# Option Delta

This is part of option Greeks tutorials. See also gamma, theta, vega, rho.

## What Is Delta

Delta is the most important of the option Greeks. It measures sensitivity of an option's price (premium) to changes in underlying price. For example, if an option has delta of 0.45, it means that when the underlying stock's price increases by \$1, the option's price increases by 45 cents. Delta can reach values from 0 to 1 for call options and from -1 to 0 for put options.

## Call Option Delta

Call options are generally more valuable when the underlying security is more valuable. A call option's value increases when the underlying price goes up. Therefore, it makes sense that call delta is always a non-negative number. At the same time, a call option's value can't grow faster than underlying price. As a result, call delta can never be greater than 1.

Call delta value range is from zero to positive one.

### Call Option Delta Example

Consider a \$55 strike call option on a stock. The stock is currently trading at \$57 (underlying price) and the option at \$2.60 (option premium). The option's delta is 0.75. The delta tells us how the option premium will approximately change if the underlying price increases by \$1. If the stock grows by \$1 to \$58, we can expect the call option premium to grow by approximately \$0.75 to 2.60 + 0.75 = \$3.35. Delta is the ratio of option price change and underlying price change.

### Delta Inaccuracy for Big Moves

Notice the word approximately. In fact, delta is only accurate for very small price changes. As we will discuss later, delta itself also changes with underlying price (this is measured by gamma). As a result, for bigger changes in underlying price, the actual option price change may be a little different than predicted by delta. In our example above (the option has 20 days to expiration and implied volatility of 25%), the new option premium with underlying price at \$58 would more likely be closer to \$3.39, four cents higher than predicted by the delta.

## Put Option Delta

Contrary to calls, put options become more valuable when underlying price goes down, and lose value when underlying goes up. Therefore, put delta is generally negative. It can't be smaller than -1, because the speed of option price changes can't be faster than the underlying price change (other factors being equal). Same logic as calls, just opposite direction.

Put delta value range is from zero to negative one.

### Put Option Delta Example

Consider a \$55 strike put option on the same stock as in our call example. With the stock trading at \$57, the put option's premium is \$0.52 and its delta is -0.25. If the stock's price grows by \$1 to \$58, the put option's premium goes down by approximately \$0.25 to 0.52 – 0.25 = \$0.27. The delta is negative, which indicates the put option's premium moving in opposite direction from the underlying price.

## Delta and Option Moneyness

Delta is closely related to option moneyness. You may already know that in the money options are generally more sensitive to underlying price changes than out of the money options. In general, the deeper in the money, the greater sensitivity to underlying price. Conversely, the further out of the money, the smaller sensitivity. At the money options are something in between.

### Delta and Call Option Moneyness

At the money calls have delta close to 0.50 (moderate sensitivity to underlying price).

In the money calls have delta from 0.50 to 1.00 (high sensitivity). The deeper in the money (lower strike), the higher delta.

In fact, the underlying security could (with some simplification, e.g. dividends) be considered an extremely deep in the money call option on itself with strike price of zero. Its delta is 1 (because its price is the same thing as underlying price, and therefore must change at the same rate).

Out of the money calls have delta from 0 to 0.50 (low sensitivity). The higher strike, the smaller delta.

In other words, call option delta is inversely related to strike price.

### Delta and Put Option Moneyness

At the money puts have delta close to -0.50 (moderate sensitivity, opposite direction).

In the money puts have delta from -1.00 to -0.50 (high sensitivity, opposite direction). The deeper in the money (higher strike), the lower (more negative) delta. A short position in the underlying security could, with some simplification, be considered an extremely deep in the money put option on itself (with infinitely high strike price) and has delta equal to -1.

Finally, out of the money puts have delta from -0.50 to 0 (low sensitivity, opposite direction). The further out of the money (lower strike), the closer delta is to zero.

Put option delta is also inversely related to strike price: close to zero for low strikes, close to -1 for high strikes.

