May 31
Options trading part 3: Gamma/curvature risk
Gamma, often known as the option’s “curvature risk,” is our second risk consideration for trading options and delta hedging with options trading.
“Gamma Γ” is the change in “delta” of an option contract for every dollar change in the underlying (i.e., spot). Gamma is the sensitivity of “delta Δ” relative to a change in the underlying.
The “delta” is dynamic. The option’s “delta” might alter over time. The “delta” changes when the underlying/spot changes.
As the underlying moves, the delta moves because of Gamma.
Gamma is the risk of a large move, regardless of direction. Gamma measures how much the delta changes in response to a change in the underlying asset’s price.
Follow this link if you haven’t read part 1 of this article series on options trading.
In part 2 of the options trading article, we covered “delta Δ” and “delta hedging.”
Introduction video:
Therefore, if the underlying (spot) has a large movement, the position whose delta is most sensitive to the underlying security (gamma) will experience the greatest change in value.
Gamma measures the option’s delta’s curvature. Gamma is positive when delta is concave (curving upward) and negative when convex (curving downward).
High gamma option contracts are more sensitive to underlying asset price fluctuations.
“Gamma Γ” is also known as the convexity of the delta.
“Γ” is the symbol used to represent “gamma.”
This all may sound a bit confusing, but it will make sense over time if you manage to read the whole article.
Don’t feel intimidated if you don’t understand it after the first time reading. It took me a while too.
You’ll probably need to read this and the previous articles multiple times over a few weeks or months. Don’t try to understand it within one day. “Sleep on it.” Sleep can help your brain form connections.
Let’s go over an example:
Tesla shares are trading at $700
We buy an “At the money” call option at $48 with a delta of 0.5Δ and a Gamma of 0.02 Γ
If the underlying asset increases by $1, the option contract value increases by $0.50.
That’s due to the option’s delta of 0.5Δ
If Tesla shares rise by $1, our call option will gain value too. How much?
delta Δ * dollar change in underlying = option value change in $0.5 Δ * $1 = option value change $0.5
Our call option increased by $0.50, so our call option is now worth $48.50
Now, what’s the new “delta Δ” of our call option now the price has changed?
The new delta of this call option is simply adding the Gamma of 0.02 Γ to the old delta of 0.5
old delta + gamma = new delta
0.5 + 0.02 = 0.52The “gamma Γ” is an addition to, it’s a derivative of “delta Δ,” it’s the addition to “delta Δ.”
Gamma measures the “speed” at which an option’s delta will change in reaction to a move in the underlying asset. Positive gamma means that the option contract will become more and more valuable at an increasing rate as the underlying asset goes up in value.
You can see that the share price of Tesla increased by ~0.143%
Our “call option” increased by ~1.036%
Let’s continue. Tesla shares are currently trading at $701, and the new delta of our call option is now $48.50 with a delta of 0.52Δ.
The price of Tesla stock increases again but by $1 and is now trading at $702
We can calculate the change for our call option again
delta Δ * dollar change in underlying = option value change in $
0.52Δ * $1 = $0.52Now our call option’s value increases by $0.52 instead of $0.50
Now the value of our call option contract is
$48.50 + $0.52 = $49.02Tesla's share price increased from $700 to $702, which is an increase of 0.285%
Our “call option” value increased from $48 to $49.02, which is an increase of ~2.1%
I’m not sure how great my audience is with maths and calculating percentages difference… I’ll provide a link to help: https://www.calculatorsoup.com/calculators/algebra/percent-difference-calculator.php
As you can see, this “leveraged return” is driven by gamma. Gamma is one of the primary reasons traders choose to buy options contracts instead of shares (or spot)
Understanding the dynamic connection between “delta” and “gamma” unlocks the full potential of options trading.
Delta value convention
Although delta value ranges from 0 to 1 for call options and 0 to -1 for put options
It has become common among options traders to express “delta Δ” values as a whole number. They drop the decimal point.
That makes it so that the
“delta” of a call option falls within a range of 0 to 100 Δ
“delta” of a put option falls within a range of -100 to 0 Δ
The underlying contract (ie spot) always has a delta of 1 or, using this convention, a delta of 100 Δ.
