02-implied-volatility.md

Implied Volatility

Black-76 takes volatility in and returns a price. Implied volatility (IV) is the inverse: the volatility that makes the model match the market quote.

$$ \text{Black76}(F, K, T, \sigma) = P_{market} $$

IV is the common language for options because it is comparable across strikes and expiries in a way raw prices are not.

Solver

Option price is monotone in volatility because vega is non-negative:

$$ \nu = df \cdot F \cdot \varphi(d_1) \cdot \sqrt{T} $$

That makes the solution unique when the price is inside no-arbitrage bounds. The implementation uses Brent's method with a [1e-4, 5.0] volatility bracket because it is robust near deep ITM/OTM strikes.

Bounds

For a call under this convention:

$$ df \cdot \max(F-K,0) \le P \le df \cdot F $$

If a feed sends a stale price outside the feasible range, the solver returns NaN instead of inventing a volatility.

Deribit Validation

The validation flow pulls option mark prices and matching futures marks, converts coin premiums to USD with mark_price * F, solves IV, and compares against mark_iv / 100. Liquid expiries typically land well below one vol point of error; same-day expiries are noisier because tiny price moves map to large IV moves.