## Delta as Probability of Expiring in the Money

One interpretation of delta is that its absolute value indicates the approximate probability of the option expiring in the money.

For instance, a deep in the money call option (\$30 strike with underlying price at \$40) is under normal circumstances (I am using 30% volatility and 50 days to expiration) almost certain to expire in the money, as its delta of 0.996 also suggests. A \$35 strike call on the same underlying is still very likely to expire in the money, although slightly less likely that the \$30 strike call. It has delta of 0.90, indicating a 90% probability. On the contrary, a \$50 strike call is far out of the money, has delta of 0.025, and is unlikely to expire in the money (the underlying would have to increase by more than 25% from its current level).

It works the same with puts – you just need to ignore the minus sign. A deep in the money put with delta of -0.95 has approximately 95% probability of expiring in the money.

## Relationship between Call and Put Delta

It is no coincidence that call and put deltas (same underlying, same expiration, same strike) seem to always sum up to approximately one if you ignore the put delta minus sign. For instance, in our first examples the \$55 strike call had delta 0.75 and the \$55 strike put had delta -0.25.

This also fits well with the probability interpretation explained above: the underlying security can either end up above a strike (only the call expires in the money) or below a strike (only the put expires in the money). The probabilities of two mutually exclusive outcomes when no other outcome is possible must add up to 100%. We disregard the possibility of underlying price exactly equal to the strike price, when both the call and the put would expire worthless – after all, the probability of such outcome is infinitely small in the abstract world of continuous price models and infinite precision.

## Delta Units

Being a ratio of two price changes, delta has no units, but you can think of it as "how many dollars (of option price change) for one dollar (of underlying price change)".

Some market practitioners prefer to express delta as a percentage (call delta of 0.75 from our example is expressed as 75%; put delta of -0.25 as -25%). This fits nicely with the probability interpretation.

Some may also omit the percentage sign and use delta multiplied by 100 (delta of 0.75 becomes 75; -0.25 becomes -25). One reason for this format is that some people are not that comfortable/quick counting with small decimal numbers. Another reason arises from the fact that US traded equity options trade in contracts representing 100 shares. Delta in this format measures the relative change in option premium in cents per one share or in dollars per one contract (of 100 shares).

When we said delta can't be greater than +1 (+100%) for calls or smaller than -1 (-100%) for puts, we were only considering a single option. A position consisting of multiple options can have delta far beyond these limits – theoretically (positive or negative) infinite. You may hear a trader say he is "long 600 deltas" or "short 350 deltas".

An important property of delta (and all the other option Greeks) is that it is additive. If you hold two options, one has delta of 0.75 and the other has delta 0.85, your entire position has total delta of 1.60. If underlying price increases by \$1, your position's value grows by \$1.60 per share, because the individual option prices increase by \$0.75 and \$0.85, respectively. Calculating total delta of even a complex option position is quite simple: add deltas of all long options and subtract deltas of all short options. Let's see a few examples.

A straddle is a popular option strategy that consists of a call option and a put option, both with the same strike and expiration date. If you buy a straddle and a call option has delta +0.49 and a put option has delta -0.51, the total delta is 0.49 + (-0.51) = -0.02. It is very close to zero – such position or strategy is often called delta neutral or non-directional (its value is not sensitive to changes in underlying price direction).

Another popular option strategy is the bull call spread, which consists of one long call option with lower strike price (and higher delta, for instance +0.70) and one short call option with the same expiration date but higher strike price (and lower delta, e.g. +0.30). When calculating total delta, make sure you subtract (not add) the short option's delta: 0.70 – 0.30 = 0.40. Selling the higher strike call option effectively makes the position less sensitive to underlying price changes (reduces delta) than if you just bought the lower strike call.

### Covered Call Delta Example

Some option strategies also include a long or short position in the underlying security. The most popular is covered call, which consists of a long underlying position and a short call option. The underlying always has a delta of +1; a short position in the underlying has delta of -1.

