Gamma Explained

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Gamma

The option’s gamma is a measure of the rate of change of its delta. The gamma of an option is expressed as a percentage and reflects the change in the delta in response to a one point movement of the underlying stock price.

Like the delta, the gamma is constantly changing, even with tiny movements of the underlying stock price. It generally is at its peak value when the stock price is near the strike price of the option and decreases as the option goes deeper into or out of the money. Options that are very deeply into or out of the money have gamma values close to 0.

Example

Suppose for a stock XYZ, currently trading at $47, there is a FEB 50 call option selling for $2 and let’s assume it has a delta of 0.4 and a gamma of 0.1 or 10 percent. If the stock price moves up by $1 to $48, then the delta will be adjusted upwards by 10 percent from 0.4 to 0.5.

However, if the stock trades downwards by $1 to $46, then the delta will decrease by 10 percent to 0.3.

Passage of time and its effects on the gamma

As the time to expiration draws nearer, the gamma of at-the-money options increases while the gamma of in-the-money and out-of-the-money options decreases.

The chart above depicts the behaviour of the gamma of options at various strikes expiring in 3 months, 6 months and 9 months when the stock is currently trading at $50.

Changes in volatility and its effects on the gamma

When volatility is low, the gamma of at-the-money options is high while the gamma for deeply into or out-of-the-money options approaches 0. This phenomenon arises because when volatility is low, the time value of such options are low but it goes up dramatically as the underlying stock price approaches the strike price.

When volatility is high, gamma tends to be stable across all strike prices. This is due to the fact that when volatility is high, the time value of deeply in/out-of-the-money options are already quite substantial. Thus, the increase in the time value of these options as they go nearer the money will be less dramatic and hence the low and stable gamma.

The chart above illustrates the relationship between the option’s gamma and the volatility of the underlying security which is trading at $50 a share.

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Gamma Definition

What is Gamma

Gamma is the rate of change in an option’s delta per 1-point move in the underlying asset’s price. Gamma is an important measure of the convexity of a derivative’s value, in relation to the underlying. A delta hedge strategy seeks to reduce gamma in order to maintain a hedge over a wider price range. A consequence of reducing gamma, however, is that alpha will also be reduced.

Gamma

Basics of Gamma

Gamma is the first derivative of delta and is used when trying to gauge the price movement of an option, relative to the amount it is in or out of the money. In that same regard, gamma is the second derivative of an option’s price with respect to the underlying’s price. When the option being measured is deep in or out of the money, gamma is small. When the option is near or at the money, gamma is at its largest. All options that are a long position have a positive gamma, while all short options have a negative gamma.

Gamma Behavior

Since an option’s delta measure is only valid for short period of time, gamma gives traders a more precise picture of how the option’s delta will change over time as the underlying price changes. Delta is how much the option price changes in respect to a change in the underlying asset’s price.

As an analogy to physics, the delta of an option is its “speed,” while the gamma of an option is its “acceleration.”

Gamma decreases, approaching zero, as an option gets deeper in the money and delta approaches one. Gamma also approaches zero the deeper an option gets out of the money. Gamma is at its highest when the price is at the money.

The calculation of gamma is complex and requires financial software or spreadsheets to find a precise value. However, the following demonstrates an approximate calculation of gamma. Consider a call option on an underlying stock that currently has a delta of 0.4. If the stock value increases by $1, the option will increase in value by $0.40, and its delta will also change. After the $1 increase, assume the option’s delta is now 0.53. The 0.13 difference in deltas can be considered an approximate value of gamma.

Gamma is an important metric because it corrects for convexity issues when engaging in hedging strategies. Some portfolio managers or traders may be involved with portfolios of such large values that even more precision is needed when engaged in hedging. A third-order derivative named “color” can be used. Color measures the rate of change of gamma and is important for maintaining a gamma-hedged portfolio.

Key Takeaways

  • Gamma is the rate of change for an option’s delta based on a single-point move in the delta’s price.
  • Gamma is at its highest when an option is at the money and is at its lowest when it is further away from the money.

Example of Gamma

Suppose a stock is trading at $10 and its option has a delta of 0.5 and a gamma of 0.1. Then, for every 10 percent move in the stock’s price, the delta will be adjusted by a corresponding 10 percent. This means that a $1 increase will mean that the option’s delta will increase to 0.6. Likewise, a 10 percent decrease will result in corresponding decline in delta to 0.4.

UNDERSTANDING GAMMA CORRECTION

Gamma is an important but seldom understood characteristic of virtually all digital imaging systems. It defines the relationship between a pixel’s numerical value and its actual luminance. Without gamma, shades captured by digital cameras wouldn’t appear as they did to our eyes (on a standard monitor). It’s also referred to as gamma correction, gamma encoding or gamma compression, but these all refer to a similar concept. Understanding how gamma works can improve one’s exposure technique, in addition to helping one make the most of image editing.

