This page explains option delta as proxy for probability of expiring in the money for call and put options.

## Delta interpretation and possible values

Delta is a measure of directional exposure, mathematically the first derivative of option premium with respect to underlying price. Simply said, an option's delta represents the dollar amount by which the option's value changes when underlying price rises by one dollar.

Delta of a call option can reach values from 0 to +1. It is never negative, as call option prices increase when underlying price rises. It is never greater than 1, because an option's value can't change by more than the underlying price change (see why for ITM and OTM options).

Delta of a put option ranges from -1 to 0, as put options tend to appreciate when underlying stock goes down. Again, the rate at which the option price moves is never greater than the underlying price change, therefore put delta is never lower than -1.

## Delta and moneyness

Just by looking at the delta, you can tell if the option is in the money, out of the money, or just about at the money.

- Far out of the money options have delta close to zero (far out of the money options have little value and they hardly move).
- Deep in the money call options have delta close to +1 (the option's market price moves almost as much as the underlying's price).
- Deep in the money put options have delta close to -1 (the option's market price moves almost as much as the underlying's price, but in the opposite direction).
- At the money options have delta about 0.50 (or -0.50 for puts).

Therefore, if the absolute value of an option's delta is lower than 0.50, the option is out of the money. If it is higher than 0.50, it is in the money. This simple rule doesn't work 100% of time, especially for deltas very close to 0.50 and options with longer time to expiration. In reality, an option can be exactly at the money and have a delta of 0.54. But when you see an option's delta of 0.80, you can be virtually sure that this option is in the money.

## Call and put delta relationship

If you have a call and a put option, both for the same underlying, with the same strike price, and the same time to expiration, the sum of absolute values of their deltas is 1.00. For example, you can have an out of the money call with a delta of 0.36 and an in the money put with a delta of -0.64.

## Delta as probability proxy

Sometimes delta is used as a proxy for the probability that an option will expire in the money. According to this technique, an out of the money call with a delta of 0.36 has a probability of expiring in the money of 36%. An in the money put with a delta of 0.64 has a 64% chance of expiring in the money (for puts you take the absolute value of delta).

This is in line with the above mentioned relationship between call and put delta (their absolute values summing up to 1 for the same strike). For any pair of call and put options on the same underlying and with the same strike price, one of them will always expire in the money and the other will expire out of the money. Therefore, the sum of the probabilities should be 100% (and the sum of the absolute values of deltas should be one).

Note: We do not assume the possibility of underlying price ending up exactly at the strike, whose probability is infinitely small when we work with continuous (as opposed to discrete) prices, which most option pricing theory does.

## Delta probability vs. reality

Using delta as probability proxy is only an estimate. It assumes random market movement and rational (unbiased) valuation of options – conditions rarely met in practice.

We should always keep in mind that even 99.99% probability does not mean certainty and 0.01% probability does not mean impossible.