Implied Volatility Calculator

Option type

How to use it

  1. Choose a call or a put.
  2. Enter the option’s market price, the underlying price S and the strike price K.
  3. Enter the calendar days to expiration and the risk-free rate, in percent a year. Use a Treasury bill yield for the same term.
  4. If the underlying pays dividends, enter its dividend yield in percent a year. Leave it blank for none.
  5. The implied volatility, what it implies about the price at expiration, and the greeks appear and update as you type.

To see it filled in, choose “Show an example”. The calculator prices European options, which can only be exercised at expiration. Most US stock options are American, and for calls on non-dividend stocks the two are worth the same. For puts, and for calls just before a large dividend, the early-exercise right adds value, so the European model reads a slightly higher volatility than the market is pricing. For comparing options and for study; not financial advice.

Result

Enter the option’s market price.

European exercise, rates and yield continuously compounded, time as calendar days over 365. Vega and rho are per 1 % change; theta is per calendar day.

Methodology, equations and examples

Implied volatility is the volatility σ that makes the Black-Scholes price of an option equal to its market price. Every other input to the formula can be observed: the underlying price, the strike, the time to expiration and the interest rate. Volatility, the annualized standard deviation of the underlying's log returns, cannot. Solving the formula backwards for it gives the market's consensus estimate of how much the underlying will move before expiration.

The Black-Scholes-Merton formula

With a continuous dividend yield q (Merton's extension), the prices of a European call C and put P are:

C = S e^(−qT) N(d₁) − K e^(−rT) N(d₂)

P = 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

  • S – Underlying price
  • K – Strike price
  • T – Time to expiration in years, taken as calendar days / 365
  • r – Risk-free rate, continuously compounded
  • q – Dividend yield, continuously compounded
  • σ – Volatility, per year
  • N(x) – Cumulative standard normal distribution, the probability that a standard normal variable is less than x

Solving for σ

There is no closed form for σ, so it is found numerically. The option price rises steadily with σ, so at most one volatility fits any price. The calculator starts from the Brenner-Subrahmanyam estimate σ ≈ √(2π/T) × C/S and refines it with Newton's method:

σ_new = σ − (BS price(σ) − market price) / vega

Vega, the derivative of the price with respect to σ, is S e^(−qT) N′(d₁) √T. Each step also narrows a bracket around the answer, and any Newton step that lands outside the bracket is replaced by bisection. This keeps the method stable for deep in- and out-of-the-money options, where vega is close to zero. It stops when the model price matches the market price to about 1 part in 10¹².

A price has an implied volatility only if it lies within the option's no-arbitrage bounds. The lower bound is the discounted intrinsic value, max(0, S e^(−qT) − K e^(−rT)) for a call and max(0, K e^(−rT) − S e^(−qT)) for a put. The upper bound is S e^(−qT) for a call and K e^(−rT) for a put. The calculator says so when a price falls outside them.

Example

A three-month call (T = 0.25, or 91.25 days) with a strike of $20 trades at $1.875. The stock is at $21, pays no dividend, and the risk-free rate is 10 %. This is the implied volatility example in Hull, Options, Futures, and Other Derivatives.

The lower bound is $21 − $20 e^(−0.1 × 0.25) = $1.4938, so the price has $0.38 of time value and a volatility exists. At σ = 20 % the formula gives $1.7647, too low. At σ = 25 % it gives $1.9268, too high. The answer lies between, and Newton's method settles on σ = 23.45 %. At that volatility d₁ = 0.6879, d₂ = 0.5707, and

C = $21 × N(0.6879) − $20 e^(−0.025) × N(0.5707) = $1.875

so the market price is reproduced exactly.

What the price implies

The implied volatility also gives the market's view of where the underlying could be at expiration. These figures are risk-neutral: they assume the underlying grows at the risk-free rate less the dividend yield, not at its expected return. Treat them as what the option price implies, not as a forecast.

  • Chance of finishing in the money is N(d₂) for a call and N(−d₂) for a put.
  • Expected move is S σ √T, one standard deviation of the price at expiration.
  • The 68 % and 95 % ranges come from the lognormal distribution Black-Scholes assumes: S e^((r − q − σ²/2)T ± z σ √T), with z = 1 and 1.96. They are skewed upward, since a price can rise without limit but cannot fall below zero.
  • Breakeven at expiration, for the buyer, is K plus the premium for a call and K minus the premium for a put.
  • Intrinsic value is what the option would be worth exercised today, max(0, S − K) for a call and max(0, K − S) for a put. Time value is the rest of the premium. A deep in-the-money European put can have negative time value, because it cannot be exercised early and the strike is received only at expiration.

In the example, the call has a 71.6 % chance of finishing in the money and an expected move of ±$2.46. The 68 % range is $19.02 to $24.04 and the 95 % range is $16.99 to $26.91. The buyer breaks even at $21.875, and the $1.875 premium is $1.00 of intrinsic value and $0.875 of time value.

The greeks

The results also give the greeks, the option's sensitivities, at the implied volatility. Delta is the change in option price per $1 change in the underlying, e^(−qT) N(d₁) for a call. Gamma is the change in delta per $1. Vega is the change in price per 1 percentage point of volatility. Theta is the change in price per calendar day that passes. Rho is the change in price per 1 percentage point of the risk-free rate. In the example, delta is 0.7543 and vega is $0.0331, so a one-point rise in volatility to 24.45 % would raise the call by about 3.3 cents.

Top