It’s worth mentioning it since you might hear traders say “50 deltas” instead of 0.5
(This I also mentioned in part 2)
Calculate Gamma
I mentioned before that the gamma of the option contract was 0.02Γ
We can calculate the gamma for an option contract if we don’t know the gamma.
- old delta
- new delta
- old price
- new price
Using delta convention.
So instead of 0.5Δ, we say 50 Δ( old delta Δ - new delta Δ ) / (old price - new price) = gamma Γ( 0.5 − 0.52 ) /( 700 − 701) = 0.02 Γ
(50−50.02)÷(700−701) = 0.02 Γ
Quantify gamma Γ
We can quantify “gamma Γ” as the change in “delta” for a 1% change in the underlying (i.e., spot)
An “at the money” call option typically has a delta of 0.5Δ or 50%.
If the underlying (i.e., spot) increases by 1%, and our delta rises from 50% to 53%
That would mean the call option has 3% “gamma” because the change in delta for a 1% change in the underlying is 3%
If the underlying asset (i.e., spot) falls 1%, the delta will go from 50% to 47% because that change in “delta” is 3%.
However, this time the delta is going down because the underlying is down. The gamma is 3%.
Since gamma refers to the rate of change in an option’s delta, it can be thought of as a measure of an option’s exposure to realized volatility.
In part 2 of the options trading article about delta hedging,
As we saw in the options trading article, the delta is not static but changes due to the options’ gamma.
Delta hedging allows us to take advantage of these fluctuations in the stock price by making trades to offset our delta. This way, we can earn money from the stock price movements without taking too much risk.
Long Gamma
Once an option contract is delta-hedged, the resulting gamma exposure is what drives changes in the “net delta” position. The more the market moves, the greater the potential profit from trading the gamma.
Long gamma exposure is the result of holding long options contracts. We can capitalize on the market volatility by selling the underlying asset when its price increases and repurchasing it when its price decreases.
The long “gamma” exposure comes from buying/long options contracts since we delta-hedge our option position by selling the underlying asset (i.e., spot) when the price rises and repurchasing it when the price falls.
We constantly delta-hedge to remain delta-neutral. We avoid the directional risk of the underlying asset moving.
If you buy an option, you are banking on rising underlying asset prices. That’s because the option’s price curve is convex. The option’s delta (a measure of the option’s price sensitivity to changes in the underlying asset price) becomes more positive as the underlying asset price rises.
As the underlying price increases, the “delta Δ” slope of the call option curve becomes more positive. As the price declines, the delta becomes more negative.
As the underlying price increases, the slope of the delta of a “put option” contract likewise gets more positive. Why?
Because the delta of the contract for this “put option” is turning less negative.
Gamma graph
The delta of an option indicates how much the underlying asset’s price will fluctuate in response to a change in the option’s price.
If you remember the “delta Δ” graph of an option, it was just an “S” curve.
The “delta” of an option is a measure of how the option’s price will change in response to movements in the underlying asset’s price.
However, “Gamma” measures how delta will change in response to the underlying asset’s price changes.
In other words, gamma measures the sensitivity of an option’s delta to changes in the underlying asset’s price.
Similar to how “delta “ was the slope of the option value curve, “gamma “ is the gradient of the “delta” curve since it indicates how much the delta varies in response to a change in the underlying (or spot) price.
The delta curve for a “out of the money” option is relatively flat. That’s because the option is still extremely “out of the money,” and as a result, minor market fluctuations have little effect on the delta.
When the option contract is “at the money,” the slope of the delta curve is at its maximum, indicating that the gamma is likewise at its maximum.
The “delta” changes the most when the option is at these prices.
The “gamma Γ” is the rate of change of the delta, which starts at a low rate.
The “gamma Γ” gradually increases as the underlying (i.e., spot) approaches the strike price.
Maximum “gamma Γ” is reached at the “strike price” or “At the money” (ATM)
Finally, it turns zero when the option becomes “deep in the money.”
The “gamma Γ” is zero when we are deeply “In the money” (ITM).
When an option’s strike price is reached, the gamma and delta change the most. That is because the option contract is transitioning from “out of the money” to “in the money,”
The likelihood of achieving “in the money” is greatest at the strike price.
