Implied Volatility Calculator
OPTIONS · BLACK-SCHOLES SOLVER · NEWTON-RAPHSON WITH BISECTION FALLBACK
Solves for the volatility Black-Scholes needs to reproduce an observed option price, step by step — not an approximation.
Option Type
Enter the price you actually saw quoted or traded — the solver backs out the volatility Black-Scholes needs to reproduce that exact price.
What is implied volatility, and why solve for it?
Black-Scholes takes volatility as an input and produces a price. Implied volatility runs that backwards: given a price you actually saw quoted or traded, what volatility would Black-Scholes need to reproduce it? There's no closed-form formula for this — it has to be solved numerically, which is exactly what this tool does, step by step, rather than approximating.
The algorithm
- Check the price against its no-arbitrage bounds first — reject impossible prices outright
- Newton-Raphson: volPct = volPct − (price(volPct) − target) / vega(volPct)
- Fall back to bisection if vega is too flat to converge (deep ITM/OTM, very short-dated)
Worked example
A call trades at $10.4506 with S=$100, K=$100, r=5%, 1 year to expiry:
- Newton-Raphson starts at a 50% vol guess
- Converges to 20.00% implied volatility in a handful of iterations
Why it sometimes says "no volatility solves this"
Every option price has hard bounds regardless of volatility: a call can never be worth more than the spot price, or less than its discounted intrinsic value; a put is bounded the mirror way. A price outside those bounds isn't explainable by any volatility — it's either a data error or a real arbitrage, and this tool says so instead of returning a number that looks plausible but isn't real.
Where to go next
Have a volatility already and want the theoretical price and Greeks instead? Use the Black-Scholes Calculator.
Use via API or MCP
This calculation is available as a deterministic API call — useful for bots or AI agents that need to back out volatility from an observed price without approximating it.
Why can't implied volatility be solved with a plain formula?
Black-Scholes's price formula isn't algebraically invertible for volatility — there's no way to isolate it on one side of the equation. It has to be found numerically, iterating toward the volatility that makes the formula's output match the observed price.
What's the difference between Newton-Raphson and bisection here?
Newton-Raphson converges fast using the option's vega as a derivative, and is used first. When vega is too flat to give a useful direction (common for deep in/out-of-the-money or very short-dated options), the solver falls back to bisection, which is slower but always converges as long as a valid solution exists.
What does "no volatility solves this" mean?
Every option price has a hard no-arbitrage range regardless of volatility (a call can never exceed spot, for example). A price outside that range has no real implied volatility — it's a data error or an arbitrage, not something to force a number onto.
Is the risk-free rate really 0% for crypto options?
That's the standard convention this site uses by default for crypto options, since there's no direct equivalent to a government bond yield embedded in the pricing. You can override it manually if you have a specific rate assumption.