Black-Scholes Options Calculator
Theoretical value and the full set of Greeks for a European call or put, computed from the inputs you supply. No market data is used and none is displayed — change any input and every figure below recalculates from the formula.
Option parameters
Values you supply. Nothing here is fetched from a market data source.
Current price of the underlying
Exercise price of the option
Calendar days, converted to years as days ÷ 365
Annualised volatility you want to price at
Continuously compounded annual rate
European exercise, no dividends
Theoretical value
$3.0626
$306.26 per 100-share contract
Delta (Δ)
0.5371
Value change per $1 of spot
Gamma (Γ)
0.0554
Delta change per $1 of spot
Vega (ν)
0.1139
Per +1 point of volatility
Theta (Θ)
-0.0544
Per calendar day elapsed
Rho (ρ)
0.0416
Per +1 point of interest rate
Value breakdown
Per share, for one long call bought at the theoretical value above and held to expiry.
Intrinsic value
$0.0000
max(spot − strike, 0)
Time value
$3.0626
theoretical value − intrinsic
Break-even at expiry
$103.0626
strike + premium
Moneyness
At the money
spot $100.00 vs strike $100.00
Max loss (long)
$3.0626
Premium paid, if it expires worthless
Max gain (long)
Unbounded
Payoff rises with spot without limit
d₁
0.0932
Black-Scholes intermediate term
d₂
0.0215
d₁ − σ√T
Profit and loss vs spot price
One long call, per share. At expiry is intrinsic value minus the premium. Today is the Black-Scholes value at that spot minus the premium, holding days to expiry, volatility, and rate at the values entered above.
Delta vs days to expiry
Black-Scholes delta recomputed at each expiry, with spot, strike, volatility, and rate unchanged.
Gamma vs days to expiry
Black-Scholes gamma recomputed at each expiry, with spot, strike, volatility, and rate unchanged.
The formula used
d₁ = [ ln(S / K) + (r + σ² / 2) · T ] / (σ · √T)
d₂ = d₁ − σ · √T
Call = S · N(d₁) − K · e^(−rT) · N(d₂)
Put = K · e^(−rT) · N(−d₂) − S · N(−d₁)
Δ = N(d₁) for a call, N(d₁) − 1 for a put
Γ = φ(d₁) / (S · σ · √T)
ν = S · φ(d₁) · √T (shown ÷ 100, per 1 volatility point)
Θ = −S · φ(d₁) · σ / (2√T) ∓ r · K · e^(−rT) · N(±d₂) (shown ÷ 365, per day)
ρ = ± K · T · e^(−rT) · N(±d₂) (shown ÷ 100, per 1 rate point)
S is spot, K is strike, T is time to expiry in years (days ÷ 365), r is the continuously compounded risk-free rate, σ is annualised volatility, N is the standard normal cumulative distribution function and φ its density. The upper sign applies to calls, the lower to puts.
At S = 100, K = 100, r = 5%, σ = 20%, T = 1 year this returns a call value of $10.4506 and a put value of $5.5735, the standard published reference values for those inputs. Call and put outputs satisfy put-call parity, C − P = S − K·e^(−rT), to floating-point precision.
This is the European, no-dividend Black-Scholes model. It assumes constant volatility and interest rates and no early exercise, so it will not match the traded price of an American option, an option on a dividend-paying underlying, or any option whose market volatility differs from the figure entered above. All figures are per share; a standard US equity contract covers 100 shares.
Frequently asked questions
Where does this calculator get its data?+
It does not use any market data. Every figure on the page is computed from the spot price, strike, days to expiry, volatility, and risk-free rate you type in. That means you can check the output against any Black-Scholes reference, and it also means the volatility used is the one you chose, not one observed in the market.
How accurate is the pricing?+
For the standard reference case of spot 100, strike 100, rate 5%, volatility 20%, and one year to expiry, it returns a call value of $10.4506 and a put value of $5.5735, matching published Black-Scholes values. Call and put outputs satisfy put-call parity to floating-point precision, and the Greeks match numerical differentiation of the price to at least six decimal places.
Does it handle American options or dividends?+
No. This is the European, non-dividend-paying Black-Scholes model. American options carry early-exercise value and dividend-paying underlyings need an adjusted model, so the value shown here will differ from the traded price in both cases.
What units are the Greeks in?+
Delta is the change in option value per $1 change in spot, and gamma the change in delta per $1 change in spot. Vega is per one percentage point of volatility, rho is per one percentage point of interest rate, and theta is per calendar day. All values are per share; a standard US equity contract covers 100 shares.
What is the difference between the two P/L lines?+
The "at expiry" line is the payoff once all time value has gone: intrinsic value minus the premium. The "today" line is the Black-Scholes value at that spot price with the days to expiry, volatility, and rate you entered left unchanged, minus the premium. The two converge as days to expiry approaches zero.
Related tools
Position Size Calculator
Work out a share count from account size, risk per trade, entry, and stop.
Risk/Reward Ratio Calculator
Reward-to-risk ratio and the breakeven win rate it implies.
Multi-Leg Options Payoff Calculator
Combine this leg with others — spreads, straddles, iron condors — and see max profit/loss.
Backtesting Engine
Test a strategy against historical market data.
All Trading Tools
Browse every free tool on QuantEngines.
This calculator performs and displays arithmetic on the inputs you provide. It is not investment advice, does not recommend any option, strike, or position, and makes no claim about what any option is worth in the market or how any trade will perform.