# Delta of Calls vs. Puts and Probability of Expiring In the Money

## What delta means

Delta, the best known of the option Greeks, is a measure of directional exposure of an option. It is the first derivative of option’s market price with respect to the underlying’s price. Simply said, an option’s delta represents the dollar value by which the market price of the option changes when the underlying asset’s price rises by 1 dollar.

## Values of delta: calls vs. puts

Delta of a call option can reach values from 0 to +1. It is never negative, as call options increase when underlying asset’s price rises (see why). It is never greater than 1, as the rate of the option’s price movement is never greater than the rate of the underlying’s movement (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’s market price moves is never greater than the underlying’s price change, therefore a put option’s 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.

## Relationship between call and put delta

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.

## Option’s 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).

We have said above that the sum of absolute values of delta of a call and a put with the same strike is one. This is in line with the probability idea. When you have a call and a put on the same underlying and with the same strike price, you can be sure that one of them will expire in the money and the other will expire out of the money (unless, of course, the underlying stock ends up exactly equal to the strike price and both options expire exactly at the money). Therefore, the sum of the probabilities should be 100% (and the sum of the absolute values of deltas should be one).

## Delta is only an indication, not a guarantee of probabilities

Using delta as a probability proxy is only an estimate and in practice it is not precise. It assumes random market movement and rational (unbiased) valuation of options – conditions rarely met in practice. An option’s delta results from the market (that means people) valuing options as related to the underlying asset. We all know that market expectations are often wrong.

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