May 24
Options trading part 2: delta hedging
A professional options trader/dealer or market maker employs an option strategy to control his profit and loss with the hope of gaining a “theoretical edge.”
The “theoretical edge” is the amount of predicted profit based on analyzing current market circumstances and numerous risk variables. There are numerous different risk considerations.
This article explains the delta of options and how to hedge them. The delta is one of the option’s Greeks, and it measures the rate of change of the option price with respect to the underlying asset.
The delta Δ can be used to determine how much an option’s price will change in response to a change in the underlying asset's price. Delta hedging is a strategy used to mitigate the risk associated with an option’s delta.
See part 1 of my options trading article
You have learned in school, on television, or YouTube how to visualize atoms, protons, neutrons, electrons, etc.
This model is entirely inaccurate, yet we use it because it helps us visualize the specifics of these abstract subjects.
Consider everything in this article to be an oversimplification to assist you with more advanced reading about options trading
Disclaimer: Please don’t start delta hedging immediately after reading this. There are multiple risk considerations.
When buying an option, you also need to consider Theta! Your option will lose value over time.
When shorting an option you need to consider Gamma risk. The magnitude of a move
Which we will discuss in part 3
part 3:
https://romanornr.medium.com/options-trading-part-3-gamma-curvature-risk-9f6dd4b3db5b
Options Delta Δ
The Greek option “delta Δ” is known as “directional risk.” The symbol to denote “Delta” is “Δ.”
Every options contract has a delta
between 0 and 1
or 0 and -1
If our call option contract has a delta of 0.5 Δ,
our call option contract will gain a $0.50 increase in value for every $1 that the underlying moves up.
The option greek “delta Δ” measures the change in the price of an option contract relative to the change in the asset's spot price. The symbol to denote “delta” is “Δ.”
The textbook definition: “The delta (Δ) is a measure of an option’s risk with regard to the direction of the change in the underlying contract. A positive delta indicates a desire for upward movement.”
The graph again reveals that the “call option” is more valuable when the “spot price” is higher. The “hockey stick” payment indicates the option’s “intrinsic value,” which is the value of the underlying “spot price” upon expiry.
However, if we add the “time value,” the curve above the hockey stick represents the overall value of the option contract.
Delta Probability
Suppose we ignore the delta signs (positive for call options, negative for put options). In that case, the delta is approximately equal to the probability of the option finishing “in the money” (ITM).
A call option with a delta of 0.3Δ has an approximately a 30% chance of finishing “in the money” (ITM).
Aput option with a delta of -0.3Δ has an approximately a 30% chance of finishing “in the money” (ITM).
Assuming that markets fluctuate at random, there is a 50/50 chance that the market will rise or fall. As the delta approaches 0, the probability that the option will expire “in the money” decreases.
That also explains why “at the money” options often have a delta around 50.
Of course, the delta is only an approximation of the probability because interest rates and dividends may distort this interpretation.
If a trader buys a call option with a delta of 0.1Δ, he could be correct, but 9/10 times, he may lose
Out the money (OTM) “Delta” example
When the stock is “out of the money,” the option will have a near-zero slope, and the “delta” will also be close to zero.
The shallow slope of the change in the “call option” value indicates that they are not very sensitive to underlying asset price changes.
Delta measures this sensitivity, and the low delta value of call options indicates that they are not very sensitive to underlying price changes.
The slope gets steeper when one is “out of the money.”
In the money (ITM) “Delta” example
If the underlying asset moves to a point where the “call option” contract is deeply “in the money,” then the option's total value will no longer change based on small movements in the underlying asset.
The option will have a delta of 1, meaning that it will move dollar for dollar with the underlying asset, which is the same delta 1 position as if you hold a spot position.
At the money (ATM) “Delta” example
If we examine an “At the money” call option, the “strike price” represents the point of greatest uncertainty.
At “the strike,” the slope is flatter than when it’s entirely “at the money.”
Your “delta Δ” is 0.5 as the price might go in either direction.
These examples demonstrated “call options,” which always have a positive “Delta” value. This curve’s slope will always be positive, between 0 and 1.
The payoff associated with “put options” is on the downside.
This graph shows that the slope is negative. That indicates that the change in the value of a “put option” contract in response to a change in the “spot price” is negative.
As the “spot price” increases, the “put option” value decreases. It’s a “bearish” position to be “long puts,” resulting in a negative “delta Δ.”
Our “delta Δ” is 0 when we are out “Out the money.”
Our “delta Δ” is -0.5 when we are out “At the money.”
Our “delta Δ” is -1 when we deeply “In the money.”
“Call option” contracts always have a positive delta Δ
“Put option” contracts always have a negative delta Δ
You can also use an option contract’s “delta Δ” practically to estimate the probability of an option contract expiring “in the money.”
For an “At the money” option contract, the “delta Δ” is around 0.5. That is because it’s a 50/50 chance of whether it will be “in the money” or not.
