i What this calculator does
Quick answer: The Black-Scholes model prices a European option from five inputs: the underlying price, strike, time to expiry, volatility and interest rate; volatility is the only one you have to estimate. With the default inputs the calculator returns fair value per share: 1.17.
An option has two parts to its price: intrinsic value, what it would be worth if exercised now, and time value, what the market pays for the chance that it ends up worth more before expiry. The Black-Scholes model, published by Fischer Black and Myron Scholes in 1973 and extended by Robert Merton to include dividends, prices that time value from the underlying price, strike, days to expiry, interest rate and volatility. This calculator returns the fair value for a call or a put, the five Greeks that tell you how that value changes, and the probability the model assigns to finishing in the money.
Enter the price an option actually trades at and the calculator works backwards to the implied volatility, the number options traders use to compare prices across strikes and expiries. For the profit or loss at expiry of single options and spreads, use the options payoff calculator; the background is in how options trading works.
This calculator is for educational purposes only. Results are estimates and may vary depending on market conditions, spreads, commissions, platform settings, and exchange rates. It should not be considered financial advice.
Common results
What the calculator returns as the days to expiry changes, with every other input at its default (option type Call, underlying price 100, strike price 105, volatility 25, risk-free rate 4, dividend yield 0, market price of the option 0, contracts 1).
| Days to expiry | Fair value per share | Delta | Theta per day |
|---|---|---|---|
| 90 days | 3.31 | 0.401 | -0.0307 |
| 60 days | 2.33 | 0.357 | -0.0352 |
| 30 days | 1.17 | 0.275 | -0.0427 |
| 14 days | 0.45 | 0.173 | -0.0467 |
| 7 days | 0.13 | 0.085 | -0.0395 |
| 1 days | 0.00 | 0.000 | -0.0003 |
The Greeks in plain language
Delta is how much the option price moves for a one-unit move in the underlying, and roughly the probability of finishing in the money. Gamma is how fast delta itself changes, highest for at-the-money options close to expiry. Theta is the time value lost each calendar day, which accelerates in the final weeks. Vega is the change in price for a one percentage point change in volatility, largest for long-dated options. Rho is the change for a one point change in interest rates, usually the smallest. Multiply each by the number of contracts and the multiplier to see the exposure of a whole position.
Implied volatility and why it matters
Every input to the model is observable except future volatility. Turning the model around, the volatility that makes the model price equal the market price is the implied volatility, and it is how the market expresses whether options are cheap or expensive. Implied volatility usually rises before earnings and events and falls after them, which is why an option can lose value even when the underlying moves the right way. Comparing implied volatility with the historical volatility of the underlying is a common first check before buying or selling options.
Where the model falls short
Black-Scholes assumes constant volatility, continuous trading, no transaction costs and price changes that follow a log-normal distribution. Real markets have fat tails, jumps and a volatility smile, where out-of-the-money options trade at higher implied volatility than the model would use. It also prices European options, exercisable only at expiry; American options on non-dividend-paying stocks have the same value for calls, while American puts are worth slightly more. Treat the result as a fair-value reference, not as the price you will get.
Spotted an outdated figure? If a rate, limit or formula in this calculator no longer matches its source, email us the page and the source. A material correction is dated on the Updates Log.
See also: FX Options Calculator (Garman-Kohlhagen)
Frequently asked questions
What inputs does the Black-Scholes model need?
The underlying price, the strike, the time to expiry, the risk-free interest rate, the volatility of the underlying and, in the Merton version used here, the dividend yield.
What is a good delta for buying options?
There is no single good delta. Around 0.50 is at the money; 0.30 and below are cheaper out-of-the-money options with lower odds of paying off; 0.70 and above behave more like the underlying itself.
Why does theta increase near expiry?
Because time value is roughly proportional to the square root of time left. The last few weeks contain a large share of the decay, especially for at-the-money options.
How is implied volatility calculated?
By finding the volatility that makes the Black-Scholes price equal the market price of the option. There is no direct formula, so the calculator solves it numerically.
Does Black-Scholes work for American options?
For calls on stocks that pay no dividend, yes, because early exercise is never optimal. American puts and calls on dividend-paying stocks are worth a little more than the model shows.
Is the probability of finishing in the money a real probability?
It is the risk-neutral probability N(d2), which assumes the underlying grows at the risk-free rate. It is a useful guide, not a forecast.
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