Implied Volatility
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
For these inputs, Volatility is 30.002595% under the stated payment, compounding and accounting assumptions.
Example 2: OTM Put IV
Stock at $100, 3-month put with Strike $90 trades at $1.50, Rf = 4%.
Given
Step-by-Step
For these inputs, Volatility is 27.848359% under the stated payment, compounding and accounting assumptions.
Frequently Asked Questions
For a non-dividend-paying European option, the lower bound is discounted intrinsic value and the upper bound is stock price for a call or discounted strike for a put. Prices below the lower bound or at/above the upper bound have no finite model solution.
At the lower price bound this calculator returns the zero-volatility limit. With rounded or deep out-of-the-money prices, volatility can be poorly identified. Expiry has no uniquely recoverable implied volatility.
A bracketed root search verifies that the theoretical model price matches the entered price. The result is a model-implied parameter, not a prediction of realized volatility.