Call option — delta Δ
“Out the money” call option contract it’s near 0 Δ
“At the money” call option contract it’s near ~0.5 Δ
“In the money” call option contract it’s near ~1 Δ
Nonetheless, you can see that the “delta Δ” of the call option contract does not change linearly but instead is non-linear.
Examining how the slope becomes steeper as we approach “in the money,” we see that the increase gradually begins. The pace begins to accelerate only once the “strike price” has been reached.
As soon as we are deeply “in the money,” acceleration slows down.
The red curve in this diagram represents the different values of “delta Δ” for different “spot prices,” and resembles an “S” curve centered on the “strike price.”
The “delta Δ” is a measure of the rate of change of the option contract price with respect to the underlying asset price (ie spot), so this curve can be used to visualize how the option contract price will change as the underlying asset price changes.
Notice that the “delta Δ” starts relatively low, close to 0.
However, the “delta Δ” starts to climb and accelerates; the velocity increases. The option contract value will only change as the underlying asset price changes
However, the “delta Δ” starts to climb and accelerates; the velocity increases.
When the underlying asset price (spot) is close to the “strike price”, that’s when the “delta” is 0.5 and achieves its maximum velocity.
However, the rate at which the “delta Δ” changes will start to slow down as the option becomes more and “more in the money” and reaches a limiting value of 1
Now let’s examine the “delta Δ” profile of a “put option”
Imagine we are trading a Tesla put option with a “strike price” of $700. If Tesla is trading at $700 (“At the money”), then the “delta Δ” is -0.5.
If Tesla’s share price is far higher, say $900, the “delta” is close to 0 since we are “out of the money.”
If Tesla shares are trading for $100 and are plummeting, then the “delta Δ” is close to -1 as we are getting deeply “in the money” for this put option contract.
The “delta Δ” of a put option decreases from 0 to -1 when the stock price falls and the option becomes “in the money.”
The shape of the “S” curve for a put option is similar to that of a call option, except that the delta is negative. Put options have a negative delta Δ
You can see how the “delta Δ” moves, and it has acceleration as the share price moves down.
As we approach “strike price” (at the money), “delta Δ” moves the most rapidly.
When the “put option” is deeply “in the money,” the “delta Δ” decreases in velocity and reaches a maximum of -1.
Delta Δ on the option chain (Deribit example)
You have already seen an options chain in a previous article.
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 will examine the options chain for the Deribit exchange. This options chain has ‘call options’ on the left, ‘put options’ on the right, and “strike prices” in the center.
The “bid” and “ask” columns represent the prices at which someone is willing to buy or sell the option. However, we will now focus on the “delta Δ” column.
Notice that the current bitcoin market price is $30171.
We will look at the “delta Δ” column of the left-hand “call options” table.
We need to look at a “strike price” that’s the same or near the “spot price.”
If we examine a “call option” with the same “strike price,” we find that the “delta Δ” is 0.56, which is close to 0.5, as expected for an “at the money” call option.
If we examine a “put option” with the same “strike price,” we find that the “delta Δ” is -0.46, which is close to -0.5, as expected for an “at the money” put option.
If bitcoin moves up by $1, the “call option” will increase by 0.56.
If bitcoin moves up by $1, the “put option” will decrease by 0.46
That’s because the “put option” has a negative “delta Δ,” and the call option has a positive “delta Δ.”
Now, if we examine options with different “strike prices,” we see that the “delta Δ” is decreasing for “call options” with higher “strike prices.”
Now for the very same “put options” with higher “strike prices,” that would be “in the money,” those have a “delta Δ” greater than 0.5, so you can see that the “delta Δ” for a “strike price” of $39k has a delta of -0.98
For lower “strike prices,” the “delta Δ” of “put options” decreases.
At a “strike price” of $24k, that’s when we have a negative “delta Δ” of -0.07, that’s a “put option” that’s “out the money.”
If we look at a “call option” with a “strike price” of $24k, we see that it has a positive “delta” of 0.93.
A call option “in the money” has a “delta Δ” greater than 0.5.
The options chain with varying “strike prices” demonstrates what we’ve learned.
Option contracts that are “out of the money” have lower “delta’s Δ,” while option contracts that are “in the money” have higher “delta’s Δ.”
Trading platform: Delta exchange
An alternative to Deribit for altcoin option contracts is “Delta exchange” which has options trading for multiple altcoins. You could trade these options contracts to hedge your portfolio for altcoins.
At the moment we see bitcoin, ethereum, XRP, Solana, Avax, Matic, BNB & LINK
If you’re seeking to signup and want a 10% discount
You can use my referral link
What is Delta Hedging? Δ
Numerous options traders and market participants use “delta hedging,”—particularly market makers and liquidity providers, including dealers and banks.
Instead of just trading an “option contract” and leaving the “delta Δ” exposure unchanged, they trade shares against their “option position” to hedge their “delta exposure.”
Market makers hedge their inventory for a certain asset, position, or shares while also having an equal and opposite hedging position in the market to protect against unfavorable price swings.
