Implied volatility calculator
Turn an option market price, underlying price, strike, time to expiry and risk-free rate into a Black-Scholes implied volatility using a hand-written bisection plus Newton solver, with the model price, vega, intrinsic value, time value and the no-arbitrage price band — and an explicit error when no solution exists.
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Option parameters
All money is handled as integer cents; the model is a Black-Scholes European option with a continuous dividend yield, and the implied volatility is solved by bisection plus Newton iteration.
Result
Solver: bisection for the bracket, then Newton iteration, with a price tolerance of 1e-7 cents.
20.01%4.768.813.950.8142.000.7690570.627590What this tool does
- You have a quote and want to know what volatility the market is charging: enter the premium, underlying price, strike and time to expiry to solve for the implied volatility.
- Compare implied volatilities across strikes and expiries on the same underlying to read the skew and the term structure and see which contract looks expensive.
- Sanity-check quotes before backtesting an options strategy: if the implied volatility comes out well below realised volatility, the premium may be too cheap.
- Teach the model: price an option at 20% volatility and then invert the price — a quick way to show that Black-Scholes pricing and implied volatility are consistent.
Example
Input
Underlying 42.00, strike 40.00, 0.5 years to expiry, 10% risk-free rate, 0% dividend yield, call trading at 4.76
Output
Implied volatility 20.01%; model price 4.76; vega 8.81; discounted intrinsic value 3.95; time value 0.81
This is the classic example from Hull, Options, Futures, and Other Derivatives: with S=42, K=40, T=0.5, r=10% and σ=20% the call is worth 4.7594, so inverting the rounded 4.76 premium returns 20.01% — the 0.01 point gap is the one-cent rounding of the quote.
Frequently asked questions
Why does a premium sometimes report that no implied volatility exists?
Because the price falls outside the no-arbitrage band. A call is always worth between its discounted intrinsic value and the present value of the underlying: below the floor it is cheaper than exercising immediately, at or above the ceiling it costs more than the underlying itself. Neither case has a matching volatility, so the tool errors out instead of inventing a number.
Is the solver bisection or Newton iteration?
Both. The Black-Scholes price is strictly increasing in volatility, so bisection always converges — but it would need hundreds of steps for 1e-7 cent accuracy. The tool first narrows the bracket with bisection, then converges quickly with Newton iteration (whose derivative is vega). If vega is tiny or a Newton step leaves the bracket, it falls back to bisection so it can never diverge.
Is the normal distribution hand-written, and is it accurate enough?
Yes. The standard normal CDF uses the rational approximation from Abramowitz & Stegun, Handbook of Mathematical Functions, formula 26.2.17, with an absolute error below 7.5e-8. That is several orders of magnitude finer than the one-cent precision of the premium, so it never disturbs the solution.
What are the units of the vega column?
Vega is the change in the model price when volatility moves by 1.00, i.e. 100 percentage points, in the same currency as the price. In the example vega is 8.81, so going from 20% to 21% volatility adds roughly 0.088 to the theoretical value. In practice you will scale it down by 100 for a one-point move.
How should the dividend yield be entered?
Use the continuous annualised dividend yield of the underlying over the option life — a 2% yield is entered as 2. Non-paying stocks and most ETFs are entered as 0. Mixing a zero yield with a paying underlying changes the implied volatility, so keep the convention consistent when you compare skews.
Keywords：implied volatilityblack-scholesoption pricingvegagreeks隐含波动率期权定价布莱克舒尔斯希腊字母期权波动率