Implied volatility calculator

What is implied volatility and how is it calculated?

Implied volatility is the one Black-Scholes input you cannot read off a screen. The strike and expiry come from the contract, the underlying price and interest rate from the market. Volatility is the yearly spread of returns the model must assume, and implied volatility is the value that makes the model's price equal the option's actual price.

No algebra rearranges the formula to give volatility directly, so the answer is found by search: try a volatility, price the option, compare with the market price, adjust, and repeat. Because a European option's value always rises as volatility rises, each price inside the allowed range matches exactly one volatility.

With the default inputs, a call with a $105 strike, 60 days left, the underlying at $100, a 4.5% rate and no dividend yield trades at $2.35. The volatility that reproduces $2.35 is 24.98% a year. Re-pricing at 24.9775% returns $2.350000, and vega there is 0.1517, meaning the price would change by about $0.15 for one point of volatility.

How the solver works: Newton's method with a safety net

Newton's method uses the slope of the price against volatility, which is vega, to jump toward the answer: new sigma = sigma - (model price - market price) / vega. Near the solution each step roughly doubles the number of correct digits.

For the default option the search starts at 70.97%, worked out as √(2 x |ln(F / K)| / T) from the forward price F, the strike K and the years to expiry T. At 70.97% the model price is $9.69, far above $2.35. One Newton step lands on 25.60%, the next on 24.98%, and the third on 24.9775%, with the price then matching to better than a billionth. The table under the calculator lists every step for your own inputs.

Newton's method can misbehave where vega is tiny, for example far from the strike with little time left, because dividing by a tiny slope throws the next guess a long way. So the calculator keeps a bracket: one volatility known to give too low a price and one known to give too high a price. Any Newton step that would land outside the bracket is replaced by the midpoint, a bisection step. Bisection is slower but cannot fail, so the search always ends with an answer between 0% and 1,000% a year whenever one exists.

Implied volatility examples in dollars, pounds and euros

Each of these prices was first set from the Black-Scholes formula at a round volatility, rounded to the cent, and then fed back to the solver. The small gaps from the round figure come from that rounding.

The pound put. The £42 put is out of the money, so all of its £1.31 is time value. Its vega is small, 0.0759, so a penny on the price moves the implied volatility by about 0.13 points; quotes on cheap options give noisy volatilities.

The euro call. With 120 days left and a higher price, vega is 0.4024, so the same euro cent moves the answer by only about 0.025 points.

The Bitcoin call. This is the 30-day $65,000 call valued on the Black-Scholes calculator at 55% volatility. Entering its $1,981.82 value here returns 55.00%, which shows the two pages use the same formula and the same conventions. For the crypto view of volatility, read crypto implied volatility explained.

How the option price, days and rate move implied volatility

Holding the default option and changing only the price shows how quickly the answer moves. Each row is solved separately:

A $0.10 rise from $2.35 to $2.45 lifts the implied volatility from 24.98% to 25.64%, a move of 0.66 points, which is close to $0.10 divided by the vega of 0.1517.

Days to expiry. The same $2.35 means very different things at different expiries. With 30 days left it implies 36.48%; with 60 days, 24.98%; with 90 days, 19.72%; with 180 days, 12.46%; and with 365 days, 6.34%. A longer option has more time to move, so a given price needs less volatility to justify it.

Interest rate. For a call, a higher rate lowers the implied volatility of the same price: at 0% the default price implies 26.59%, at 2% it implies 25.88%, and at 6% it implies 24.43%. For a put the effect runs the other way. A dividend yield works in the opposite direction to the rate.

When does a price have no implied volatility?

Only prices strictly inside the no-arbitrage range have an answer. For a European call that range runs from max(S e^-qT - K e^-rT, 0) up to S e^-qT; for a put it runs from max(K e^-rT - S e^-qT, 0) up to K e^-rT. The price range panel under the calculator shows both ends for your inputs.

• Too low. A call with a $100 strike, 30 days left, the underlying at $110 and a 4.5% rate is worth at least $10.37 at zero volatility. A price of $10.00 is below that, so the calculator says no volatility fits and shows the bound. Often the cause is a stale quote, a dividend that should have been entered, or a put typed in as a call.
• Too high. A call can never be worth as much as the discounted underlying, and a put never as much as the discounted strike. A call priced at $100 with the underlying at $100 has no answer.
• Beyond the search range. A one-day at-the-money call priced at $30 would need well over 1,000% a year; even at 1,000% the model gives $20.65. The calculator stops there and says so.

