Black-Scholes Calculator

Inputs

Loads the price, the ~30-day expiry and the at-the-money strike, with live r and q.

Stock price

Strike

Time to expiration

days

Calendar days (years = days ÷ 365), as in the calculator.

Volatility

Risk-free rate

Dividend yield

Continuous; 0 if none.

Results

Call: $3.02; Intrinsic $0.00 · time value $3.02
Put: $2.69; Intrinsic $0.00 · time value $2.69

Greeks per share for the call and the put
Greek (per share)	Call	Put
Delta (Δ)Price change per $1 move in the stock	0.5326	−0.4674
Gamma (Γ)Delta change per $1 move	0.0555	0.0555
Theta (Θ)Price change per calendar day	−0.0530	−0.0421
Vega (ν)Price change per vol point (1% IV)	0.1140	0.1140
Rho (ρ)Price change per 1% rate move	0.0413	−0.0406

d₁: 0.0817
d₂: 0.0100
Time (years): 0.0822
C − P vs parity: $0.33 = $0.33

How the Black-Scholes model works

The Black-Scholes model, published by Fischer Black and Myron Scholes in 1973 and extended by Robert Merton the same year, gives a closed-form price for European calls and puts. It takes five inputs (six with dividends): the stock price, the strike, the time to expiration, the risk-free rate and the volatility, plus the dividend yield. Volatility is the one you can't observe directly; in practice the market's implied volatility stands in for it.

CallC = S·e^−qT·N(d₁) − K·e^−rT·N(d₂)

PutP = K·e^−rT·N(−d₂) − S·e^−qT·N(−d₁)

d₁ = [ln(S/K) + (r − q + σ²/2)·T] / (σ·√T), d₂ = d₁ − σ·√T

ParityC − P = S·e^−qT − K·e^−rT

N(x) is the standard normal cumulative distribution, φ(x) its density, T the time to expiration in years and q the continuous dividend yield (Merton's extension; q = 0 is the original 1973 model).

Worked example

The standard textbook case (Hull): a six-month option on a stock at $42 with a $40 strike, 20% volatility, a 10% risk-free rate and no dividend.

d₁ = [ln(42/40) + (0.10 + 0.2²/2) × 0.5] / (0.2 × √0.5) = (0.0488 + 0.0600) / 0.1414 = 0.7693

d₂ = 0.7693 − 0.1414 = 0.6278; N(d₁) = 0.7791, N(d₂) = 0.7349

C = 42 × 0.7791 − 40·e^−0.05 × 0.7349 = 32.72 − 27.96 = $4.76

P = 40·e^−0.05 × 0.2651 − 42 × 0.2209 = 10.09 − 9.28 = $0.81

Inputs
Stock price (S)	$42.00
Strike (K)	$40.00
Time (T)	0.5 years (182.5 days)
Volatility (σ)	20%
Risk-free rate (r)	10%
Dividend yield (q)	0%

This calculator gives
Call	$4.76
Put	$0.81
d₁	0.7693
d₂	0.6278
Call delta	0.7791
Put delta	−0.2209

The Greeks

The results table shows each Greek per share for the call and the put, from these formulas. The option Greeks calculator scales them to a position.

Delta (Δ)

Calle^−qT·N(d₁)

Put−e^−qT·N(−d₁)

Price change per $1 move in the stock.

Gamma (Γ)

Calle^−qT·φ(d₁) / (S·σ·√T)

Put: same as the call.

Change in delta per $1 move.

Theta (Θ)

Call−S·e^−qT·φ(d₁)·σ / (2√T) − r·K·e^−rT·N(d₂) + q·S·e^−qT·N(d₁)

Put−S·e^−qT·φ(d₁)·σ / (2√T) + r·K·e^−rT·N(−d₂) − q·S·e^−qT·N(−d₁)

Per year in the formula; shown here per calendar day (÷ 365).

Vega (ν)

CallS·e^−qT·φ(d₁)·√T

Put: same as the call.

Per 1.00 of volatility in the formula; shown here per vol point (÷ 100).

Rho (ρ)

CallK·T·e^−rT·N(d₂)

Put−K·T·e^−rT·N(−d₂)

Per 1.00 of rate in the formula; shown here per 1% (÷ 100).

φ(d₁) = e^−d₁²/2 / √(2π). Positions scale these per-share figures by 100 shares per contract, with the sign flipped for a short option.

What each input means

Stock price (S): The current price of the underlying stock.
Strike (K): The price at which the option can be exercised.
Time to expiration (T): In days or years. Days are calendar days divided by 365, the same basis as the calculator.
Volatility (σ): The annualised standard deviation of the stock's log returns, usually the implied volatility.
Risk-free rate (r): The annual risk-free interest rate, continuously compounded, typically a Treasury bill yield.
Dividend yield (q): The stock's annual dividend as a share of its price, paid continuously in the model. Dividends lower call values and raise put values.

Assumptions & limitations

European exercise: the option can only be exercised at expiration. Most US stock options are American (see the FAQ).
Constant volatility and interest rate over the option's life. Real implied volatility differs by strike and expiry (the smile) and moves over time.
Dividends as a continuous yield q. A stock paying quarterly cash dividends is only approximated.
Log-normal stock prices with no jumps, so the model tends to underprice far out-of-the-money options relative to the market.
No transaction costs, taxes or bid-ask spread, and borrowing and lending at the risk-free rate.
Time in calendar days ÷ 365, measured to expiration.

Frequently asked questions

Why is my answer different from my textbook?

Almost always the time input. Textbooks give T in years (0.5 is six months); this calculator starts in days, so switch the time field to Years or enter 182.5 days. Typed into the days field, 0.5 prices a half-day option. The model counts 365 calendar days a year. If your course counts trading days, divide by 252 instead: 63 trading days is 0.25 years, the same T as 91.25 calendar days, so enter it in years to match. Rates and volatility are percentages (type 10, not 0.10), and r is continuously compounded.

Does this price American options?

No. Black-Scholes-Merton prices European options, which can only be exercised at expiration, and most US stock options are American. A call on a stock that pays no dividend is worth the same either way, because exercising it early never pays. An American put, or a call just before an ex-dividend date, can be worth more than the figure here.

What volatility should I enter?

The model needs the volatility of the stock over the option's life, which can't be observed directly. Traders use the implied volatility in market prices (the implied volatility calculator solves it from a price); a textbook problem states it. Historical volatility from past prices is a starting point, not the same thing.

Which risk-free rate should I use?

A Treasury bill yield for a maturity close to the option's expiry, as an annual percentage with continuous compounding. Fill from a stock uses the live Treasury curve at the option's tenor, as the options calculator does.

What do d₁ and d₂ mean?

N(d₂) is the model's risk-neutral probability that the call finishes in the money (N(−d₂) for the put), and e^(−qT)·N(d₁) is the call's delta. The calculator shows both so you can check your working step by step.

How can I check the result?

With put-call parity: C − P must equal S·e^(−qT) − K·e^(−rT). The calculator shows both sides under the Greeks; if they differ by more than a cent, an input is off.

References

Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637–654.
Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4(1), 141–183.
Hull, J. C. Options, Futures, and Other Derivatives. Pearson (11th ed., 2021).