# Delta

Delta is the estimated change in the price of an option for a rise of $1 in the price of the underlying stock. The delta of a call lies between 0 and 1 and the delta of a put between -1 and 0.

Intermediate · Concept · published 29 September 2026 · last reviewed 3 October 2026
Canonical page: https://optionsbeacon.pro/learn/delta
Learn Options > Pricing and volatility

## Key points

- A delta of 0.50 means that the premium is expected to rise by about $0.50 when the stock rises by $1, and to fall by about $0.50 when the stock falls by $1.
- Options at the money have a delta near 0.50 for calls and near -0.50 for puts. Options deep in the money approach 1 or -1, and options far out of the money approach 0.
- Delta times 100 is the number of shares one contract behaves like at this moment.
- Delta changes when the stock moves, as time passes and when implied volatility changes. It is an estimate of a pricing model, not a fact of the market.

[Delta](https://optionsbeacon.pro/learn/glossary.md#delta) is the first of the [Greeks](https://optionsbeacon.pro/learn/glossary.md#greeks), the measures that say how
the price of an option reacts when one thing changes. Delta answers the most direct question: what happens to
the premium when the stock moves.

## What the delta of an option measures

Delta is a theoretical estimate of the change of the premium, in dollars per share, for a move of $1 in the
stock, with everything else unchanged.

A call with a delta of 0.53 and a premium of $1.80 is expected to cost about $2.33 after the stock has risen by
$1, and about $1.27 after it has fallen by $1. One contract covers 100 shares, so the contract gains or loses
about $53.

The delta of a call is positive, because a call gains when the stock rises. The delta of a put is negative,
because a put loses when the stock rises. The sign belongs to the option that is held. A written option has the
opposite exposure: a written put with a delta of -0.30 gains about $0.30 per share when the stock rises by $1.

## How delta changes with the strike price

The table shows 30-day options on a stock at $50, from the Black-Scholes model with a volatility of 30%, an
interest rate of 4% and no dividend.

| Strike price | Call premium | Call delta | Put premium | Put delta |
|---:|---:|---:|---:|---:|
| 45 | $5.35 | 0.90 | $0.20 | -0.10 |
| 47.50 | $3.32 | 0.75 | $0.67 | -0.25 |
| 50 | $1.80 | 0.53 | $1.63 | -0.47 |
| 52.50 | $0.83 | 0.31 | $3.16 | -0.69 |
| 55 | $0.33 | 0.15 | $5.15 | -0.85 |

An option deep in the money moves almost dollar for dollar with the stock, so its delta is close to 1 or -1. An
option far out of the money hardly reacts, so its delta is close to 0. At the money the delta is near one half.
In this model the delta of a call and the size of the delta of the put with the same strike price add up to 1.

## What delta says about a position in shares

Delta times 100 is the number of shares that one contract behaves like for a small move of the stock. A call
with a delta of 0.53 moves like 53 shares. A written put with a delta of -0.30 moves like 30 shares held,
because its writer gains when the stock rises.

The deltas of several positions on the same stock can be added. 100 shares have a delta of 100. 100 shares and
one written call with a delta of 0.31 behave like 100 - 31 = 69 shares.

## Why the delta of an option does not stay the same

Delta holds for a small move and for this moment. It changes for three reasons.

- **The stock moves.** As a call goes into the money its delta rises toward 1. The rate of that change is
  [gamma](https://optionsbeacon.pro/learn/glossary.md#gamma). The 30-day 50 call of the table has a delta of 0.35 with the stock at $48 and
  0.70 with the stock at $52.
- **Time passes.** As expiration comes closer, the deltas of options in the money move toward 1 or -1 and the
  deltas of options out of the money move toward 0.
- **Implied volatility changes.** A higher [implied volatility](https://optionsbeacon.pro/learn/glossary.md#implied-volatility) brings the
  deltas of more strikes closer to one half. With a low implied volatility, options in the money have higher
  deltas and options out of the money lower ones.

