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# Option Delta: Explanation & Calculation

Updated: Mar. 31, 2023Written By: Gordon B ScottReviewed By:

Delta is a measure related to options that traders can use to predict option price movements based on the change in the underlying asset. It can also be used to assess the probability that a given option will expire in the money. It's useful to understand delta, how it is calculated, and the ways option traders use delta in their trading.

## What Delta Measures

Delta is one of the variables used to describe the different dimensions of risk involved in taking options positions. Delta measures the sensitivity of an option's price to movement in the underlying stock. Specifically, delta designates the amount an option’s price is expected to move based on a \$1 change in the underlying security.

The variables used to predict changes in option values are known as the “Greeks,” and each assists traders assess the opportunity and risk associated with a given option position. Delta is one of the key variables because it helps investors determine how option prices are likely to change as the underlying stock price varies. The calculation of delta is done in real-time by computer algorithms that continuously publish delta values to broker clientele. The delta value of an option is often used by traders and investors in assessing their options strategy.

Key Takeaway: Delta measures the sensitivity of an option’s price movement in an underlying stock. It is a useful tool for investors to assess their options strategy or existing options positions.

## Delta Values

Delta values can be positive or negative depending on the type of option.

### Deltas for Call Options

Deltas for owning call options always range from 0 to +1, because there is a positive relationship between changes in the underlying stock price and the value of the call option. A jump in the price of the underlying security should result in an increase in the value of the call option (assuming implied volatility and time-to-expiry remain essentially flat).

As a simple example, if a call option has a Delta of 0.25 and the underlying stock increases by \$1, the value of the call option should increase by about \$0.25. (note that we're speaking of dollars and cents here. In performance terms the call option movement would deliver a higher % change than the % move of the underlying).

### Deltas for Put Options

Conversely, Deltas for owning put options always range from -1 to 0 because when the underlying security increases in price, the value of put options decreases (again, assuming essentially unchanged implied volatility and time-to-expiry)

### In-The-Money vs. Out-Of-The-Money Option Delta

Option delta behavior depends on the relative position of the strike price in comparison to the current price of the underlying asset. An in-the-money option, one that could be currently exercised for value, will have a higher Delta score than an out-of-the-money contract. An out-of-the-money option has no intrinsic value, meaning that it would make no financial sense to exercise an out-of-the-money option.

Note: Option contracts have both intrinsic and extrinsic value. For in-the-money options, the intrinsic value of both call and put options is the difference between the underlying stock’s price and the strike price, while extrinsic value (or time value) will account for the remaining portion of the option price.

For out-of-the-money options, there is no intrinsic value regardless of how near or far the underlying security is trading from the strike price. In such case, 100% of the option price is related to time value (extrinsic value), which is the value assigned to the possibility that the option moves into the money.

### At-The-Money Call & Put Options

The delta of an “at-the-money” call option (where the strike price is near to the currently traded price of the stock) is typically around 0.5. The deeper a call option is in-the-money, the closer the delta will be to +1, and the more the option price will move uniformly in price to the underlying security.

The same is true for put options, but with negative numbers for delta scores. The delta of an in-the-money put option gets closer to -1 the deeper it becomes in-the-money. The delta of out-of-the-money put options approaches 0 as it becomes deeper out-of-the-money.

## How Delta is Calculated

Professional option sellers determine how to price their options based on sophisticated models that often resemble the Black-Scholes model: a mathematical equation that estimates the theoretical value of options by taking into account the impact of time and other risk factors. The formula for delta can be derived by dividing the change in the value of the option by the change in the value of its underlying stock.

Mathematically, it is represented as:

Delta = (Of - Oi) / (Sf - Si) where

• Of: the new value of the option
• Oi: the initial value of the option
• Sf: the new value of the underlying stock
• Si: the initial value of the underlying stock

For example, suppose stock XYZ was trading at \$520 per share and a call option with a strike price of \$500 was trading for \$45. This call option is in-the-money because the stock price is above the strike price. If the price of XYZ stock rises to \$523, and the value of the call option rises to \$46.80, the delta of this option was:

Delta = (\$46.80 - \$45.00) / (\$523 - \$520) = +0.6

Important: Delta is used to measure the theoretical change in the value of option price given a change in the price of the underlying security. In reality, an option price, as reflected by its more recent trade price, may not change exactly as predicted by the delta, and could possibly not move at all if there is a wide bid-ask spread or limited trading volume.

## Option Delta Values & Uses

In addition to predicting option price movement, delta values can also be used as a probability measure. Delta measures the expected probability that an option will end in-the-money at expiration. Remember, the deeper a call option is in-the-money, the closer the delta value will be to +1. Delta values are important to consider when assessing the risk and price volatility of option contracts.

