Black-Scholes Options Calculator
Price a call or put with Black-Scholes and see all five Greeks, intrinsic value, time value and break-even.
An option is the one instrument whose price is itself the output of a model. A share is worth what it trades at. An option's value depends on the strike, the time remaining, the volatility expected over that time and the interest rate — and the way those interact is not something anyone works out in their head. This Black-Scholes options calculator does that arithmetic for you, returning the option's theoretical value together with every Greek behind it.
Formula: call = S x e^(-qT) x N(d1) - K x e^(-rT) x N(d2)
Your figures
Calendar days. The model converts to years internally.
Annualised, as a percentage.
Annualised. Roughly the yield on a short government bond.
Optional — annualised continuous yield on the underlying.
Results
Theoretical call price
US$2.49
per share, 30 days to expiry
- Intrinsic valueWhat the option is worth if exercised right now.
- US$0.00
- Time valueWhat is paid for the possibility of further movement.
- US$2.49
- Break-even at expiry
- US$102.49
- DeltaPrice change per 1.00 move in the underlying.
- +0.5400
- GammaChange in delta per 1.00 move in the underlying.
- 0.06923
- Theta, per dayValue lost to one calendar day passing, all else equal.
- -US$0.0450
- Vega, per 1% of volatility
- US$0.1138
- Rho, per 1% of rate
- +US$0.0423
- Probability of finishing in the moneyA risk-neutral model probability, not a real-world forecast.
- 51.72%
What this means
Black-Scholes values this call at US$2.49 per share, of which US$2.49 is time value that decays to nothing by expiry — currently at US$0.0450 a day. A delta of 0.540 means the price moves roughly that much for each 1.00 the underlying moves. This is a model value under specific assumptions, not a prediction of what the option will trade at.
- Prices European exercise. American options can be exercised early, which adds a premium the model does not capture — most significant for puts and for calls on dividend-paying shares.
- Assumes one constant volatility. Real markets price a different implied volatility at each strike and expiry, which is why a single model price rarely matches every quote on the chain.
Theoretical call price: US$2.49
The link reopens this calculator with the same numbers. Nothing you enter is sent to a server.
Continue with these figures
About the Black-Scholes Options calculation
Black-Scholes-Merton is the standard call and put pricing model, and it earned Scholes and Merton the 1997 Nobel prize in economics. It gives a theoretical price and, more usefully, the Greeks: the sensitivity of that price to each input moving.
The Greeks are what make options manageable. Delta, gamma, theta, vega and rho each answer one question: delta says how much the price moves with the underlying, gamma how fast delta itself changes, theta what one day of time costs, vega what a point of volatility is worth, and rho what interest rates contribute. A position is understood through those five numbers rather than through the price alone.
Black-Scholes Options formula
d1 = [ln(S/K) + (r − q + σ²/2) × T] ÷ (σ × √T)
d2 = d1 − σ × √T
call = S × e^(−qT) × N(d1) − K × e^(−rT) × N(d2)
put = K × e^(−rT) × N(−d2) − S × e^(−qT) × N(−d1)
delta(call) = e^(−qT) × N(d1)
gamma = e^(−qT) × φ(d1) ÷ (S × σ × √T)
vega = S × e^(−qT) × φ(d1) × √T- S
- Spot price of the underlying.
- K
- Strike price.
- T
- Time to expiry in years. 30 days is 30 ÷ 365.
- σ (sigma)
- Annualised volatility, as a decimal.
- r
- Annualised risk-free interest rate.
- q
- Annualised continuous dividend yield.
- N(x)
- Cumulative standard normal distribution — the probability of a draw below x.
- φ(x)
- Standard normal probability density.
The call formula reads as the expected value of receiving the share, weighted by the probability of exercise, less the expected cost of paying the strike, similarly weighted and discounted. N(d2) is the risk-neutral probability the option finishes in the money; N(d1) is the probability-weighted value of the share if it does.
