#17: Options Greeks: Gamma as "Curvature"
 
In the previous 3-part series, I went into detail about options Delta and how they are used for trading.

In this next 3-part series, i'm going to be talking about the second of the four greeks, Gamma. 

Gamma is pretty kick-ass and requires a lot of scrutiny. I will be breaking down Gamma into the following 3-parts:

Part One: Gamma as "Curvature"

Part Two: Factor's affecting Gamma

Part Three: Using Gamma in Strategies 

Ok, buckle down and let's get going!

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You might be familiar with the concepts of distance, speed, and acceleration (or displacement, velocity and acceleration for you physics Nazis).

In the options trading world, distance/speed/acceleration are analogous in the following way:

- Distance : How much profit or loss you are going to make

- Speed : How quickly you are going to make that profit or loss (Delta)

- Acceleration : How quickly the speed of your profits or losses escalates! (Gamma)

That's right. If you understand your gamma correctly, you will be able to quickly determine which type of options trades can potentially explode with profits!

And no, this is no scam. The numbers are real, and they have always been there. 

You can make use of this information right after reading this post to put on trades that might turn into 500% returns within minutes.

But if something sounds too good to be true, it usually comes with a big catch. Well... what's the catch? 

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If we look purely at the math, Gamma is a very simple entity. It's simply the rate of change of Delta.

Put simply, Gamma tells us how quickly delta is changing.

If you have a bullish position, you will have positive delta. That means stock goes up, your position makes money.

Now let's split that example up into two different scenarios: the position is made up of long stock, versus the position made up of a long call option.

Scenario A: Long 100 shares of SPY (long stock)

SPY is trading at 273 right now, which is the orange vertical line in the center of the image. That diagonal pink line is our profit/loss profile. 

As the SPY price increases bullishly to 283 (the horiontal green arrow), our profit for this position will be $1000 (the $994 shown there is due to the decimal point differences).

The critical thing to note here is this: The profit/loss profile is LINEAR.

For every $1 that SPY rises, our profit will be $100. SPY goes from 273 to 283, our profit will be $1000. It's fixed. 

The slope of the pink line tells us our Delta, which is 1.0 ($1 rise gives $1 profit x 100 shares).

The delta of this stock position will never change. The slope is always a straight line. 

Yawn.... let's look at something more interesting shall we?

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Scenario B: Long 1 contract of at-the-money Call options of SPY (long call option)

Obviously, the profit/loss profile is completely different with options!

Looking at the pink line in the center of the image, we can see that it is no longer a straight line, but a curve

This implies a few important things:

1. As SPY rises bullishly, the slope of the pink line gets steeper and steeper till it becomes parallel with the blue line on the right. 

The blue line on the right represents the call option expiring in-the-money into 100 shares of stock, which means it has a delta of 1.0 

This means the delta of the option at 273 is NOT 1.0, but less than 1.0 since the slope is gentler. In fact, the delta at 273 is 0.5 since it is at-the-money.

2. As SPY falls bearishly, the slope of the pink line becomes more gentle until it becomes horizontal.

This time, the horizontal blue line on the right represents the call option expiring out-of-the-money and worthless, which means a delta of 0.

That makes sense because my maximum loss is whatever I paid for the option. I will never lose more than that, so my losses are limited.

Scenario B has very big implications for the profitability of using long options. 

Firstly, it means that if I made a correct directional bet, my profits no longer increase linearly, but instead will accelerate!

Secondly, it means that if I made a wrong directional bet, my losses is also no longer linear, but instead will decelerate

What sorcery!! Accelerating profits if i'm right, decelerating losses if i'm wrong!!

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So how does Gamma show up in any of the above scenarios?

Let's use a word to visually represent Gamma, then it would quickly become obvious.

If Delta can be represented by the word "slope" or "gradient"...

... then Gamma can be represented by the word "curvature" (or "concavity" for you math-lovers).

I'm going to show you a series of profit/loss profiles, and see if you can tell which of the lines has sharper or gentler curvature and, hence, higher or lower Gamma.

The pink line has the gentlest curvature while the yellow line has the sharpest curvature, which means that the pink line has the smallest Gamma, and the yellow line the most Gamma.

That also means that we stand to profit much more if our position follows the yellow line.

The sharper the curvature, the more quickly the delta accelerates to 1.0, and the faster the profit explodes.

Let's verify this with the actual Gamma of the positions.

The pink and yellow lines were both created by buying a call and a put at the same strike (known as a "straddle"). The Gamma values can be found in the options chain and are circled in white.

The Gamma of the pink line are both these Gammas added together to give us 0.11. 

The Gamma of the yellow line on the other hand has a Gamma of 0.40. The larger the value of Gamma, the sharper the curvature of the profile, and quicker the acceleration of delta!

While on this note, I should mention that, unlike Delta, there is no limit to the value of Gamma

Delta has a maximum of 1.0, but Gamma can go into the hundreds, if not thousands. Just imagine the letter "V". How sharp is the curvature at the connecting point of the two straight lines? 

The sharper and more abrupt the change in gradient, the larger the Gamma!

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So in this post, I have introduced you to the concept of Gamma, and the use of the word "curvature" to visualize Gamma in the profit/loss profile.

But how do you find options with high (or low) Gamma?

What are the factors that affect Gamma? 

And what is that "catch" that makes high Gamma too good to be true?

Look out for my next post where I will elaborate on the points above.

Till then, happy trading!

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