Black-Scholes calculator
Black-Scholes prices an option from five inputs: the underlying price, the strike, the time left, the volatility and the risk-free rate, with a sixth for dividend yield. This calculator returns the theoretical price, all five Greeks, and the split between intrinsic and extrinsic value.
Five inputs, and only one you cannot look up
Spot, strike, days, rate and dividend yield are all observable. Volatility is not. It is the one input you have to supply an opinion about, and every disagreement about what an option is worth is a disagreement about that number.
Which is why the model gets used backwards more often than forwards. Nobody computes a price and compares it to the market. They take the market price and solve for the volatility that produces it, and then argue about whether that volatility is too high or too low. That is what the implied volatility calculator does.
What each Greek is worth to a seller
- Delta is share equivalents. A 0.30 delta short call behaves like being short 30 shares, and that is the reading that matters for a portfolio, not the probability one.
- Gamma is how fast delta moves. Small and boring at 45 days out. Vicious in the last week, which is the real argument against selling weeklies.
- Theta is your paycheck, quoted per day per share. Multiply by 100 for the contract. It is not constant, it accelerates, and roughly the last 21 days is where most of it lives.
- Vega is what one volatility point does to the contract. Sell into high IV and vega works for you when it collapses. Sell into low IV and you are short the thing that has the most room to move against you.
- Rho is rate sensitivity. It mattered again after 2022, and on a 45-day contract it is still the smallest number on the page.
Where the model is wrong
Black-Scholes prices a European option, exercisable only at expiration. American equity options can be exercised any day, which matters most for in-the-money puts and for calls around an ex-dividend date. It assumes constant volatility, which is why the volatility smile exists as a permanent correction pasted on top of the model. It assumes lognormal returns, which understates the frequency of large moves, and the understatement is worst exactly where a premium seller gets hurt.
Use it as a reference point. The market is the price. The model tells you what the market is implying, and that is a genuinely useful thing to know.
Questions people actually ask
What risk-free rate should I use?
The Treasury yield closest to your expiration. A 45-day contract wants roughly the 1 to 2 month bill. Precision here barely moves the price on a short-dated equity option, so anything in the right neighborhood is fine.
Why does my result differ from my broker?
Almost always the volatility input, and after that the model. Brokers price American options with a binomial or trinomial tree, which handles early exercise and prices in-the-money puts slightly higher than Black-Scholes does. The gap is small on out-of-the-money contracts.
Is theta shown per day or per year?
Per day, per share, which is the convention every broker uses. Multiply by 100 to get what the contract decays in a day, holding everything else still. Nothing ever holds still.
How accurate is the math on this page?
It is the same Cephes normal-distribution port the OptionsKing engine uses, matched to the reference implementation to full double precision. The calculator and the app will not disagree with each other.