Make sure you get the units right when working with positions including both the underlying and options. Let's say we own 100 shares of a stock and sell one call option contract (that represents 100 shares) with delta of 0.30. Total delta of this covered call position is 1 – 0.30 = 0.70 per share, or alternatively 100 – 30 = 70 per 100 shares. When the stock price increases by \$1, each of your 100 long shares makes \$1 (total \$100 gain), but your short call option's value decreases by \$0.30 per share (\$30 per contract). In total, your position makes \$70, or \$0.70 per share.

## Using Delta in Practice

At this point you should understand what delta means, how to read its values, and how to easily calculate total delta of a position. Now we will discuss why we should care about delta in the first place: what it is used for and how it can help our trading and risk management.

The role of all option Greeks is to measure risk – quantify how an option position's value (and thereby profit or loss) would change under various possible market developments (a move in underlying price in case of delta, or volatility, interest rates, or passage of time in case of other Greeks). Quantifying these effects enables us to do several useful things:

• Choose the most suitable option strategy, expiration(s) or strike(s) for a potential trading idea.
• Decide the size (number of contracts) of a potential trade.
• Know what would happen under different scenarios and plan in advance how we would react.
• Make adjustments to an existing position – keep it exposed to the risks we want to take (and profit from) and at the same time reduce or eliminate (hedge) its exposure to the risks we are not willing to take.

## Hedge Ratio and Delta Hedging

An alternative name for delta is hedge ratio; it refers to another way we can interpret delta.

We already know that delta is the ratio of an option price change to underlying price change. For instance, if a call option has delta of 0.60, we can expect its price to grow approximately by \$0.60 if underlying price increases by \$1.

Suppose we are holding this 0.60 delta call option, but we are worried that the underlying stock might fall. We want to keep the option (perhaps we believe it is underpriced based on our volatility estimate and a "fair" price calculated using an option pricing model), but we want to somehow insure (hedge) it against an accidental drop in underlying price.

The easiest way how to do that is to sell short the underlying stock. If the stock falls, gains on the short position will offset any losses on the call option. If, on the contrary, the stock turns out going higher, we will lose money on the short stock, but gains on the call option will compensate for the losses. The objective of hedging is to make our profit or loss zero / unchanged, regardless of the stock price direction. In other words, eliminate our position's sensitivity to underlying price – make the total delta zero.

Holding the call option, our current delta is +0.60 per share, or 60 for the one option contract. We already know that a short position in the underlying stock always has a delta of -1 per share. How many shares we need to short to make the total delta zero?

The answer is of course 60. If we are short 60 shares (-60 delta) and long one call option contract with delta of +0.60 (+60 delta), our total delta is zero. The position should be immune to small moves in the stock price.

In fact, we can immediately see the number of shares to short just by looking at the option's delta, or hedge ratio. 0.60 or 60:100 – sell 60 shares for one option contract of 100 shares.

This single option example was quite trivial, but imagine you have a portfolio of 50 options of different types, expiration dates, and strikes. A seemingly complex task of hedging such portfolio against underlying price moves becomes very simple using delta: just add up the individual option deltas to find the portfolio's total delta – and you have the number of shares you need to sell short. Of course, if total delta of the portfolio is negative, you need to buy (rather than sell) that many shares to hedge it.

One final, but very important note about delta hedging: Remember delta is only accurate for small changes in underlying price. In case of larger moves, a portfolio's delta itself may also increase or decrease. An option position with 30 delta may suddenly become 50 delta and if you have hedged it by selling 30 shares of the underlying, you are suddenly long 20 delta. To stay hedged, you need to short additional shares. Conversely, if your portfolio's delta declines, you may find yourself overhedged and may need to close some of your shorts to bring total delta back to zero. Successful hedging requires ongoing monitoring and rebalancing.

Moreover, delta can change not only due to underlying price moves. There are several other factors which may affect it, such as volatility, interest rates, or just passing time. We will explore them below in the final part of this page.

