Black-Scholes Options Calculator

European call + put prices and the 5 Greeks (delta, gamma, vega, theta, rho) under the Black-Scholes-Merton model.

Inputs

Decimal year. 30 days โ‰ˆ 0.0822; 1 year = 1.0.

Implied or historical. S&P 500 ~15-20% historically; single stocks 25-40%; speculative ones higher.

Use 0 for non-dividend stocks; ~1.5-2% for S&P 500.

Result

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How to use this calculator

  • Enter S = current underlying price, K = strike price.
  • T = decimal years to expiry. 30 days = 30/365 = 0.0822.
  • r = annualized risk-free rate (use the 3-mo T-bill yield, ~5% in 2026).
  • ฯƒ = annualized volatility. Use historical (sample std dev ร— โˆš252) or implied from option chain.
  • q = continuous dividend yield. Use 0 for non-dividend stocks; ~1.5-2% for S&P 500.

About this calculator

The Black-Scholes-Merton model (Fischer Black, Myron Scholes, Robert Merton โ€” Nobel 1997 to Scholes & Merton; Black had died in 1995 and the prize is not awarded posthumously) prices a European-style option as the discounted risk-neutral expected payoff under a geometric-Brownian-motion stock-price process. The 5 standard Greeks measure first-order sensitivity to the inputs: ฮ” to underlying price, ฮ“ to delta itself, ฮฝ to volatility, ฮ˜ to time, ฯ to rate. The model assumes constant volatility (the most-violated assumption โ€” implied-vol smiles are exactly the market's correction), no early exercise, no transaction costs, and log-normally distributed prices. For US single-stock options (most are American-style), the European-style Black-Scholes price is a lower bound; the early-exercise premium is generally small for non-dividend-paying calls but can be material for puts and for calls on dividend-paying stocks just before ex-date.

Frequently asked

Is Black-Scholes accurate for American options?+
For European options, exactly. For American options, it's a lower bound โ€” early exercise can be optimal for puts (always) and for calls on dividend-paying stocks (just before ex-date). The premium is usually < 1% for non-dividend calls but can be 5-15% for ITM puts. Use binomial or finite-difference for American options when precision matters.
What volatility should I use?+
Historical: stdev of log-returns ร— โˆš252. Implied: back-solve from a known liquid option (preferred for relative-value trades). For risk-management or stress tests, scenario-test at ยฑ10 vol points around the current implied.
How do Greeks change as expiration approaches?+
Gamma spikes (especially for at-the-money options) โ€” small price moves cause big delta changes. Theta accelerates (more time decay per day). Vega shrinks (less time for vol to matter). This is why short-gamma positions get dangerous in the last week.
Is the erf approximation accurate enough?+
The Abramowitz & Stegun 7.1.26 series used here is accurate to ~1.5e-7 in absolute terms โ€” well within rounding error for any option price. For trading systems, use libm's erf() directly.
Source for the formulas?+
Black, F. and Scholes, M. (1973), "The Pricing of Options and Corporate Liabilities", Journal of Political Economy 81(3); Merton, R. (1973), "Theory of Rational Option Pricing", Bell Journal 4(1). Modern presentation per Hull, J., "Options, Futures, and Other Derivatives" (any edition, Ch. 15).

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