Short Gamma
The seller is " short gamma when an options contract is sold, and the seller is “short gamma.” That implies that when the market changes, the convex curvature of the option curves will work against you.
Your “delta” will become more negative on rallies in the underlying (i.e., spot) and will “delta” increase if the underlying market sells off.
Let’s go over an example.
Before continuing, you may need to review part 2 of this article series on options trading to refresh your memory.
We short-sell just 1 Tesla “At the money” call option with a strike price of $300
Remember 1 option gives us the right to trade 100 shares. Our option shares is our “delta” multiplied by the multiplier (100)
0.5 * 100 = 50 sharesIn this scenario, our delta is not 0.5Δ but -0.5Δ since we are short-selling this call option contract, which gives us the opposite exposure.
-0.5 * 100 = -50 sharesNow, this call option leaves us with an exposure a delta of -0.5Δ.
We need to neutralize our delta to zero. How do we do that? We have to buy the number of shares that the short call option exposed us to this time.
We buy 50 shares
If the price moves to $350, our call option exposes us to a delta of -0.7Δ since we are short.
Let’s look at our book.
+50 (buy shares)
______
+50 shares The call option is delta is -0.7Δ which is equal to -70 shares short50 - 70 = 20 shares short (delta -0.2Δ)
Also, what was the gamma of that call option contract?
( old delta Δ - new delta Δ ) / (old price - new price) = gamma Γ
(-0.5 − −0.7) / (300-350) = -0.004 ΓWe are left with a net delta position of -0.2 Δ.
We need to get delta neutral again. We buy 20 shares this time to get back to delta-neutral
Now, if the price of Tesla shares moves back to $300, our call option delta is -0.5Δ
Let’s look at our book.
+50 (buy shares)
+20 (buy shares)
___
+70 shares The call option is delta is -0.5Δ which is equal to -50 shares short70 - 50 = 20 shares long (delta 0.2Δ)
We need to sell 20 shares to be delta neutral.
Also, what was the gamma of the option?
( old delta Δ - new delta Δ ) / (old price - new price) = gamma Γ
(-0.5 - -0.7) / (300 - 350) = -0.004 Γ
So by being “short-gamma” due to “short-selling” an option contract, it will require selling the underlying asset when its price falls and buying when it increases to hedge even though these are losing trades.
The trades used to hedge delta When you are “short” on an option, you will sell shares when the underlying price (spot) is low and buy shares when it is high.
When you short options and you want to delta hedge, ironically you end up
Buying high
Selling low 😆
If you purchase option contracts, your delta will be pushed in the correct direction, allowing you to profit from any market movement. In contrast, the reverse occurs if you are “short-selling” an option. Both directions fail.
So you may think
“Why would somebody choose to sell options contracts short?”
Why not constantly buy options and use delta hedging to make lucrative trades?”
Well
Buying options contracts has a drawback.
This is referred to as “Time decay” or “theta”
It would be best to execute these losing trades in the underlying (i.e., spot) or futures market to offset your exposure. If you believe your volatility estimation is correct and think you have some, a “theoretical edge” or the “time decay”/“theta” will work in your favor.
I will explain our next risk consideration, “theta,” in part 4 of these option trading articles.
FTX MOVE Contracts
For anyone familiar with my article about FTX MOVE contracts
If you still didn’t make the connection, allow me to explain
If you short an FTX MOVE contract, you are “short-gamma,”
if you long an FTX MOVE contract, you’re “long-gamma.”
Now you also know at which delta the gamma is at its highest ;)
If you long BTC-MOVE-WK-0603
Your delta: 0.771
Your gamma: 0.00014
Your theta: -159.92 (time decay will erode the value)To delta hedge: short 0.771 BTC
Adjust your short when delta changes
If you short BTC-MOVE-WK-0603
Your delta: -0.771
Your gamma: -0.00014
Your theta: +159.92 (time decay will work in your favor)To delta hedge: long 0.771 BTC
Adjust your long when delta changes
I shouldn’t have to explain this since it should have been obvious after reading this post, but I chose to do it anyway because some people may not be able to grasp and draw the connection/relationship.