Every option price splits into intrinsic value and time value. Intrinsic value is what exercising right now would yield and cannot be negative. Time value is everything else — what is paid for the possibility of further movement — and it decays to exactly zero at expiry regardless of what the underlying does.
Gamma and vega are identical for a call and a put at the same strike and expiry, which follows from put-call parity: the difference between the two prices is a fixed forward value that does not depend on volatility, so the second derivative and the volatility sensitivity must match.
Theta is reported per calendar day and vega and rho per percentage point, because those are the units traders actually use. The model produces them per year and per whole unit of rate, so the conversion happens before display.
Black-Scholes Options: a worked example
A one-year at-the-money call: underlying at 100, strike 100, 20% volatility, a 5% risk-free rate and no dividend.
- σ√T0.20 × √1 = 0.20
- d1[ln(1) + (0.05 + 0.02) × 1] ÷ 0.20 = 0.35
- d20.35 − 0.20 = 0.15
- N(d1)0.6368
- N(d2)0.5596
- Call price100 × 0.6368 − 100 × e^(−0.05) × 0.5596 = 10.4506
- Matching put5.5735
10.45 per share, all of it time value since the option is exactly at the money. Put-call parity checks the pair: 10.4506 − 5.5735 = 4.8771, which equals 100 − 100 × e^(−0.05).
How to use the Black-Scholes Options calculator
- Choose call or put and enter the underlying and strike prices.
- Enter the days to expiry — the calculator converts to years internally.
- Enter the implied volatility. If you do not know it, derive it from a traded price with the implied volatility calculator.
- Enter the risk-free rate, roughly the yield on a short-dated government bond.
- Add a dividend yield for shares or indices that pay one; it reduces call values and raises put values.
- Read the Greeks alongside the price. Theta in particular makes the daily cost of holding the position concrete.
Limitations of the Black-Scholes Options calculator
Every calculation makes assumptions. These are the ones that matter here.
- Prices European exercise. American options can be exercised early, which adds a premium the model does not capture — most noticeable on puts and on calls over a dividend.
- Assumes one constant volatility. Real markets price a different implied volatility at each strike and expiry, so a single model price will not match every quote on a chain.
- Assumes the underlying moves continuously with no jumps. Gaps around earnings and news are exactly what the model cannot represent, and they are systematically underpriced by it.
- Ignores transaction costs, bid-ask spread and any borrowing cost on a short position.
- The probability of finishing in the money is a risk-neutral model probability, not a real-world forecast.
Frequently asked questions about Black-Scholes Options
What is the difference between intrinsic and time value?
Intrinsic value is what the option would be worth if exercised immediately — the amount it is in the money, or zero. Time value is the rest of the price, paid for the possibility that the underlying moves further before expiry. Time value is always zero at expiry, which is what theta measures the decay of.
Why does my broker's price differ from this option pricing calculator?
Usually because the market is pricing a different implied volatility than you entered — each strike has its own. Early-exercise value on American options, dividends, borrowing costs and the bid-ask spread account for the rest. The model gives a theoretical value, not a quote.
What does an options Greeks calculator tell me that the price alone does not?
Delta is how much the price moves per 1.00 in the underlying, and roughly how many shares the position behaves like. Gamma is how quickly delta changes. Theta is what a day costs. Vega is what a point of volatility is worth. Rho is the interest rate sensitivity, usually the least important except on long-dated options.
Is N(d2) the real probability of profit?
No. It is a risk-neutral probability — a construct within the model, computed under an artificial measure that makes the maths work. It is a reasonable rough guide but it is not a forecast, and it does not account for the premium paid, so it is not the probability of profit either.
Does a Black-Scholes calculator work for American options?
It gives a close approximation for American calls on shares paying no dividend, where early exercise is never optimal. For American puts, and for calls over a dividend, early exercise has genuine value that Black-Scholes does not price, so the model sits below the true value.
Last reviewed . See our methodology and editorial policy.
More options calculators