=0.01, not a one-percent relative change. IV is not direction, a guaranteed range, physical probability, future realized volatility or an executable option price.

A controlled workflow

Lock the exact claim: underlying, call or put, strike, expiry and timezone, last trade, American or European style, cash or physical settlement, multiplier, deliverable, currency and quote unit.
Select the target price: bid, ask, midpoint, last trade or theoretical mark. Record timestamp, size and condition; separately retained bid and ask roots form a diagnostic interval, not guaranteed execution.
Lock model and inputs: spot or forward, exact T and day count, discount curve, dividend amounts and dates, borrow or rebate, corporate actions, and an exercise-aware tree or PDE for American claims.
Check contract-specific model bounds, parity and data quality first. Mark stale, crossed, zero-size, out-of-bounds or invalid-expiry observations as invalid rather than manufacturing an IV.
Solve f(sigma)=V_model(sigma)-P_target=0 on sigma>=0 with a bracketed method or safeguarded Newton step. Store bracket, cap, tolerance, iteration count, price residual and Vega; near-zero Vega makes inversion ill-conditioned.
Report decimal, percent and volatility-point conventions, model, price side, timestamp and surface coordinate such as strike, log-forward-moneyness or Delta. Apply controlled interpolation and test strike monotonicity, butterfly convexity and calendar consistency.
Reinsert IV into the model, then full-reprice spot, time, curve, dividend, borrow and surface scenarios. Keep local Greeks, expected-move shortcuts, forecasts, actual fills and realized P/L as separate objects.

Worked examples

Unique European root: For S_0=100, K=100, T=1, r=5%, q=2% and a European call price C=11.1237619281, Black-Scholes inversion gives sigma_star=25.0000000000%, with d1=0.2450000000 and d2=-0.0050000000. Repricing under the same inputs returns the target to numerical tolerance; it says nothing about direction.
No-solution boundary: The same call has discounted spot 98.0198673307, discounted strike 95.1229424501, lower bound 2.8969248806 and upper bound 98.0198673307. A target C=2.50 is below the model lower bound, so no finite nonnegative Black-Scholes IV exists. The correct result is invalid input or bounds error, not 0%.
Quote-side envelope: Under the same inputs, bid 10.90 implies IV_bid=24.4103525576% and ask 11.35 implies IV_ask=25.5961897451%. The interval width is 1.1858371875 volatility points. A midpoint or last-trade IV is another reference, not a fill or a model-independent fact.
Vega units and full repricing: At sigma=20%, call value is 9.2270055082 and decimal Vega is 37.9011575100 per share per +1.00 volatility, or 0.3790115751 per volatility point. A +10-point linear estimate is +3.7901157510; full repricing to 30% gives 13.0202812687-9.2270055082=3.7932757605, a difference of 0.0031600095 per share or $0.31600095 for multiplier 100.

Risks and validation

Series risk: Wrong underlying, strike, expiry or option type changes the claim.
Style risk: A European model applied to an American premium contaminates IV.
Settlement risk: Cash, physical, AM, PM and adjusted deliverables are not interchangeable.
Quote risk: Last, midpoint, bid and ask represent different price objects.
Liquidity risk: Stale, crossed, zero-size or wide markets make inversion unreliable.
Timestamp risk: Unsynchronized option, underlying and curve inputs create false IV.
Coordinate risk: Spot and forward moneyness or different Delta conventions change comparisons.
Curve risk: Discount, compounding and collateral assumptions affect model price.
Dividend risk: Wrong amounts, dates or corporate-action treatment move the root.
Borrow risk: Hard-to-borrow fees and rebate assumptions affect forwards and American values.
Clock risk: Day count, expiry time and cutoff errors change T materially near expiry.
Bounds risk: A target outside model price bounds has no valid finite root.
Conditioning risk: Near intrinsic value and low Vega, tiny price errors create large IV changes.
Solver risk: Bad brackets, caps, tolerances or nonconvergence can produce plausible-looking garbage.
Unit risk: Decimal, percent, volatility-point, per-share and per-contract conventions can be mixed.
Interpolation risk: Surface interpolation or extrapolation can create unstable wings.
Arbitrage risk: Butterfly or calendar violations should not be hidden by smoothing.
Event risk: Jumps and concentrated event variance violate a constant-diffusion interpretation.
Greek risk: Local Delta and Vega do not replace full repricing for large joint moves.
Decision risk: IV alone is not a forecast, executable value, loss limit or trading rule.

Common misconceptions

“A stock has one true IV.” IV belongs to a contract, price, model, inputs and timestamp.
“IV predicts direction or a guaranteed range.” It is an annualized model scale, not a directional probability statement.
“Bid, midpoint and ask IV are interchangeable and executable.” They are different inversions of different price references.
“High IV always means sell and low IV always means buy.” Payoff, future path, surface, price and risk capacity matter.
“VIX, ATM IV, single-contract IV and model-free variance are the same.” They use different claims, weights and methodologies.

Authoritative sources

OCC Characteristics and Risks — contract rights and lifecycle risks, not an IV algorithm or value.
OIC Technical Information — annualized IV inferred from price and model, not a universal forecast.
OIC Vega — local price sensitivity and display convention, not full repricing.
Black and Scholes — the European diffusion foundation, not observed smiles, American exercise or execution.
Merton, Theory of Rational Option Pricing — continuous carry and no-arbitrage theory, not current contract rules.
Cox, Ross and Rubinstein — tree pricing and American exercise representation, not one universal implementation.
Cboe Theoretical Options Methodology — operational model, surface and input conventions, not universal desk truth or fills.
Cboe VIX Methodology — model-free 30-day SPX variance-strip methodology, not a single-option Black-Scholes IV.