Keep in mind that FTX MOVE contracts are option straddles. To trade FTX MOVE contracts, you must first understand options trading.
FTX 10% Fee Discount
My referral link will give you a 10% discount instead of the usual 5% discount for anyone who wishes to give it a go.
However, keep reading. The next topic in this article is Gamma & Convexity.
Gamma & Convexity
The contract’s gamma drives the “leveraged return” of an options contract.
Gamma is a measure of the rate of change of an option’s delta in response to changes in the underlying asset price.
The higher the gamma, the more sensitive the option’s delta is to changes in the underlying asset price.
Understanding the dynamic connection between “delta” and “gamma” unlocks the full potential of options trading.
Let’s look at a simulation.
Using Deribit tool: https://pb.deribit.com
You do need an account to use their tool.
If you don’t have an account on Deribit, you can sign up and receive a 10% discount on fees for trading futures & options: https://www.deribit.com/reg-572.9826
We buy 12 “call option” contracts with a strike price of $50k expiring July 29
This position has a total “delta” of 0.53Δ and a total “gamma” of 0.000146 Γ
The average price of this one contract is 0.005 BTC, but we have 12 contracts, so 0.005*12 = 0.06 BTC.
The current price of BTC is $30730
0.06 * $30730 = $1843.8Every time the underlying (BTC) moves, we must update our delta value by factoring in the gamma.
We saw this leveraged return in our previous example with the Tesla shares example. That return is not “linear” due to “gamma.”
The “Gamma” can drive this “convex.” increase in the option contract value. The value does not increase linearly but is “convex,” more like a curve up, quadratic.
That would give us a PNL of ~$60.2k
Now compare this if we bought opened a long “futures position” or “spot position” on bitcoin with the same delta value of 0.53Δ but without “gamma.”
You can see our PNL is not like a curve but linear. Our total PNL is around $10.2k
So you can see that the return of options due to gamma gives us a curved parabolic PNL.
However, as I mentioned before
Buying options contracts has a drawback.
This is referred to as “Time decay” or “theta”
In Part 1 of these options trading article series, I explained briefly “time value”/“time decay”/Theta.
Your option contract will either expire “in the money,” or it expires worthless. The amount of time you have to expire “in the money” slowly decreases, and the time decay is becoming larger, eroding your option contract’s value.
You can see that here.
If I use the slider to move a couple of days closer to expiration, you will see that the time will erode the option contract value.
An option derives its value from “intrinsic value” and “time value.”
Time value refers to the odds that the option contract will become “in the money” over the contract's lifetime.
The time value decays over time, meaning that as the option contract approaches expiry. Intrinsic value is whether the option contract is “in the money” or not.
We will discuss our next risk consideration (“Theta) when buying/selling options in part 4 of these options trading article series.
Buying options contracts and delta hedging is not “free money.” There’s a catch, which is the time decay or time decay.
Always remember, there’s no such thing as a free lunch.
However, keep reading. There’s a video about options contract and convexity.
Convexity bonus video
Bonus video which I found on YouTube, for those who gained a better knowledge of Gamma/curvature risk and already have a solid grasp of “convexity” and options trading.
Trading platform: Delta exchange
Another new exchange called “Delta exchange” has options trading for multiple altcoins. You could use these options to hedge your portfolio for altcoins. Here we can see the options chain for Avax.
If you’re seeking to signup and want a 10% discount
You can use my referral link
They also have a great tool to model your options position
It’s a great alternative if you can’t acquire access to Deribit due to your location.
ByBit exchange
ByBit recently offers options trading for bitcoin & ethereum. If you are trading on ByBit, you can use options to hedge with options.
ByBit Options (Discount on fees and $100 deposit bonus): https://www.bybit.com/register?affiliate_id=6776&group_id=1653&group_type=1
Looking for options trading parts 1 and 2?
part 1: https://medium.com/@romanornr/options-trading-fd4d0bffb2c5
part 2: https://romanornr.medium.com/options-trading-part-2-delta-hedging-9fa6f981883f
Final words
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Part 4 of options trading will be released soon
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If you liked this article, you will probably also love this article about FTX MOVE contracts.