Let's consider 2 different call options on XYZ stock which is currently trading at \$500/share.

1. The first call option has a strike price of \$550, and is set to expire in 2 months.
2. The second call option has a strike price of \$600, and is set to expire in 2 months.

Both call options are trading out-of-the-money here. Since the first call option, with the \$550 strike price, is closer to the market price of XYZ stock, it will carry a higher price & higher Delta than the second call option. The \$550 strike call option might have a delta value of 0.25, while the \$600 strike call option might have a delta value of 0.15. This corresponds to the fact that the \$550 strike call option has a greater probability of expiring in-the-money.

Key Takeaway: Delta can be used to assess the probability that an option will be in-the-money at the time of its expiration date.

## Shifts in Delta Scores

Since options carry delta scores that approach +1 as they move further into-the-money, and approach zero as they move further out-of-the-money, it's evident that the delta for any specific option is never fixed. In fact, delta values constantly change on the basis of:

1. Where the underlying security price current stands relative to the option strike price
2. The remaining time to expiration
3. The current implied volatility

The option delta for an initial \$1 move in the underlying stock will likely be different than an additional \$1 move in the same direction, although perhaps only marginally so, depending on various other factors, like the absolute level of the share price. For instance, if an in-the-money call option rises in value by \$1.80 on the basis of the underlying security price increasing from \$500 to \$503, its delta score was +0.6.

\$1.80 / \$3.00 = +0.6

However, if this underlying security price increases by \$200 (from \$500 to \$700), the option price should rise by more than the \$120, as that same delta score would imply

\$200 x 0.6 = \$120

The reason for this is due to the fact that a delta score increases as an option becomes further in-the-money. So as the underlying security rises higher and higher, the delta itself increases. In the above scenario of a \$200 increase in the underlying security, the effective delta for this move might be +0.80. Furthermore, after the underlying security price reached \$700, the delta of the call option at this point might be very close to +1.0, since it is heavily in the money at that stage.

## Delta for Short vs. Long Options

Options, just like stocks, can be bought or sold. Depending on which side of an option trade an investor is on, the delta of that option will adjust accordingly.

For long options, delta values are positive for calls and negative for puts. A bought (long) call will have a delta between 0 and +1, rising as the option becomes more in-the-money. A purchased put option will have a delta between 0 and -1, with delta falling the further the put is positioned in-the-money.

The inverse is true for shorting options. When selling call options, delta scores will be a negative value, between 0 and -1. This is true because a short call option position will increase in value as the underlying security falls - the writer of a call option will benefit as the underlying security falls. The other way to look at this is to understand that a call option has a positive delta, but that the seller/writer of that call option has the inverse exposure.

Similarly, put options, which provide a delta exposure of -1 to 0 for the owner, expose the seller/writer of the put option to a positive delta between 0 and +1.

## Position Delta

A fundamental understanding of delta for individual options can help investors grasp the concept of position delta. Position delta estimates the profit or losses on an entire option position relative to \$1 changes in the stock price, and is helpful when deploying trading strategies that involve multiple options at once.

Position delta can be calculated using the following formula:

Position Delta = Option Delta x Number of Contracts Traded x 100

For example, suppose a trader sold two \$120 call options of stock XYZ, that is trading at \$120 per share. It is possible to calculate this trader’s position delta this way:

-0.5 (estimated option delta) x 2 (number of contracts) x 100 = -100

For this trade, the investor in the example can use delta as a tool to help minimize risk. Remember, delta is a ratio that compares the change in the value of an option to the change in the price of an underlying security. For this reason, delta is sometimes referred to as a hedge ratio. The hedge ratio compares the value of a position protected through the use of a hedge with the size of the entire position itself.

Owning a stock has a delta of +1 by default, and shorting a stock has a delta of -1 by default. For the trader to hedge their delta risk in the example above, where their exposure is -100 they would need to add positive delta to their portfolio. They could do this by buying 100 shares of the underlying stock, which would bring their delta position to zero. Conversely, they could also establish a delta neutral position by selling put options to offset the negative delta, or buy call options to achieve the same result.

Key Takeaway: Traders and investors can hold several different security types that relate to the same underlying asset, and calculate a delta for their combined exposure to this asset. This delta value could then be used in devising possible strategies to mitigate the risk of their position, if that is in the investor's interest.

## Bottom Line

Delta measures the change in the value of an option based on a change in the price of an underlying security. A useful tool for hedging positional risk, delta can also be thought of as the probability that an option will expire in the money.