## Example of delta over a move of two dollars

> **Example**
>
> XYZ trades at $50 and a 50 call with 30 days left costs $1.80 with a delta of 0.53, in the model of the table.
> Delta alone predicts a premium of $1.80 + 2 × $0.53 = $2.86 with XYZ at $52. The model gives $3.04, because the
> delta rose on the way up. With XYZ at $48, delta alone predicts $1.80 - 2 × $0.53 = $0.74, and the model gives
> $0.92, because the delta fell on the way down. For the holder the error is in the holder's favour in both
> directions. For the writer of the call it is the reverse.
>
> *Hypothetical example for illustration only. One contract covers 100 shares; commissions, fees, dividends and taxes are ignored unless stated.*

## What can go wrong when delta is relied on

Delta is computed by a pricing model from inputs that include a volatility nobody can observe, and its value
changes with the model and with the inputs. It changes all through the trading day. Close to expiration it
becomes unstable for options at the money, where gamma is highest: a small move of the stock takes the delta
from near 0 to near 1.

Delta says nothing about a large move. A position built to be neutral for a move of $1 is not neutral for a
move of $10, and a stock can open far from where it closed.

Delta is also not a forecast. It describes how the option is priced today, under the assumptions of the model,
and not what the stock will do.

## Questions and answers

### Is delta the probability that an option expires in the money?

It is an approximation that many traders use, and no more. In the Black-Scholes model the chance of ending in the money is a separate figure, computed under the assumptions of the model. For the 30-day 50 call on this page the delta is 0.53 and that figure is 0.50, and for the 55 call they are 0.15 and 0.13. Neither is a measured frequency of what stocks do.

## Terms used on this page

- **Delta**: The estimated change in the price of an option for a rise of $1 in the underlying. The delta of a call lies between 0 and 1 and the delta of a put between -1 and 0.
- **Gamma**: The estimated change in the delta of an option for a rise of $1 in the underlying. It is highest for options at the money that are close to expiration.
- **Greeks**: The measures of how the theoretical price of an option changes when one of its inputs changes. They are delta, gamma, theta, vega and rho, and each is an estimate of a pricing model.
- **Implied volatility**: The volatility that, put into a pricing model, gives the market price of an option. It is quoted as a yearly percentage and says how much movement the price of the option pays for.
- **Black-Scholes model**: The option pricing model published by Fischer Black and Myron Scholes in 1973. It gives a theoretical price from the stock price, the strike price, the time left, the interest rate and the volatility.

## Sources

1. [Delta](https://www.optionseducation.org/advancedconcepts/delta), The Options Industry Council (read 29 September 2026).
2. [Gamma](https://www.optionseducation.org/advancedconcepts/gamma), The Options Industry Council (read 29 September 2026).
3. [Understanding Options Greeks](https://www.optionseducation.org/advancedconcepts/understanding-options-greeks), The Options Industry Council (read 29 September 2026).
4. [Technical Information](https://www.optionseducation.org/referencelibrary/faq/technical-information), The Options Industry Council (read 29 September 2026). Frequently asked questions.
5. [The Pricing of Options and Corporate Liabilities](https://www.journals.uchicago.edu/doi/10.1086/260062), Fischer Black and Myron Scholes, Journal of Political Economy 81 (3), 1973 (read 29 September 2026).
6. [Additional background material on the Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel 1997](https://www.nobelprize.org/prizes/economic-sciences/1997/advanced-information/), The Royal Swedish Academy of Sciences (read 29 September 2026).

## Related topics

- [Theta and time decay](https://optionsbeacon.pro/learn/theta-time-decay.md)
- [Implied volatility](https://optionsbeacon.pro/learn/implied-volatility.md)
- [How to read an option chain](https://optionsbeacon.pro/learn/how-to-read-an-option-chain.md)
- [Covered call](https://optionsbeacon.pro/learn/covered-call.md)

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