Gamma (Option Greeks)- One of the four Greeks.
Today, I’ll talk about gamma, which helps us understand how our option positions are expected to change, based on changes in specific environmental factors such as changes in stock price over time and changes in implied volatility.
I’ll focus on Gamma. It’s an option metric that tells us how an option delta is expected to change as the stock price moves.
I suggest you learn about option delta first, then go straight back to this post so everything here makes more sense.
While an option delta indicates how much the option price will change if the stock price changes by $1,
The options Greek gamma indicates how much the option delta will change if the stock price changes by exactly that $1.
I know this is confusing,
So let’s go through a series of examples, and by the end of this post, you’ll know very well what gamma represents.
As I’ll go through numerous examples to demonstrate each of the concepts I’ll discuss. If you like to read more about options trading (Greeks) check those other posts I wrote
Delta Options Greeks, The best options trading strategies and Selling put options for Income.
Table of Content
- Gamma ( How it works )
- Calculation Delta Formula – Gamma option Greeks
- Google Stock Example
- RISK Gamma
- TLDR
Gamma ( How it works )
Let’s consider different options with different delta and gamma values expected,
if the stock price increases by $1 and if the stock price decreases by $1.
For example, let’s consider a call option with an initial
Delta = 0.5
Gamma of 0.05
If the stock price increases by $1, the delta of the call option is expected to be 0.55,
and if the stock price decreases by $1, the delta of the option is expected to be 0.45.
So, the delta of the option is expected to change by the amount of gamma,
and in this case, the gamma at the beginning of the option is 0.5, which means that the price will change up or down by $1.
and the share price is expected to change the delta of the option by the amount of gamma, which is 0.05.
Another scenario is if we have a put option with an initial
Delta = – 0.35
Gamma = 0.03
then the delta of this put option is expected to change to negative 0.32 when the stock price increases by $1, while the delta of this put option is expected to change to negative 0.38 when the stock price decreases by $1.
so in this scenario, the put option’s delta is expected to change by the amount of its gamma, which is 0.03
Calculation Delta Formula – Gamma option Greeks
The expected delta value of an option after a $1 increase in the stock price is equal to the delta of the initial option plus the gamma of the option.
On the other hand, to calculate an expected option delta after a $1 decrease in the stock price,
we take the delta of the initial option and subtract the option gamma from this value to get the expected option delta value after a $1 decrease in the stock price.
Since call options have positive deltas and positive gamma,
the deltas of the call options will increase and approach 1 as the stock price increases,
they will approach 0, which means that they will fall when the stock price falls.
since put options have negative deltas and positive gamma, the opposite is true.
for put options, the delta increases when the share price increases
This means that the option’s delta gets closer to 0 as the share price falls.
the put option’s delta becomes more and more negative, approaching 1.
This means that as the share price falls, the put option deltas fall to more negative values and approach 1.
Gamma explained
Changes in option deltas in rising and falling stock prices.
All option deltas increase after a rise in the stock price and decrease after a fall in the stock price.
However, since put options have negative deltas,
adding gamma to the negative put deltas means that the put deltas actually get closer to 0.
This means that their prices are less sensitive to future stock price movements when the stock price increases.
The opposite is true for call options
The deltas of call options grow towards 1 as the stock price increases,
which means that their prices become more sensitive to future stock price movements when the stock price increases.
Google Stock Example
Let us look at the changes in a call option and a put option on exactly the same Google stock price,
as Google’s stock price changes over time.
Imagine the change in the stock price and the strike price of the options.
The strike prices are: call $120 and put $120
As the stock price of Google fluctuates over time, we can see
Google’s share price is declining throughout the period.
At the beginning of the period, the price of Google was at
$130 and fell to $105 by the end of the period.
This means that the 120 calls started in the money and were out of the money at the time of expiration.
The 120 Put started out of the money and ended in the money at the end of the period, which means that the option expired in the money.
As explained earlier, a decrease in the stock price causes the call delta and the put delta to decrease.
The call delta moves towards zero and the put delta moves towards negative 1.
