Implied Volatility
The volatility input that, plugged into the Black-Scholes-Merton formula, makes the model price equal to the observed market price. Solved numerically via Newton-Raphson on price-vs-σ, using vega as the derivative.
When to use: Use to back out the market's consensus volatility expectation from an observed option price — the standard way to compare implied volatilities across strikes (the volatility smile/skew) and over time. The single most-watched figure in options trading after the underlying price itself.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| Sigma | Volatility | Annualized volatility of the underlying as a decimal (e.g. 0.25 for 25%) | % |
| S | Stock Price | Current price of the underlying | $ |
| Strike | Strike Price | Exercise price of the option contract | $ |
| Rf | Risk-Free Rate | Continuously compounded annual risk-free rate as a decimal | % |
| T | Time to Expiration | Time to expiration in years (e.g. 0.25 for 3 months) | years |
| OptionPrice | Market Option Price | Observed market price of the call or put | $ |
| OptionType | Option Type | Call (1) or Put (-1) — enter 1 for a call, -1 for a put | integer |
Real-Life Examples
Example 1: ATM Call IV
Stock at $100, 3-month ATM call (Strike $100) trades at $6.46, risk-free rate 4%. What volatility is the market pricing in?
Given
Step-by-Step
IV of 30% — the market is pricing this option as if the underlying will move ~30% annualized. Compare to historical realized vol to see if options are over- or under-priced.
Example 2: OTM Put IV
Stock at $100, 3-month put with Strike $90 trades at $1.50, Rf = 4%.
Given
Step-by-Step
IV of 27.85% on a 10% OTM put. In equity markets, OTM puts often trade at higher IV than ATM (the volatility skew) — driven by demand for downside protection. A volatility skew chart plots IV against strike to visualize this pattern.
Frequently Asked Questions
Because option prices are dominated by volatility (it's the only Black-Scholes input that isn't directly observable). Traders quote, hedge, and trade options in IV terms — "buy this at 28 vol" rather than dollar prices. IV across strikes/expirations forms the volatility surface, the central object in derivatives risk management.
The pattern of IV across strikes at the same expiration. In equity index options, IV is typically higher for OTM puts (downside protection demand) than for OTM calls — the "skew." Currency options often show a symmetric "smile." The shape encodes the market's view of tail risks beyond what Black-Scholes assumes.
Volatility itself cannot be negative (variance is squared). The solver may fail or hit zero when the market price is at or below the option's intrinsic value (no time premium left), or for arbitraged-away pricing. Such cases return NaN — verify the price is > intrinsic.
Newton-Raphson with vega as the derivative converges quickly for liquid, ATM-ish options. For deep OTM/ITM options or near-expiration, vega approaches zero and Newton can fail; this implementation falls back to bisection in those cases for robustness.