How to use the implied volatility solver
- Choose a call or a put and enter the stock price, the strike price and the time to expiry with its unit.
- Enter the yearly risk-free rate and, if the stock pays dividends, a dividend yield in the advanced options.
- Enter the market price of one option.
- Read the implied volatility, the model price at that volatility and the lowest and highest price that is possible.
Formula
call price = S e^(-qT) N(d1) - K e^(-rT) N(d2)
d1 = (ln(S/K) + (r - q + v^2/2) T) / (v sqrt(T)), d2 = d1 - v sqrt(T)
find v so that the model price equals the market price
The model price rises steadily with volatility, so exactly one volatility fits any price between the lowest and highest possible price. The page halves a bracket around it until the answer is fixed to many digits.
A one year at the money call
A call with a stock price of $100, a strike of $100, one year to expiry and a rate of 5 percent is priced at $10.450584 by the model at 20 percent volatility. Entering that price returns an implied volatility of 20.0000 percent, with a model price of $10.450584 and a vega of about $0.3752 per volatility point.
A put with a strike of $90, six months to expiry, a 4 percent rate and a price of $1.50 returns an implied volatility of 20.7437 percent.
Prices that have no solution
A call cannot be worth less than the stock price (after dividends) minus the discounted strike, and cannot be worth more than the stock price. A put has the matching limits. If the price you enter is outside them, no volatility fits and the page says so with the limits for your inputs.
A price right at a limit would need a volatility of zero or one with no upper bound, so it is refused as well.
What this is and is not
Implied volatility is the volatility that the model needs to match a price. It summarizes what the market price implies under the model’s assumptions, and it is not a forecast.
The model prices European options, which can be exercised only at expiry. American options, early exercise and bid and ask spreads are not modeled. This is educational and not trading advice.
Assumptions and limits
- The option is European and the model is Black-Scholes with a constant rate and dividend yield.
- The rate and the yield are continuously compounded yearly rates.
- The market price is for one share of the option.
- The time unit is converted to years using 12 months or 365 days.
- The search covers volatilities up to 2,000 percent.
Frequently asked questions
Why does the page say no volatility fits?
The price is below the lowest value the option can have or above the highest. That usually means a wrong input, such as the wrong type, strike or expiry, or a stale quote.
What is vega?
Vega is how much the option price changes when volatility rises by one point. The page shows it at the solved volatility so you can see how sensitive the price is.
Can I use this for American options?
It will give a number, but the model ignores early exercise, so the result is only an approximation for American options, and a poor one for deep in the money puts.
Does a higher implied volatility mean a higher price?
Yes, for both calls and puts, all else equal. That steady rise is what lets the page find exactly one answer.