Vanilla / binary options

Black-Scholes prices and Greeks for European vanilla and cash-or-nothing binary options. Enter market inputs, then calculate call and put values side by side.

Spot, strike, volatility, the risk-free rate, and days to expiry are enough for a closed-form price. Choose vanilla for the usual call or put payoff, or binary for a cash-or-nothing that pays one unit if the option finishes in the money. Figures are theoretical and for information only.

Type a symbol, then Load or leave the field. Spot, ~30-day historical realized vol, and a US risk-free proxy are filled when the ticker is found. Strikes and target are recentered near the new spot.

Delayed/free data. Historical realized vol is not implied vol from an options chain. Not investment advice.

Call and put results

Metric Call Put
Option Price — —
Delta — —
Gamma — —
Vega — —
Theta — —
Rho — —

How the Black-Scholes price is calculated

Black-Scholes assumes the underlier follows a geometric Brownian motion with constant volatility and a constant risk-free rate, and that the option can be exercised only at expiry. There is no dividend yield in this implementation.

Time is converted to a year-fraction T by dividing days until expiration by 365. Volatility and the risk-free rate are entered as percentages and converted to decimals. N is the standard normal cumulative distribution:

C = S N(d1) − K e−rT N(d2)

P = K e−rT N(−d2) − S N(−d1)

d1 = [ln(S / K) + (r + σ2 / 2) T] / (σ √T)

d2 = d1 − σ √T

  • S — spot price
  • K — strike price
  • σ — volatility (decimal)
  • r — risk-free rate (decimal)
  • T — time to expiry in years (days ÷ 365)

A cash-or-nothing binary that pays one unit if it finishes in the money uses the same d2 but a simpler payoff: the call is e−rT N(d2) and the put is e−rT N(−d2). Option prices are rounded to three decimals. Vega and rho are reported per one percentage point; theta is per calendar day.

Worked examples

These three cases use the same formula and rounding as the calculator above. Type the inputs in and you should get the same call and put prices.

ATM vanilla, 1 year

Spot 100, strike 100, 30% volatility, 1% rate, 365 days. This is the default form.

Call price
12.368
Put price
11.373
Call delta
0.57273

ATM vanilla, 90 days

Same spot, strike, vol, and rate, but only 90 days to expiry. Less time means a lower premium.

Call price
6.054
Put price
5.808
Call delta
0.53627

ATM binary, 1 year

Same market as the first example, with type set to binary (cash or nothing). The payoff is one unit of cash, not the vanilla difference.

Call price
0.449
Put price
0.541
Call delta
0.01308

Black-Scholes is the closed-form equity-style case. FX options use two interest rates, implied volatility inverts the same vanilla formula, and the converter rescales a vol number before you type it in here. Path-dependent Asians and lookbacks, and a long call or put payoff chart, sit next to that vanilla engine.

Option strategies guide

Plain-language straddle, strangle, spread, butterfly, and iron condor — with a calculator for each payoff.

Read the guide

FX options

Garman-Kohlhagen prices and Greeks with foreign (base) and domestic (quote) rates.

Open calculator

Implied volatility

Back out Black-Scholes implied vol from an observed vanilla price.

Open calculator

Volatility converter

Translate daily, weekly, monthly, or annual volatility figures.

Open calculator

Gamma Optimizer

Best long call or put on a strike grid — maximize return on premium at a target expiry price.

Open calculator

Value at Risk

Estimate theoretical maximum loss at a chosen confidence level.

Open calculator

Asian options

Monte-Carlo pricing for fixed- and floating-strike Asian options.

Open calculator

Lookback options

Monte-Carlo lookback pricing with in-the-money statistics.

Open calculator

Long call / put

Charted expiry payoff and live P&L sliders for a long option.

Open calculator

Option pricing FAQ

It uses the Black-Scholes closed-form for a European call or put. Spot, strike, volatility, the risk-free rate, and time to expiry (days divided by 365) go into d1 and d2. The call is spot times N(d1) minus the discounted strike times N(d2). The put uses the complementary terms. Prices are rounded to three decimals, matching the results panel.

A vanilla call pays the difference between spot and strike if that difference is positive; a vanilla put pays the opposite difference. A cash-or-nothing binary here pays one unit of cash if the option finishes in the money, and nothing otherwise. The binary price is therefore the discounted risk-neutral probability of finishing in the money, not a share of the underlier.

European only. Exercise is assumed at expiry, which is what the Black-Scholes formula prices. Early exercise (American style) is not modelled, and there is no dividend yield on the underlier.

Put-call parity says call minus put equals spot minus the discounted strike. With a positive risk-free rate the discounted strike is below spot, so the call is slightly more valuable than the put at the same strike. The default one-year example (call 12.368, put 11.373) shows that gap.

Delta is the change in option value for a one-unit move in spot. Gamma is the change in delta for that same spot move. Vega and rho are scaled per one percentage point of volatility or rate. Theta is the change in value per calendar day. Call and put gamma and vega are the same for a vanilla option.

Enter calendar days. The model converts that to year-fractions by dividing by 365, so 365 days is exactly one year. There is no business-day calendar or day-count switch.

Live quotes include bid-ask spreads, dividends, borrow, early-exercise premia, and a volatility smile that Black-Scholes (a single flat vol) does not capture. Rounding and the 365-day year can also differ. Figures here are theoretical and for information only.

Disclaimer: the contents of this website are for informational purposes only and do not constitute any investment recommendation. The visitor acts at his own risk.