# Option Greeks: delta, gamma, vega, and theta

## The option Greeks

The options Greeks delta, gamma, vega, and theta give investors and trader more insight into the price formation of options and are therefore crucial for option traders. An option premium, for example, is determined by the price of the underlying asset, the maturity of the option and its volatility. These three variables are not constant. For example, volatility can increase as uncertainty rises, causing the option premium to go up.

The option Greeks provide a trader with more insight into the price change of an option premium when the variables change. The image below shows an example of a trading platform demonstrating the option Greeks. Unfortunately, the option Greeks isn’t the most exciting matter to discuss, even though it is basic option theory. In this article I will discuss de most important option Greeks step by step.

## 1. Delta

Delta is the most important option Greek and refers to the price change of an option with respect to the changes in price of the underlying asset. Delta indicates to what extent the option premium fluctuates due to a price increase or decrease of the underlying asset. A movement in de the underlying asset is generally indicated as an increase / decrease of 1 point.

Premium Call option Delta
€2.50 0.50

In the theoretical example above a call option has a premium of €2.50 and delta equals 0.5.

When the underlying asset increases with €1, the value of the option increase with €0.50 to €3.

Premium call option €2.50 → Delta 0.50 → Premium call option €3.00

The other way around it works exactly the same. If the share decreases with €1.00, then the option premium decreases with €0.50 and falls to €2.00

Premium call option €2.50 →  Delta 0.50 →  Premium call option €2.00

The working of delta for put options is identical, except that the put delta is written with a – (minus) sign. If the underlying value decreases by 1 point, the premium of the put option increases with delta. If the underlying value increases by 1 point, the premium of the put option decreases the delta.

By the way, delta has a second definition. Delta also indicates the probability that an option expires in-the-money. This is not a scientific approach, but a rule of thumb. In this way, an option trader is able to quickly estimate if an option has a large or small probability to expire in-the-money. Suppose the delta of an option is 0.80, then the probability that the option will expire in-the-money is 80%. If an option has a delta of 0.10, then this probability is only 10%. An at-the-money option therefore has delta 0.50, which gives a probability of 50% that it expires in-the-money.

## 2. Gamma

When the underlying value of an option moves, an option either gets more in-the-money or more out-of-the-money. This has the result that delta is constantly changing as well. This change of delta is measured based on the option Greek gamma. Gamma indicates the change in value of delta when the underlying value moves. The gamma is highest for at-the-money options, since movements in the underlying asset than have the biggest impact on the delta of these options.

The following table lists the delta and gamma for call options on AEX (Amsterdam Stock Exchange) with a maturity of one month. The AEX in this example is at around 490 points. The values of Gamma are rounded to two decimals to make it easier to comprehend.

Strike price Delta Gamma AEX + 1 New Delta
480 0.65 0.01 0.66
490 0.51 0.02 0.53
500 0.35 0.02 0.37
510 0.21 0.01 0.22

The at-the-money option with strike price 490 has the largest gamma of 0.02. If the AEX rises by 1 point, delta changes with 0.02. The calculation is simple: add gamma to delta when the AEX rises by 1 point or subtract it if the AEX decreases by 1 point. The delta of the call option 490 changes from 0.51 to 0.53 after a 1 point increase.

When the options have a shorter maturity, gamma get bigger. The explanation is quite simple. Short-term options have less time value than longer-term options. Movement in the underlying security is thus of greater influence on the option premium than with long-term options. For example, if you look up the gamma of a daily or weekly option, you will see that these are many times greater than in the example we discussed.

## 3. Vega

The third option Greek is vega. Volatility plays an important role in option pricing because the price of an option is partly determined by the volatility of the underlying asset. To understand the meaning of vega, it is necessary to realize that volatility relates to the expected volatility of the underlying value until the expiration date. In option terms this is called implied volatility.

The vega of an option reflects the extent to which the price of an option changes if the volatility changes. A decrease in volatility causes less movement in the price of the underlying asset, making options cheaper. An increase in volatility leads to an increase expected movement of the underlying asset, making options more expensive.

The vega varies for each strike price and maturity of the option. A short-term option is less sensitive to changes in volatility than a long term option. Like all other option Greeks, Vega is quoted as a decimal, which relates to a one point change in the implied volatility of the option.

The table below shows the vega for at-the-money options on the AEX with different maturities.

Maturity Strike price Implied volatility Vega
32 days 490 17.77% 0.58
60 days 490 17.82% 0.79
95 days 490 17.39% 1.01
214 days 490 16.76% 1.49

Vega in absolute terms is always lower for short-term options than for long-term options. It should however be observed that it is precisely at these short term options that changes in implied volatility occur faster. The implied volatility indeed indicates the expected volatility of the option until its expiration date. Without going too deep into this matter, it is important to realize that volatility is constantly changing and this has to do with supply and demand and market expectations. When market expectations change due to a sudden event, it will have the greatest impact on short-term options. So even though vega in absolute terms is smallest for options with a term of one month, for example, fluctuations in implied volatility are the highest.

If the implied volatility of a call option AEX 490 which expires in 32 days increased from 17.77% to 19.77%, then the option premium increases by € 1.16 (2 x 0.58). On the other hand, if the volatility decreases, the value of the option goes down. Volatility is thus a crucial concept when trading options. Therefore, I will further elaborate on this concept in the next article.

## 4. Theta

The fourth and final option Greek we discuss is theta. Theta refers to the time value of options. Purchased options give an option trader the right to buy or sell shares up to the expiration date. For example, when an out-of-the-money call option is bought, the premium paid for the option is simply the expectation value. Out-of-the-money options do not have intrinsic value. When the option has no intrinsic value at the expiration date, it is not exercised and the option is worthless. From the moment of purchase until the expiration date, the time value is getting smaller and smaller. Theta indicates to what extent the time value of the option is getting smaller and smaller.

Theta thus shows how much time value the option loses every day. The shorter the duration of the option, the faster the option loses its time value. Fortunately, this is quite easy to understand. For two options with the same strike price but different expiration dates, the short-term option loses more time value per day than the long-term option. From about one month before expiration the time value is decreasing exponentially.

The table below shows the theta of at-the-money AEX options with different maturities.

Maturity Theta
32 days -0.16
60 days -0.12
95 days -0.09
214 days -0.06

The option which expires in one month, loses about € 0.16 in option premium per day. As expiration approaches, theta will increase. The option which expires in nearly seven months to loses “only” € 0.06 of time value each day. The loss of time value is a constant process. During the day, the decline in time value is virtually invisible. Of course, the loss of time value is greatest during the weekend.

Theta is also linked to volatility. When the volatility of options is relatively high, the option will lose more time value. The expected volatility is than higher. As indicated previously, in the next blog I will explain the concept of volatility in more detail, so that after this “dry” theory we can take the next step to the various option strategies such as covered call / covered put, call and put spread, straddle, strangle , butterfly and iron condor.