Black-Scholes calculator
Enter the underlying, strike, days, volatility, rate, and dividend yield. This prices a European call or put and returns delta, gamma, vega, theta, rho, vanna, and charm, each labelled with the unit it is quoted in.
Worked example — A 30-day at-the-money call, 28% volatility, prices at: $3.2979. Delta +0.5256, theta -0.0564 a calendar day. $100 strike, 4% rate, 1.5% dividend yield.
30-day, at the money: The page's own silent defaults, named as an explicit starting point.
Contracts and your quote 2
$0.00 of that is intrinsic and $3.30 is time value. Breakeven at expiry: $103.30.
European exercise, one lognormal volatility for the whole life of the option, a continuous dividend yield, and no fees or bid-ask spread. Nothing on this page is quoted from a market.
Every greek, with the unit it is quoted in
At the default inputs ($100 spot and strike, 30 days, 28% volatility):
| Greek | Value | Unit |
|---|---|---|
| Delta | +0.5256 | option $ per $1 of underlying |
| Gamma | +0.0495 | delta per $1 of underlying |
| Vega | +0.1140 | option $ per 1 volatility point |
| Theta, per calendar day | -0.0564 | option $ per calendar day |
| Theta, per trading day | -0.0817 | option $ per trading day |
| Rho | +0.0405 | option $ per 100bp of rate |
| Vanna | +0.00021 | delta per 1 volatility point |
| Charm | -0.00041 | delta per calendar day |
What the time value does between now and expiry
Time value only ever falls to zero, but not in a straight line: the loss accelerates as expiry approaches.
| Days left | Model price | Time value |
|---|---|---|
| 30 | $3.2979 | $3.2979 |
| 22.5 | $2.8458 | $2.8458 |
| 15 | $2.3133 | $2.3133 |
| 7.5 | $1.6260 | $1.6260 |
| 3 | $1.0228 | $1.0228 |
| 0 | $0.0000 | $0.0000 |
Worked example: the tested benchmark case
A one-year at-the-money option, 5% risk-free rate, 20% annualised volatility, no dividend, prices at $10.4506. This is the exact case the test suite asserts to four decimals, computed here by the same module the calculator above runs, so you can load the "tested benchmark" preset and check this tool against a known answer.
Why theta per calendar day and per trading day differ
Most greek confusion is a units problem, and theta is the clearest case: the same annual decay, spread over 365 calendar days instead of 252 trading sessions, prints a different daily number.
| Unit | Theta |
|---|---|
| Per year | -20.5975 |
| Per calendar day | -0.0564 |
| Per trading day | -0.0817 |
The trading-day figure is 45% larger in magnitude than the calendar-day figure, because the same annual decay is divided by fewer sessions. A position held over a weekend answers to the calendar version; a desk marking daily P&L against trading sessions uses the other.
This assumes European exercise, one constant volatility for the whole life of the option, continuous trading, no transaction costs, and lognormal returns with no jumps. Real markets break all five: volatility varies by strike and expiry, American options can be exercised early, and returns have fatter tails than the normal distribution allows. Treat the price as the model's answer under its own assumptions, not as fair value. To back out the volatility a quoted price implies, use the implied volatility calculator; to turn a volatility into a range and a set of probabilities, use the expected move calculator; for several legs at expiry, the option spread calculator.
The next question after this one
Implied volatility calculatorCalculatorRun the model backwards: from a quoted price to the volatility that produces it, with bid and ask as a band.Open nextOther questions
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American exercise with a binomial tree, term structure by expiry, and a volatility surface built from your own quotes are Pro features. The launch list sends one email at launch.