A European put can legitimately trade below intrinsic value. With the underlying at $60, a $100 strike, one year and a 5% rate, the lowest possible put value is $35.12, so a price of $36.00 is valid and implies 32.68% even though intrinsic value is $40.

How to calculate implied volatility in Excel or Google Sheets

Neither spreadsheet has a built-in implied volatility function, but both can run the search. First build the Black-Scholes price: S, K, T in years, r, a volatility guess and q in cells B1 to B6, with d1, d2 and the call value in B7 to B9, exactly as on the Black-Scholes page. Put the market price in B10.

• Excel Goal Seek. On the Data tab, in the Forecast group, choose What-If Analysis, then Goal Seek. Set cell B9, to value equal to the market price, by changing cell B5. Excel adjusts the volatility until the price matches. With the default inputs it should settle near 0.2498.
• Newton steps by hand, in either program. In the row below, compute vega as =B1*EXP(-B6*B3)*NORM.S.DIST(B7,FALSE)*SQRT(B3) and the next guess as =B5-(B9-B10)/vega. Copy the block down five or six rows, each row reading the guess above it. Starting from 0.30, the column reads 0.250341 after one row and 0.249775 after two.

Enter rates and volatility as decimals, so 4.5% is 0.045, and count days over 365 to match this calculator. A different day count gives a slightly different answer for the same price.

A quick estimate for an at-the-money option

When the strike is close to the forward price, the call value is roughly 0.4 x S x sigma x √T, because 1 / √(2π) is about 0.3989. Turned round, sigma is roughly price / (0.4 x S x √T), a common at-the-money shortcut.

For a 30-day call with S and K both at 100, no interest or dividend, and a price of $2.30, √T is 0.2867. The shortcut gives 20.06% (20.11% with 0.3989 in place of 0.4), and the exact implied volatility is 20.1124%. The estimate is close at the money and grows worse as the strike moves away, which is why the calculator uses it only to choose a starting point when the forward sits right at the strike.

Why the call and the put can show different implied volatility

In the Black-Scholes model a call and a put with the same strike and expiry must imply the same volatility, because put-call parity ties their prices together. Market data often shows a gap, for four ordinary reasons:

• Rounding. The Black-Scholes default call and put are worth $10.4506 and $5.5735. Rounded to $10.45 and $5.57, they imply 19.998% and 19.991%, already a small gap.
• Dividends and rates. A dividend or rate that differs from the one the market uses moves the call one way and the put the other.
• Early exercise. If the contract allows exercise before expiry, feeding its American price into a European formula overstates the implied volatility of a deep in-the-money put.
• Bid and ask. Using the bid for one and the ask for the other builds the spread into the gap.

Across strikes the pattern goes further: market prices usually imply different volatilities at different strikes, the smile or skew. That shape is a feature of real prices that a single-volatility model does not produce. For an example from crypto markets, see Bitcoin options skew explained.

What this implied volatility calculator leaves out

It inverts the European Black-Scholes-Merton formula and nothing else. It does not price early exercise, discrete dividends paid on set dates, jumps, or a volatility that changes with the strike or over time. It reports what volatility the price implies under those assumptions, which is not a forecast of how much the underlying will move and says nothing about whether the option is cheap or dear.

The answer is only as good as the price you enter. A last-traded price from hours ago, a wide spread, or an option that rarely trades can each produce an implied volatility that would not survive a fresh quote. For profit and loss at expiry once you hold a position, see the options profit calculator.

Common implied volatility mistakes and key terms

• Entering the contract price. The option price is per unit. A $235 contract covering 100 units is a $2.35 price.
• Mixing call and put. A put price entered as a call often falls below the call's lowest possible value and returns no solution.
• Forgetting the dividend yield. On a stock with a meaningful yield, leaving it out lowers the implied volatility of calls and raises that of puts.
• Using trading days for T. This page counts calendar days over 365, like its Black-Scholes sibling.

Implied volatility: the volatility at which the model value equals the market price. Historical volatility: the annualized standard deviation of past returns. Vega: the change in option value for one point of volatility. Time value: the part of the price above intrinsic value, the only part volatility explains. No-arbitrage bounds: the lowest and highest prices consistent with the model at any volatility. Newton's method: a search that uses the slope to step toward a root.

Frequently asked questions

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