Option Delta Calculator
Delta is the first-order sensitivity of an option's price to a $1 change in the underlying asset's price. It is one of the primary "Greeks" used in options trading and risk management. A call option with a delta of 0.60 gains approximately $0.60 in value for each $1 rise in the stock. Delta also approximates the probability that the option expires in the money. Under Black-Scholes, call delta equals N(d1), where d1 is a function of the stock price, strike, interest rate, time to expiration, and volatility. Put delta equals N(d1) minus 1, which is always negative, reflecting that puts gain value when the stock falls. Traders use delta to construct hedged positions, to estimate directional exposure, and to size trades.
Delta formula
d1 = [ln(S/K) + (r + sigma^2/2) * T] / (sigma * sqrt(T))
Call Delta = N(d1)
Put Delta = N(d1) - 1
Where N() is the cumulative standard normal distribution, S is the current stock price, K is the strike price, r is the risk-free rate (decimal), T is years to expiration, and sigma is annualized volatility (decimal).
Interpreting delta
- At-the-money calls (S = K) have delta near 0.50; at-the-money puts have delta near -0.50.
- Deep in-the-money calls approach delta = 1.00; deep in-the-money puts approach delta = -1.00.
- Delta approximates the hedge ratio: to hedge 100 short call contracts (each representing 100 shares) with delta 0.60, buy 6,000 shares.
- As expiration approaches, delta moves sharply toward 0 or 1 for near-the-money options.
- Higher volatility causes at-the-money options to have delta closer to 0.50 and reduces delta for deep in-the-money options.
Frequently asked questions
What is option delta?
Delta measures how much an option's price is expected to change for a $1 change in the price of the underlying asset. A call delta of 0.60 means the option price rises approximately $0.60 for every $1 rise in the stock.
What is the range of delta values?
Call deltas range from 0 to +1. Put deltas range from -1 to 0. Deep in-the-money options approach delta of +1 (calls) or -1 (puts). At-the-money options have delta near +0.50 (calls) or -0.50 (puts).
How is delta calculated using Black-Scholes?
For a European call: delta = N(d1), where d1 = [ln(S/K) + (r + sigma^2/2)*T] / (sigma*sqrt(T)) and N() is the cumulative standard normal CDF. For a European put: delta = N(d1) - 1.
What does it mean to be delta neutral?
A delta-neutral position has a net delta of zero, meaning small moves in the underlying asset do not change the portfolio value. This is achieved by combining options and shares so that positive and negative deltas cancel out.
Does delta change over time?
Yes. Delta is not constant; it changes as the stock price moves, time passes, and volatility changes. The rate of change of delta with respect to the stock price is called gamma. Near expiration, deltas of near-the-money options move sharply.
Official sources
- Black, F. and Scholes, M. (1973). "The Pricing of Options and Corporate Liabilities." Journal of Political Economy 81(3): jstor.org/stable/1831029.
- CBOE Options Institute: cboe.com/education.
- Options Clearing Corporation: theocc.com.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.