## Factors Affecting Option Delta

Like option premium, delta generally depends on several factors. The exact formula for calculating delta can be slightly different under different option pricing models, but the inputs which almost always appear include the following:

• Underlying price
• Strike price
• Time to expiration
• Volatility
• Interest rate

Depending on underlying security type, other factors may also be important, such as dividend yield (for stock options) or foreign interest rate (for forex options).

Because all these inputs enter delta calculation, if any of them changes, the delta may also change.

### Delta and Underlying Price

We have already mentioned that delta changes with underlying price. As a result, delta is only valid for small underlying price changes. As we have seen in our first example, when the underlying stock increased from \$57 to \$58, a \$55 strike call option with delta 0.75 (valid at stock price of \$57) grew by \$0.79. The new delta at the \$58 underlying price was in fact 0.83; you can see the actual increase in option price was approximately average of the two deltas.

The fact that delta changes with underlying price should also be obvious from the relationship between delta and moneyness. We have explained that in the money call options have higher delta than out of the money call options. As an option changes from out-of-the-money to in-the-money (which can only happen due to underlying price going up), its delta must increase. Similarly for puts, with increasing underlying price puts generally move out of the money and their deltas also increase (become less negative – get closer to zero).

In sum, both call and put deltas increase as underlying price goes up. Calls become more sensitive to underlying price (delta gets closer to +1), while puts become less sensitive (delta gets closer to zero from below).

Moreover, the speed of delta changes (which is measured by gamma) also changes with underlying price: it is generally fastest (highest gamma) at the money and approaches zero both deep in the money and far out of the money. For more details see option gamma.

### Delta and Time to Expiration

Delta also changes with passing time, even when nothing else happens in the markets.

In the money options become more sensitive to underlying price changes as they approach expiration and their delta gets closer to +1 for calls or -1 for puts.

Out of the money options become less sensitive to the underlying price moves and their delta gets closer to zero near expiration.

In general, the longer time to expiration, the closer an option's delta is to the middle of its range (0.50 for calls, -0.50 for puts), because with plenty of time remaining, it is still unclear whether the option eventually ends up in the money or out of the money. As time to expiration becomes shorter, the delta is pushed to one of the extremes: towards zero for out of the money options, towards +1 for in the money calls, and towards -1 for in the money puts.

### Delta and Volatility

The effect of volatility on delta is very similar to the effect of time to expiration. After all, these two factors combined are what allows markets to move – a big move is more likely in a more volatile market, over a longer time horizon, and most likely when both these conditions work together. More volatility and longer time to expiration generally make options more valuable, and they also affect the delta in similar ways.

Increasing volatility pushes call delta closer to 0.50 and put delta closer to -0.50. Decreasing volatility pushes delta to the extremes – out of the money call and put delta closer to zero, in the money call delta closer to +1, and in the money put delta closer to -1.

## How to Calculate Delta

Mathematically, option delta is the first derivative of option price with respect to underlying price (graphically it is the slope of a chart where underlying price is the X-axis and option premium is the Y-axis). In this tutorial I have tried to avoid the complexities of option pricing mathematics and focused instead on the logic and practical considerations. Those who are interested can see the exact Black-Scholes Greeks formulas and their Excel implementation.

## Summary

• Delta measures how option price will change if underlying price increases by \$1.
• Call option delta is from 0 to +1. Put option delta is from 0 to -1.
• Out of the money options have delta near zero. In the money options near +1 (calls) or -1 (puts).
• Delta itself changes with underlying price (this is measured by gamma). Therefore, delta is only accurate for small underlying price changes.
• Like all other Greeks, delta is additive. Total delta of a position with multiple options is the sum of all options' deltas.
• Delta hedging makes delta zero – makes a position immune to small underlying price changes. It requires ongoing monitoring and rebalancing.
• Delta also changes with volatility and passing time. Lower volatility or lower time to expiration push delta closer to the extremes (0 or +1 or -1).

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