This means that in this example, the price of the call option becomes less sensitive to changes in the share price over time because the delta of the call option shrinks, i.e. it approaches more and more the value 0.
while the put option price was more sensitive to changes in the stock price, as the put option delta became significantly more negative when the stock price fell.
How can we intuitively understand these changes in an options Delta?
the options delta is used as an estimate of the probability that this option will expire in the money.
if we consider a call option with a delta of 0.55, this call option has an estimated 55% probability of expiring in the money.
if we consider a put option with a delta of -0.35, this put option has an estimated probability of 35% to expire in the money.
Since Gamma ( option greeks )estimates how much an option’s delta will change with a $1 movement in the stock price.
gamma essentially tells us the change and the option’s probability of expiring in the money if the stock price changes.
gamma estimates the change and the estimated probability of an option to expire in the money if the stock price changes.
Visualizing the changes
First, Google is at $130 and we are looking at a call option with a strike price of $120.
Since the call option is $10 in the money at the beginning of the period,
it makes sense that the option’s delta is 0.75.
indicating a 75% probability that it will be in the money at expiration.
The stock price can fall $10 and the $120 call option will still be in the money at expiration.
On the other hand, the 120 put is $10 out of the money at the beginning of the period.
That is, the stock price must fall by more than $10 for the 120 put to be in the money at expiration.
so it makes sense that the put option delta is negative 0.25,
indicating an estimated 25% probability of being in the money at expiration.
If the stock price falls, we can see that the delta of the calls falls from 0.75 and approaches zero,
indicating a lower probability of being in the money at expiration.
For the put options, the delta increases from negative 0.25 to a value closer to 1,
indicating a higher probability of expiring in the money.
In our example, the delta of the calls is 0, while the delta of the puts is 1.
Probability Point of View
This implies an estimated 0% probability that the call will expire
In-the-money and an estimated 100% probability that the put will expire in-the-money.
These values should make sense because as we approach the expiration date, the call is $10 to $15 out-of-the-money and the put is $10 to $15 in the money.
The stock price would have to rise very sharply in a short period of time for the call option to be in-the-money and the put option to be out-of-the-money.
These changes are unlikely, and therefore these options have a very high probability of expiring out of the money or in the money.
these changes make sense because when the stock price rises, call options at each individual strike price have a higher probability of expiring in the money.
While put options have a lower probability of expiring in the money at each individual strike price
The opposite is true when the stock price falls, if the stock price falls,
call options have a lower probability of expiring in the money with each individual strike price.
While put options have a higher probability of expiring in the money at each and every strike price.
This explains why, when stock prices fall, the deltas of call options are closer to 0 and the deltas of put options are closer to negative 1.
RISK Gamma ( option Greeks)
You may have heard options traders sometimes talk about gamma risk and gamma risk.
The term “gamma risk” essentially refers to the increase and sensitivity of an option position to changes in the stock price as the stock price changes over time.
gamma risk refers to the increase in the delta value of an option or option position or the sensitivity to changes in the share price as the share price changes over time or as time passes.
a larger delta value of your option position means that your position will be exposed to larger price fluctuations or profit and loss fluctuations when the share price changes.
As an options trader, or especially as a seller of an option, you usually do not want large fluctuations in the profitability of your position.
The highest gamma options are options with strike prices close to the stock price and options with very little time to expiration.
and at the same time have strike prices close to the share price.
TLDR
Do you really need to worry about gamma risk or is that a bit of an exaggerated issue?
if you trade short-term options, I think gamma risk is a real threat to you.
As I just described, you trade options with very little time to expiration.
These options have strike prices close to the stock price, and if the stock price changes, the value of your option position will change significantly.
Of course, if the stock price moves against you, that’s not a good thing, because the losses can add up pretty quickly.
But if you are someone who closes out your option positions a few weeks before expiration and never holds options close to expiration, then you really do not have to worry about the gamma risk that options traders talk about.
At the end of the day, if the stock price moves against you, you will experience changes in the PNL of your position, but not as much as if you were trading short-term options and those options had strike prices close to the stock price
if you have any questions or comments about this post please leave a comment below, I will get back to you as soon as possible and would love to hear from you.
I hope you found this post helpful and now have a better understanding of option gamma, which is one of the four primary option Greeks.