Glossary

Options terms.

Plain-English definitions for the inputs, outputs, and concepts used by the calculator.

Vega: The sensitivity of an option's price to changes in implied volatility. On this site, vega is reported as the dollar change in option price for a one percentage point change in implied volatility. Vega is the same for calls and puts, is largest for at-the-money options, and grows with time to expiration.
Delta: The sensitivity of an option's price to a one-dollar change in the underlying. Call deltas range from 0 to 1 (or 0 to e^-qT with dividends); put deltas range from -1 to 0. Delta is also a rough approximation of the probability the option finishes in the money.
Gamma: The rate of change of delta. Gamma measures how much delta moves when the underlying moves by one dollar. Gamma is the same for calls and puts and is highest at the money near expiration.
Theta: The change in an option's price as time passes, with everything else held constant — the cost of holding a long option through time decay. On this site, theta is reported per calendar day. Long options usually have negative theta; short options have positive theta.
Rho: The sensitivity of an option's price to changes in the risk-free interest rate. Reported here per one percentage point change in the rate. Rho is generally the smallest of the Greeks for short-dated options and grows with time to expiration.
Implied volatility (IV): The volatility figure that, when plugged into the Black-Scholes formula, makes the model price match the observed market price. IV is a forward-looking estimate of how much the market expects the underlying to move, expressed as an annualized standard deviation. It is the input that vega is sensitive to.
Black-Scholes model: A closed-form pricing model for European options published in 1973 by Fischer Black and Myron Scholes, with extensions by Robert Merton. It assumes lognormal returns, constant volatility, continuous trading, and no frictions. Despite its simplifying assumptions, it is the foundation of nearly all modern option pricing.
Call option: The right, but not the obligation, to buy the underlying at the strike price on or before expiration. Calls gain value when the underlying rises and when implied volatility rises.
Put option: The right, but not the obligation, to sell the underlying at the strike price on or before expiration. Puts gain value when the underlying falls and when implied volatility rises.
Strike price: The price at which the option holder can buy (call) or sell (put) the underlying. Combined with the current stock price, it determines the option's moneyness.
Moneyness (ATM, ITM, OTM): How the strike compares to the current stock price. At the money (ATM) means strike equals stock price. In the money (ITM) means the option has intrinsic value: a call with strike below the stock price, or a put with strike above. Out of the money (OTM) means no intrinsic value. Vega and gamma are largest for ATM options.
Time to expiration: The remaining life of the option. On this site, days are converted to years using T = days / 365 (calendar-day convention). More time means more vega and more rho, less gamma per day, and a wider distribution of possible outcomes.
Risk-free rate: The interest rate on a default-free instrument over the option's life. In practice, a Treasury yield matched to expiration is used. The Black-Scholes formula treats it as continuously compounded.
Dividend yield: The continuous dividend yield q on the underlying. Dividends reduce the forward price, which lowers call values and raises put values. Real dividends are discrete; modeling them as a continuous yield is an approximation.
European vs American exercise: European options can only be exercised at expiration; American options can be exercised any time. Black-Scholes assumes European exercise. Most listed U.S. equity options are American, but the early-exercise premium is small for non-dividend-paying calls and is often ignored in educational pricing.
Cumulative normal distribution, N(x): The probability that a standard normal random variable is less than or equal to x. It appears throughout the Black-Scholes formula. The site computes it using a polynomial approximation accurate to about 1.5×10^-7.
Standard normal PDF, n(x): The bell-curve density function: n(x) = e^-x²/2 / √(2π). It appears in the formulas for vega, gamma, and theta.