When a trader buys or sells an asset directly, their profit changes in a linear 1:1 manner with the underlying price of that asset. For example if you purchase 1 bitcoin, and the price of bitcoin subsequently increases by $1,000, you have made a profit of $1,000. This means that for every $1 increase in the underlying price, you have made a $1 profit.

If you instead purchase 10 bitcoin, and the price of bitcoin subsequently decreases by $2,000, you have made a loss of $20,000 (10 * -2000). This means that for every $1 decrease in the underlying price, you have made a $10 loss, or a $1 loss for every bitcoin that you purchased.

When a trader trades derivatives contracts instead, this relationship may not always be 1:1, depending on the contract details. Remember, while a derivatives contract derives its value from the price of the underlying asset, it is not the same as holding the asset itself. The details of the contract may cause it to vary based on more than just the underlying price, and this is certainly the case with option contracts. This means that a $1 increase in the underlying price, may not lead to a $1 increase in the value of the contract.

Delta definition

Delta is a measure of the option price’s sensitivity to movements in the underlying asset price. Specifically, the delta tells you how much the option price is expected to change if the underlying asset price increases by $1.

Delta can be positive or negative. Before learning options, most traders will only be familiar with delta one products. That is, they only trade instruments that have a delta of 1 (or -1 if they are short).

Delta one products are any instrument that has a 1:1 relationship (or very close to 1:1) with the underlying price of the asset. For example the bitcoin or ethereum futures contracts on Deribit would be described as delta one products. If the underlying price of bitcoin increases by $1, so too will the price of all the bitcoin futures contracts. There may be some slight variations, and some backwardation or contango in the market, but in general that 1:1 relationship will be very close to accurate for those products.

Not all financial derivatives have a delta of one though. The delta of an option will be between -1 and 1. All other things being equal, for every $1 increase in the underlying price, an option’s value will change according to it’s delta.

For example:

A delta of 0.30 means the price of the option is expected to increase by $0.30 if the underlying price increases by $1.

A delta of -0.25 means the price of the option is expected to decrease by $0.25 if the underlying price increases by $1.

Call options have a positive delta, because all other things being equal, an increase in the underlying price will make all call options more valuable. Call options have deltas between 0 and 1, with:

-Deep in the money calls having delta approaching 1
-At the money calls having delta of roughly 0.5
-Far out of the money calls having delta approaching 0

Put options have a negative delta because all other things being equal, an increase in the underlying price will make all put options less valuable. Put options have deltas between 0 and -1, with:

-Deep in the money puts having delta approaching -1
-At the money puts having delta of roughly -0.5
-Far out of the money puts having delta approaching 0

Delta is always quoted as the rate of change in option value relative to an increase of $1 in the underlying asset price. However, as you may have inferred, if instead of increasing, the underlying asset price decreases by $1, then options will change value according to the negative of their delta.

For example, if a call option has a delta of 0.30, this means the price of the call will decrease by $0.30 if the underlying asset price decreases by $1.

Or if a put option has a delta of -0.25, this means the price of the put will increase by $0.25 if the underlying asset price decreases by $1.

All other things being equal, as the price of the underlying asset decreases, you would expect call option prices to fall and put option prices to rise.

It’s worth mentioning that deltas are sometimes referred to without the decimal. For example a delta of 0.50 is sometimes referred to as a 50 delta, and a delta of 0.25 is sometimes referred to as a 25 delta. So if you see someone write that they are looking at the 25d call, that means they are looking at the call option with a delta of 25 (or 0.25 when written as a decimal). I usually prefer to work with decimals as it makes calculations easier, so I will continue to use them in these examples.

Delta for option sellers

Call options have positive delta. If the underlying price increases, all other things being equal, the value of call options will also increase. Put options have negative delta. If the underlying price increases, all other things being equal, the value of put options will decrease. So being long a put option has a negative delta.

This is not the only way to get a negative delta with options though, because the seller of a call option also has a negative delta. If someone buys a call option with a delta of 0.75 for example, the seller of that option will have a delta of -0.75. If the underlying price increases by $1, the buyer of the call option can expect it to have increased in value by $0.75, meaning a $0.75 profit. For the seller this increase in value represents a loss of the same amount.

More generally:

The buyer of a call option has a positive delta.

The seller of a call option has a negative delta.

The buyer of a put option has a negative delta.

The seller of a put option has a positive delta.

Summary

An option’s delta tells you how much the option price is expected to change if the underlying asset price increases by $1. Unlike futures contracts, an option’s delta will normally not be 1, though it can be for deep ITM options.

Call options have a positive delta between 0 and 1. Put options have negative delta between 0 and -1. This is because, all other things being equal, if the underlying price increases, call options will gain value, and put options will lose value. Similarly, if the underlying price decreases, call options will lose value and put options will gain value.

While an option’s delta is a measure of the sensitivity of the option price to underlying price movements, delta itself has sensitivity to various parameters. We will look at these in the rest of this section.