Asteria Essentials #5 — Options Gamma

👋Hello Asterians,

We’re back with another essential concept under the options-hood.

Following up on the segment of Greeks, today we’re working with Options Gamma.

Warning- this blog post does involve a bit of number crunching (ok, a lot of numbers!). Just so you are prepared.

Summary of Options Delta

In the last edition (#4), we learned in-depth about one of the five Greeks — Options Delta.

Here are some pointers from the last segment since some concepts from that last blog do trickle into this one:

We are now aware that Delta measures the change in the option’s premium price relative to the change in the underlying asset price.

Let’s jog your memory back with an example in Asteria Style,

Let’s say that the price of 🥔token is $90. This means that if there is a call option (called Option A) priced at $92, it’s certainly out-of-the-money (meaning the option’s underlying price is trading below strike price). So the value of Delta for Option A is ~0.2.

Let’s say that price of 🥔token appreciates by $3, coming up to $93 the same day. This means Option A is no longer out-of-the-money, and it has become somewhat in-the-money (the option’s market price is above the strike price). This means that the Delta will change as well, moving anywhere between 0.5 to 1. For the sake of this example, let’s just say that the Delta is ~0.8.

Key Takeaway:

From the example above, it can easily be seen that if the value of the underlying asset (🥔token) changes, the Delta value changes as well.

Hence, Delta is a variable value relative to a) Price of the underlying asset and b) the Option’s premium.

Now let’s get back to the main topic of this post -

Options Gamma — What is it?

Gamma measures Delta’s sensitivity to a $1 movement in the underlying asset price, and it is identical for both call and put options.

In simpler terms, the Gamma measures the change in delta for a given change in the price of an underlying asset.

Question- So — How will I measure the delta for x change in the price of 🥔token?

Answer - Through Gamma

Corresponding Delta with Gamma

In order to gain a more practical understanding of Gamma, we’ll have to draw some parallels between Gamma and Delta. And for that, we’ll have to touch base with some high school concepts like velocity and acceleration.

Delta is a first-order Greek/Derivative, while Gamma is a second-order Greek/Derivative.

While that statement might cause some confusion, it is a very simple analogy:

  • Change in option’s premium captured by delta — 1st order derivative
  • Change in delta captured by Gamma — 2nd order derivative

Now that we know Delta is variable and its change is relative to the change in the underlying asset price.

To progress to the next concept stage, please pay attention to the graph below, which represents the Delta vs. Spot price.

⚠️ Warning! Calculations ahead!

Let us ask — why does delta change even matter? And how do you calculate the potential change in Delta? (that question is enough to make you question why you’re even reading this article in the first place, right? Don’t worry, everything you learn here matters!)

Let’s answer the 2nd question first,

The value of Gamma is simply telling you how fast the Delta will move in the case the underlying asset experiences a $1 oscillation. Let’s assume we have an ATM call option on 🥔token with a Delta of +0.5 while futures prices are moving around $100 and Gamma is 0.08. What does that imply? The interpretation is rather simple: a 0.08 gamma tells us that our ATM call, in the case of the underlying moves by $1 to $101, will see its Delta increasing to +0.58 from +0.5.

Gamma is also called an option’s curvature, and you will be expressing Gamma as — deltas lost or deltas gained per unit change in the underlying asset price. Let me explain it through a scenario:

Let’s use a scenario to ponder better:

Example 1

🍅token is trading at $8,300 with a strike price of $8,400. So we can say that the call option for 🍅token is kind of out-of-the-money.

  • Option premium: $30
  • Option Delta: 0.3
  • Option Gamma: 0.0025
  • Change in underlying asset price: $70
  • New price: $8,300 + $70 = $8,370

— Assignment: So now we have to calculate the new option premium, new delta, and new moneyness.

  • Change in options premium

= delta x change in underlying asset price

= 0.3 x 70

= 21

  • New option premium

= 22 + 26

= 47

  • Gamma [Rate at which delta changes]

= by 0.0025 for every $1 change in the price of an underlying asset

  • Change in option delta

=Gamma x change in underlying asset price

= 0.0025 x 70

=0.175

  • New option delta

= original delta + change in option delta

= 0.3 + 0.175

= 0.475

  • New option moneyness

= At-the-money [ATM], meaning option strike price is identical to the current market price of the underlying asset.

Trick question

Here’s a trick question for you — if the change in option’s premium is calculated by delta, and change in delta is calculated by gamma. What calculates the change in delta!?🤯

The change in gamma caused by the change in the underlying asset’s price is calculated by ‘Speed’ — the 3rd derivative.

Let’s work on it with another example,

EXAMPLE 2

We have an at-the-money put option, and the gamma for this put option is 0.005. If the price of the underlying asset fluctuates and increases by 20 points. What will the new delta be?

So,

  • Delta: -0.5 (for ATM options, the delta value is always as 0.5, and for put options, that value is always in the negative. )
  • Gamma: 0.005 (gamma of any option is always positive)
  • Change in underlying asset price: + $20

Change in options delta

= Gamma x change in underlying asset price

= 0.005 x 20

= 0.1

New option delta

-0.5 + (0.1)

= -0.4

So that is how you’d calculate the change in delta if you had gamma.

And that will be it for today! We know there were a lot of numbers in this post, but we also hope that we were able to take away some practical knowledge.

Thank you for sticking so far. We’ll be back soon!

The future is Options. The future is Asteria.

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