WHY GAMMA IS USEFUL

1. Our eyes do not perceive light the way cameras do. With a digital camera, when twice the number of photons hit the sensor, it receives twice the signal (a “linear” relationship). Pretty logical, right? That’s not how our eyes work. Instead, we perceive twice the light as being only a fraction brighter — and increasingly so for higher light intensities (a “nonlinear” relationship).

Reference Tone Select:

Perceived as 50% as Bright
by Our Eyes
Detected as 50% as Bright
by the Camera

Refer to the tutorial on the photoshop curves tool if you’re having trouble interpreting the graph.
Accuracy of comparison depends on having a well-calibrated monitor set to a display gamma of 2.2.
Actual perception will depend on viewing conditions, and may be affected by other nearby tones.
For extremely dim scenes, such as under starlight, our eyes begin to see linearly like cameras do.

Compared to a camera, we are much more sensitive to changes in dark tones than we are to similar changes in bright tones. There’s a biological reason for this peculiarity: it enables our vision to operate over a broader range of luminance. Otherwise the typical range in brightness we encounter outdoors would be too overwhelming.

But how does all of this relate to gamma? In this case, gamma is what translates between our eye’s light sensitivity and that of the camera. When a digital image is saved, it’s therefore “gamma encoded” — so that twice the value in a file more closely corresponds to what we would perceive as being twice as bright.

Technical Note: Gamma is defined by Vout = Vin gamma , where Vout is the output luminance value and Vin is the input/actual luminance value. This formula causes the blue line above to curve. When gamma 1.

2. Gamma encoded images store tones more efficiently. Since gamma encoding redistributes tonal levels closer to how our eyes perceive them, fewer bits are needed to describe a given tonal range. Otherwise, an excess of bits would be devoted to describe the brighter tones (where the camera is relatively more sensitive), and a shortage of bits would be left to describe the darker tones (where the camera is relatively less sensitive):

Note: Above gamma encoded gradient shown using a standard value of 1/2.2
See the tutorial on bit depth for a background on the relationship between levels and bits.

Notice how the linear encoding uses insufficient levels to describe the dark tones — even though this leads to an excess of levels to describe the bright tones. On the other hand, the gamma encoded gradient distributes the tones roughly evenly across the entire range (“perceptually uniform”). This also ensures that subsequent image editing, color and histograms are all based on natural, perceptually uniform tones.

However, real-world images typically have at least 256 levels (8 bits), which is enough to make tones appear smooth and continuous in a print. If linear encoding were used instead, 8X as many levels (11 bits) would’ve been required to avoid image posterization.

GAMMA WORKFLOW: ENCODING & CORRECTION

Despite all of these benefits, gamma encoding adds a layer of complexity to the whole process of recording and displaying images. The next step is where most people get confused, so take this part slowly. A gamma encoded image has to have “gamma correction” applied when it is viewed — which effectively converts it back into light from the original scene. In other words, the purpose of gamma encoding is for recording the image — not for displaying the image. Fortunately this second step (the “display gamma”) is automatically performed by your monitor and video card. The following diagram illustrates how all of this fits together:

1. Depicts an image in the sRGB color space (which encodes using a gamma of approx. 1/2.2).
2. Depicts a display gamma equal to the standard of 2.2

1. Image Gamma . This is applied either by your camera or RAW development software whenever a captured image is converted into a standard JPEG or TIFF file. It redistributes native camera tonal levels into ones which are more perceptually uniform, thereby making the most efficient use of a given bit depth.

2. Display Gamma . This refers to the net influence of your video card and display device, so it may in fact be comprised of several gammas. The main purpose of the display gamma is to compensate for a file’s gamma — thereby ensuring that the image isn’t unrealistically brightened when displayed on your screen. A higher display gamma results in a darker image with greater contrast.

3. System Gamma . This represents the net effect of all gamma values that have been applied to an image, and is also referred to as the “viewing gamma.” For faithful reproduction of a scene, this should ideally be close to a straight line (gamma = 1.0). A straight line ensures that the input (the original scene) is the same as the output (the light displayed on your screen or in a print). However, the system gamma is sometimes set slightly greater than 1.0 in order to improve contrast. This can help compensate for limitations due to the dynamic range of a display device, or due to non-ideal viewing conditions and image flare.

IMAGE FILE GAMMA

The precise image gamma is usually specified by a color profile that is embedded within the file. Most image files use an encoding gamma of 1/2.2 (such as those using sRGB and Adobe RGB 1998 color), but the big exception is with RAW files, which use a linear gamma. However, RAW image viewers typically show these presuming a standard encoding gamma of 1/2.2, since they would otherwise appear too dark:

If no color profile is embedded, then a standard gamma of 1/2.2 is usually assumed. Files without an embedded color profile typically include many PNG and GIF files, in addition to some JPEG images that were created using a “save for the web” setting.

