Implied Volatility Calculator

Inputs

Loads the price, the ~30-day expiry and the at-the-money strike, with live r and q.

Option

Option price

Per share — the mid of the bid and ask is usual.

Stock price

Strike

Time to expiration

days

Calendar days (years = days ÷ 365), as in the calculator.

Risk-free rate

Dividend yield

Results

Implied volatility

42.36%

1σ daily move: ±$2.22; 2.22%
1σ weekly move: ±$5.87; 5.87%
1σ to expiration: ±$12.14; 30 days
Model price: $5.00
Vega: $0.11/pt
Time value: $5.00; Intrinsic $0.00

Moves use calendar days, the same time basis as the calculator (a trading-day basis would use √252 and give a 20% larger daily figure).

What implied volatility is

Implied volatility is the volatility that makes the Black-Scholes-Merton price equal the option's market price. Historical volatility looks back at how the stock moved; implied volatility is forward-looking, the market's price for movement over the option's life. When demand for options rises, IV rises with it.

There's no closed-form inverse of the formula, so this calculator solves it numerically: a safeguarded Newton-Raphson iteration that uses vega as the derivative and falls back to bisection, with the same dividend yield and calendar-day time basis as the options calculator.

CallC = S·e^−qT·N(d₁) − K·e^−rT·N(d₂)

PutP = K·e^−rT·N(−d₂) − S·e^−qT·N(−d₁)

d₁ = [ln(S/K) + (r − q + σ²/2)·T] / (σ·√T), d₂ = d₁ − σ·√T

ParityC − P = S·e^−qT − K·e^−rT

N(x) is the standard normal cumulative distribution, φ(x) its density, T the time to expiration in years and q the continuous dividend yield (Merton's extension; q = 0 is the original 1973 model).

Worked example

Hull's textbook case: a European call worth $1.875 on a stock at $21, with a $20 strike, three months to expiration and a 10% risk-free rate. The textbook answer is about 23.5%; the solver finds 23.45%, and pricing the call at that volatility gives back $1.88.

Find σ such that BS(S = 21, K = 20, T = 0.25, r = 0.10, σ) = 1.875 → σ = 23.45%

Inputs
Option	Call
Option price	$1.875
Stock price (S)	$21.00
Strike (K)	$20.00
Time (T)	0.25 years (91.25 days)
Risk-free rate (r)	10%

This calculator gives
Implied volatility	23.45%
Model price at that IV	$1.88
1σ move to expiration	±$2.46
Vega	$0.03 per vol point

From IV to an expected move

IV is quoted per year. To turn it into a dollar move over a period, scale by the square root of time: one standard deviation over t days is S × IV × √(t ÷ 365). The results show it for a day, a week and the time to expiration, on calendar days (a trading-day basis divides by 252 instead, which gives a larger daily figure for the same IV).

1σ move = S × σ × √(days ÷ 365)

Volatility smile and skew

In practice IV varies across strikes. Out-of-the-money puts usually carry higher IV than at-the-money options, reflecting demand for downside protection; that pattern is pronounced in equity indexes. See it on a live chain in the options chain.

Assumptions & limitations

The IV is the one volatility at which the European Black-Scholes-Merton price matches yours. For American options (most US stock options) it can differ slightly, most for deep in-the-money puts.
Constant volatility and rate to expiration, and dividends as a continuous yield q.
The answer is only as good as the price: stale or one-sided quotes give misleading IVs.
Expected moves assume log-normal returns; real markets have fatter tails and jumps.
Time in calendar days ÷ 365.

Frequently asked questions

Why is my answer different from my textbook or broker?

Check the time input first. Textbooks give T in years (0.25 is three months); this calculator starts in days, so switch the time field to Years or enter 91.25 days. Typed into the days field, 0.25 is six hours and gives a very different IV. The model counts 365 calendar days a year; if your course counts 252 trading days, 63 trading days is 0.25 years, so enter it in years. Brokers may also use an American-exercise model, the last trade instead of the mid, or a different dividend and rate, each of which moves IV by a point or two.

Why does it say no volatility gives this price?

Under the model a European call is worth at least S·e^(−qT) − K·e^(−rT), roughly its intrinsic value, and less than the stock; a put is worth at least K·e^(−rT) − S·e^(−qT) and less than the discounted strike. A price outside those bounds has no implied volatility. It usually means a stale quote, a typo, or a time or dividend input that's off.

Which price should I enter?

The mid of the bid and ask, per share. The last trade can be hours old, and the bid or the ask alone gives a lower or higher IV than the market's consensus.

How is the expected move worked out?

One standard deviation of the stock's price over a period is roughly S × IV × √(days ÷ 365). The stock is expected to finish inside that range about 68% of the time under the model. It's a model figure, not a forecast: real moves have fatter tails.

Why do strikes on the same expiry have different IVs?

The model assumes one volatility for every strike, but markets price crash risk: out-of-the-money puts usually trade at higher IVs than at-the-money options. Plotting IV against strike gives the volatility smile or skew.

References

Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637–654.
Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4(1), 141–183.
Hull, J. C. Options, Futures, and Other Derivatives. Pearson (11th ed., 2021).