June 15, 2021
Dopex Essentials: Option GammaDopex-Essentials
Trigger Warning: Numbers… A lot of them
Hey, welcome again to another Dopex Essentials episode. If this is your first episode I would advise you to visit our medium homepage and start with episode 1 — What are Options?
In today’s episode, we are taking an in-depth look at Option Gamma. There is quite a bit of number-crunching involved and some concepts are more difficult to grasp so we have split the gamma articles in two. In this first article, we will go over some of the more simple aspects with the use of examples to make it as simple as possible.
There is a glossary of terms at the end of the article in case you come across some terms that you are unfamiliar with.
Without further ado, let’s dive right in!
So we discussed option delta in the previous articles, we know that it represents the change in premium for the given change in the underlying asset’s price.
For example, if the price of HakhoToken is $80, this obviously means that an $82 call option (let’s call it option X) would be OTM. Therefore its delta would be a value between 0 and 0.5. For the sake of this discussion let’s say the delta of Option X is 0.2.
Now, let’s assume that the price of HakhoToken pumps to $83 in a single day, this means the $82 call option (Option X) is no longer an OTM option, it becomes a slightly ITM option. Thus, because of the increase in the price of the underlying asset, the delta of Option X would no longer be 0.2 and would instead be somewhere between 0.5 and 1.0. Let’s say it’s now 0.8.
What we can take from this is that when the price of the underlying asset changes, the delta value of the option changes. So, we can conclude that Delta is a variable whose value changes in relation to the changes in the price of the underlying asset and the option premium.
Can you tell that Delta is very similar to velocity?
What is Gamma?
The Gamma of an option measures the change in delta for the given change in the price of the underlying asset. Gamma helps us answer this question — “For a given change in the price of the underlying asset, what will be the corresponding change in the delta of the option?”
Before going any further, let’s draw some parallels to Delta and Gamma. Remember velocity and acceleration from high school? No? Get your money back! I’m kidding, here’s a quick recap.
Velocity is the rate of change in distance. (i,e the change in distance travelled divided by the change in time) . In the calculus world, velocity is called the ‘1st order derivative’ of distance travelled.
Acceleration is the rate of change of velocity (i.e. the change in velocity over time, which is, in turn, the change in position over time). Therefore, we can refer to acceleration as the second derivative of the position or the first derivative of velocity.
So what does any of what I just said have to do with option Greeks? Let’s get into that
First Order Derivative
We established that the change in distance travelled (position) with respect to the change in time is captured by velocity and that velocity can be described as the 1st order derivative of position.
Likewise, in terms of options, the change in the option premium price with respect to the change in the price of the underlying asset is captured by delta, and hence delta is called the 1st order derivative of the premium.
We established that the change in velocity with respect to the change in time is captured by acceleration and that acceleration can be described as the 2nd order derivative of position.
Likewise, in terms of options, the change in delta with respect to changes in the price of the underlying asset is captured by gamma, hence gamma is called the 2nd order derivative of the premium.
I’m sure you can tell by now that calculating delta and gamma values for options requires a lot of number crunching and lots of calculus (i.e differential equations). But don’t worry we won’t get into that,
Fun fact: as we all already know, derivatives are called derivatives because derivative contracts derive their value based on the value of the respective underlying assets. This value that the derivative contracts derive is calculated using the application of “derivatives” as a mathematical concept. Now you know why futures and options are called “Derivatives”.
We have established over and over again that the delta of an option is variable and it constantly changes its value relative to the change in the price of the underlying asset.
Let’s take a look at the graph from the previous article representing Delta vs Spot
As you can see by observing the graph, the delta changes with the spot price.
Okay; but why does this matter? To answer that, we will have to ask ourselves a couple of questions:
- Why does a change in the delta value matter?
- How exactly can we estimate the likely change in the delta value?
Let’s tackle the second question first:
The Gamma also referred to as the curvature of the options is usually expressed in deltas gained or lost per one-point change in the underlying asset price — with the delta increasing by the amount of the gamma when the price of the underlying asset rises and decreasing by the amount of gamma when the price of the underlying asset falls.
Let’s Take A Look At Some Scenarios:
Let’s work it out:
Evidently, when the price of HakhoToken moved from $8,326 to $8,396, the $8,400 call option premium changed from $26 to $47, and the delta of the option changed from 0.3 to 0.475.
With the change of 70 points, the moneyness of the option transitions from slightly OTM to ATM. Which would mean, theoretically, that the option’s delta has to change from 0.3 to a value around 0.5. As we can see, this is what happened here.
Let’s go a step further. Suppose HakhoToken moves up another 70 points from $8,396 to $8,466. What would happen with our $8,400 call option (Option Y)? Well let’s see:
Okay, let’s look at one more scenario. Assuming HakhoToken falls by 50 points, what would happen with the $8,400 call option? Let’s work it out:
Speed (Gamma of Gamma)
You may be wondering why we kept the value of gamma constant in the previous examples while, in reality, the value of gamma changes along with the change in the price of the underlying asset.
This change in gamma due to changes in the price of the underlying asset is captured by a third derivative of the underlying asset called “Speed” or “Gamma of Gamma” or “change in gamma divided by the changes in the price of the UA”. However, we won’t be diving into that right now
Moving on, as you have likely noticed, unlike the delta value, gamma is ALWAYS a positive number, for both call and put options.
You may have heard traders use the term “I’m long gamma” this simply means the trader is longing options (both Calls and Puts), or conversely trader may use the term ‘short gamma” when they are shorting options (both calls and puts)
As usual, let us take a look at some examples:
Picture this, the Gamma of an ATM put option (Option A) is 0.004. If the price of the underlying asset moves 10 points, what do you think the new delta is?
Think about it…
Enough time thinking, let’s explore the solution.
First of all let us recall that for ATM options the delta value is (.5) and for put options, the delta value is always negative. Therefore, the delta value of Option A is (- 0.5). We already established that gamma is always a positive number regardless of the option type. Now the scenario says the underlying asset price moves by 10 points, but it doesn’t specify the direction, so let’s just look at both scenarios.
Underlying asset price increases by 10 points
Underlying asset price decreases by 10 points
That’s a lot of numbers today. Thank you for sticking around!
In part 2 of the Gamma episode, we will go a little more in-depth and hopefully, you will be left with an above-average understanding of gamma.
As usual, if you have any questions do not hesitate to reach out.
Glossary of terms
Underlying asset — The underlying asset, the price of which is being speculated on, for example, Bitcoin.
Expiry date — The date the option will expire and be exercised, after this date, the contract is no longer valid.
Strike price — The price at which the buyer has the right to buy or sell the underlying asset at expiry.
Option price (premium) — The price the buyer pays to the seller for the right to buy or sell the asset at the strike price on the expiry date.
In the money (ITM):
- For a call — this term is used when the strike price is lower than the current price of the underlying asset.
- For a put — this term is used when the strike is higher than the current price.
At the money (ATM):
- For both a call and a put — this term is used when the strike is equal to the current price.
Out of the money (OTM):
- For a call — this term is used when the strike price is higher than the current price of the underlying asset.
- For a put — this term is used when the strike is lower than the current price.
All options on Dopex are European style, which means they can only be exercised at expiry, unlike American style options that can be exercised any time until expiry
Dopex is a decentralized options protocol that aims to maximize liquidity, minimize losses for option writers and maximize gains for option buyers — all in a passive manner.
Dopex uses option pools to allow anyone to earn a yield passively. Offering value to both option sellers and buyers by ensuring fair and optimized option prices across all strike prices and expiries. This is thanks to our own innovative and state-of-the-art option pricing model that replicates volatility smiles.
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