Technical Note on Camera Gamma. Most digital cameras record light linearly, so their gamma is assumed to be 1.0, but near the extreme shadows and highlights this may not hold true. In that case, the file gamma may represent a combination of the encoding gamma and the camera’s gamma. However, the camera’s gamma is usually negligible by comparison. Camera manufacturers might also apply subtle tonal curves, which can also impact a file’s gamma.

DISPLAY GAMMA

This is the gamma that you are controlling when you perform monitor calibration and adjust your contrast setting. Fortunately, the industry has converged on a standard display gamma of 2.2, so one doesn’t need to worry about the pros/cons of different values. Older macintosh computers used a display gamma of 1.8, which made non-mac images appear brighter relative to a typical PC, but this is no longer the case.

Recall that the display gamma compensates for the image file’s gamma, and that the net result of this compensation is the system/overall gamma. For a standard gamma encoded image file ( ), changing the display gamma ( ) will therefore have the following overall impact ( ) on an image:

Diagrams assume that your display has been calibrated to a standard gamma of 2.2.
Recall from before that the image file gamma ( ) plus the display gamma ( ) equals the overall system gamma ( ). Also note how higher gamma values cause the red curve to bend downward.

If you’re having trouble following the above charts, don’t despair! It’s a good idea to first have an understanding of how tonal curves impact image brightness and contrast. Otherwise you can just look at the portrait images for a qualitative understanding.

How to interpret the charts. The first picture (far left) gets brightened substantially because the image gamma ( ) is uncorrected by the display gamma ( ), resulting in an overall system gamma ( ) that curves upward. In the second picture, the display gamma doesn’t fully correct for the image file gamma, resulting in an overall system gamma that still curves upward a little (and therefore still brightens the image slightly). In the third picture, the display gamma exactly corrects the image gamma, resulting in an overall linear system gamma. Finally, in the fourth picture the display gamma over-compensates for the image gamma, resulting in an overall system gamma that curves downward (thereby darkening the image).

The overall display gamma is actually comprised of (i) the native monitor/LCD gamma and (ii) any gamma corrections applied within the display itself or by the video card. However, the effect of each is highly dependent on the type of display device.

CRT Monitors. Due to an odd bit of engineering luck, the native gamma of a CRT is 2.5 — almost the inverse of our eyes. Values from a gamma-encoded file could therefore be sent straight to the screen and they would automatically be corrected and appear nearly OK. However, a small gamma correction of

1/1.1 needs to be applied to achieve an overall display gamma of 2.2. This is usually already set by the manufacturer’s default settings, but can also be set during monitor calibration.

LCD Monitors. LCD monitors weren’t so fortunate; ensuring an overall display gamma of 2.2 often requires substantial corrections, and they are also much less consistent than CRT’s. LCDs therefore require something called a look-up table (LUT) in order to ensure that input values are depicted using the intended display gamma (amongst other things). See the tutorial on monitor calibration: look-up tables for more on this topic.

Technical Note: The display gamma can be a little confusing because this term is often used interchangeably with gamma correction, since it corrects for the file gamma. However, the values given for each are not always equivalent. Gamma correction is sometimes specified in terms of the encoding gamma that it aims to compensate for — not the actual gamma that is applied. For example, the actual gamma applied with a “gamma correction of 1.5” is often equal to 1/1.5, since a gamma of 1/1.5 cancels a gamma of 1.5 (1.5 * 1/1.5 = 1.0). A higher gamma correction value might therefore brighten the image (the opposite of a higher display gamma).

OTHER NOTES & FURTHER READING

Other important points and clarifications are listed below.

  • Dynamic Range. In addition to ensuring the efficient use of image data, gamma encoding also actually increases the recordable dynamic range for a given bit depth. Gamma can sometimes also help a display/printer manage its limited dynamic range (compared to the original scene) by improving image contrast.
  • Gamma Correction. The term “gamma correction” is really just a catch-all phrase for when gamma is applied to offset some other earlier gamma. One should therefore probably avoid using this term if the specific gamma type can be referred to instead.
  • Gamma Compression & Expansion. These terms refer to situations where the gamma being applied is less than or greater than one, respectively. A file gamma could therefore be considered gamma compression, whereas a display gamma could be considered gamma expansion.
  • Applicability. Strictly speaking, gamma refers to a tonal curve which follows a simple power law (where Vout = Vin gamma ), but it’s often used to describe other tonal curves. For example, the sRGB color space is actually linear at very low luminosity, but then follows a curve at higher luminosity values. Neither the curve nor the linear region follow a standard gamma power law, but the overall gamma is approximated as 2.2.
  • Is Gamma Required? No, linear gamma (RAW) images would still appear as our eyes saw them — but only if these images were shown on a linear gamma display. However, this would negate gamma’s ability to efficiently record tonal levels.

For more on this topic, also visit the following tutorials:

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