Black-Scholes Option Calculator

What is Black-Scholes and why check a real instrument against it?

Black-Scholes is the standard model for theoretically pricing a European option from five inputs: spot, strike, time to expiry, volatility, and the risk-free rate, plus an optional continuous dividend yield for stocks and ETFs. Deribit's, Binance's and Bybit's options are European-style (no early exercise), so the model applies cleanly. This calculator does two things: prices any hypothetical option from manual inputs, or pulls a real Deribit (BTC, ETH), Binance (BTC, ETH, gold, silver) or Bybit (BTC, ETH) instrument's own spot/strike/expiry/IV and checks whether its quoted mark price is actually consistent with Black-Scholes at that volatility: a live sanity check, not an estimate.

The formulas

• 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), Delta(put) = −e^(−qT)·N(−d1); Gamma = e^(−qT)·φ(d1) / (S·σ·√T); Vega = S·e^(−qT)·φ(d1)·√T
• Theta(call) = −S·e^(−qT)·φ(d1)·σ/(2√T) + q·S·e^(−qT)·N(d1) − r·K·e^(−rT)·N(d2), Theta(put) = −S·e^(−qT)·φ(d1)·σ/(2√T) − q·S·e^(−qT)·N(−d1) + r·K·e^(−rT)·N(−d2), per year (the tool reports per day); Rho(call) = K·T·e^(−rT)·N(d2), Rho(put) = −K·T·e^(−rT)·N(−d2)

q is the continuous dividend yield, entered as percent per year (1.2 means 1.2%). Leave it at 0 for crypto: the formulas then reduce to plain Black-Scholes. It applies to the manual inputs only; the live mode does not use it.

Worked example

Classic textbook case: S=$100, K=$100, r=5%, volatility=20%, 1 year to expiry:

• d1 = 0.35, d2 = 0.15
• Call = $10.4506, Put = $5.5735
• Delta(call) = 0.6368

Coin-denominated price convention

Deribit quotes BTC/ETH option premiums in the underlying coin, not USD. This tool converts the standard USD Black-Scholes price into that convention by dividing by spot. Cross-checked live against a real deep-in-the-money Deribit instrument, where this formula's coin price matched Deribit's own quoted mark price to within normal bid-ask noise. Greeks shown here stay standard USD-denominated Black-Scholes Greeks, the universal convention every options calculator uses, rather than a further coin-adjusted variant.

Binance: USDT-settled, no coin conversion

Binance European options (BTC, ETH, gold and silver) are quoted and settled in USDT per one unit of the underlying, so the textbook formulas apply as they are and the price is shown in USDT. The inputs are the contract's own: the Binance index as spot, its mark IV, and the risk-free rate Binance states for that contract (it can be negative, and the price uses it as stated; a rate typed here is not used). Checked against an independent Python implementation on live Binance contracts, and the gap to Binance's own mark price is shown. A contract whose mark is only a cent or two can show a large percentage gap; read the absolute gap there. When the market is not trading (gold and silver outside their session), no check is shown, because the mark price and mark IV are not a consistent pair then.

Bybit: USDT-settled, priced off the forward

Bybit options (BTC and ETH) are quoted and settled in USDT per one unit of the underlying. Bybit prices each expiry off its own forward and states no interest rate, so the inputs are the contract's forward as the base, its mark IV and a rate of 0; a rate typed here is not used. Delta and gamma are per unit of that forward and rho is not offered (there is no rate in the model). The gap to Bybit's own mark price is shown. A contract whose mark is only a cent or two can show a large percentage gap; read the absolute gap there.

Checked against QuantLib

On 5,000 random European options, the price and every Greek from this calculator match QuantLib, the open-source quantitative finance library, to within 1e-10. The method, the exact differences per output and a script to run the same check yourself are in the methodology docs.

Where to go next

Want the second order Greeks (vanna, vomma, charm) too? Use the option Greeks calculator.

Already decided on a position and just want the payoff at expiry, not today's theoretical price? Use the Options Payoff Calculator. Selling premium against coin you already hold? Try the Covered Call Calculator. Have a price but need the volatility instead of the price? Use the Implied Volatility Calculator.

Use via API or MCP

This calculation is available as a deterministic API call: useful for bots or AI agents that need to price options or check a live quote against the model, not guess.