For example, if you are a market maker in gold, you would buy gold for your inventory and either buy “put options” or sell “call options” to hedge against downside risk.
A trader in options may construct a " delta neutral position.” The position’s value will remain constant while the underlying asset fluctuates within a narrow price range. If the underlying asset moves a dollar in either direction, the position value does not change much.
However, the delta is not static, and it can change.
So, when a sophisticated options trader trades option contracts, They will try to neutralize their “delta Δ” by buying or selling shares in the correct amounts. Due to buying/shares constantly to get delta neutral, they will gain profits that will offset the loss of the option contract.
Now, the trader is exposed to risk parameters which are the “greeks” of the option contract rather than the stock price direction.
As mentioned before in the previous article
The appeal of options trading is the “leverage” they provide.
Since 1 option contract controls 100 shares of the underlying asset, buying a call option contract exposes the gains and losses of 100 shares at a fraction of the price of 100 shares.
So remember that 1 option contract is the right to buy 100 shares.
That is the “multiplier” (standardized at 100)
So, 5 call option contracts would be the right to buy 500 shares.
Let’s retake the example of Tesla shares, which trade at $700.
We buy 5 “at the money” call option contracts, which gives us the right to buy 500 shares.
amount of options * multiplier = amount of shares5 * 100 = 500 shares
To calculate the notional value of that position:
amount of options * multiplier * stock price = notional value5 call options * 100 * 700 = $350 000
Remember our call option contracts are “at the money,” with a “strike price” of $700, so the “delta” for this position is 0.5 Δ
How to delta hedge my options contracts?
So now we will “delta hedge” our 5 “at the money” call option contracts which have a delta of 0.5Δ
Since our option position has a positive “delta,” we have to sell shares since doing so will have the opposite effect on the option’s delta.
We will neutralize the delta to 0.
This calculation determines how many shares we must sell to neutralize our delta to zero.
amount of options * multiplier * delta = amount of shares to sell 5 * 100 * 0.5 = 250 shares
By selling 250 shares, we are “delta hedged” or “delta neutral.”
We can calculate the notional value is of our delta hedge
amount of shares * price = notional value
250 * 700 = $175 000Which is 0.5 of the notational value of the option position.
Earning money with delta-neutral trading with options trading
Delta hedging a call option & making money on it
We buy 1 “At the money” Tesla 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 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? By selling the number of shares that the option exposed us to.
So for our next magic trick, we sell 50 shares.
Now we sell 50 shares at $300 to get delta neutral again to bring our delta to 0Δ. Our new delta after the trade is 0 Δ.
If the price moves up to $350, our option exposes us to a delta of 0.7Δ!
But we reduced our delta by 0.5 in our previous trade, so we are left with still a net delta position of +0.2 Δ
Why? We sold 50 shares of our book, our delta increased by 0.7Δ which is 70 shares
-50 + 70 = 20 shares or delta 0.2Δ
We need to get rid of that 0.2Δ, and we can do that by selling 20 shares to get back to delta-neutral. Let’s execute that trade. See diagram below
Now, if the share price of Tesla moves down to $300, our option exposes us to a delta of 0.5Δ again.
We had that initial hedge by selling 50 shares and sold another 20 shares, so our current delta is -0.7Δ
Our net delta of 0.5 from the previous trade combined with the delta of -0.7 leaves us having a “short” exposure of 20 shares.
0.5 – 0.7 = -0.2ΔWe need to fix that since we are now net short, and we can do that by rebuying back 20 shares.
Now, if the price of Tesla shares falls to $250, our call option is “Out of the money,” and our option delta becomes -0.3Δ
See the diagram below. Our new delta gave us a total position of “short” 20 shares.
We have a “short” exposure of -0.2 Δ, but we get back to delta neutral by repurchasing 20 shares.
If Tesla’s share price recovers to $300, our option delta will equal 50 shares, or 0.5Δ. The remaining exposure is 20 shares
Let’s look at our book
-50 (sell shares)
-20 (sell shares)
+20 (buy shares)
+20 (buy shares)
______
-30 shares shortThe call option is delta is 0.5Δ which is equal to +50 shares long-30 + 50 = 20 shares long (delta 0.2Δ)
To remain delta neutral again, we need to sell 20 shares again
Now we are delta neutral again, our delta is 0 Δ, and the price of Tesla is back again at $300
We are back at square one, and you may wonder.
What was the point of doing this?
The price has returned. However, don’t forget that we executed trades to maintain delta neutrality. Our total profit is $2000 by continual hedging to maintain a delta neutral position.
See the diagram below.
Even if our call option contract would expire “Out of the money” and our options contract would expire worthless, due to the trades we made, the total profits are more than the loss on the premium of the call option contract.
I know this looks pretty complicated, but on an exchange platform such as Deribit, it’s easier to understand because they show you your net delta since they show that.
If your option gives you a negative delta exposure, you know that you have long BTC-perp to neutralize your delta. You won’t need a pen and paper to write down a table to recalculate your new delta of the option.
So don’t worry. You won’t need a pen and paper to calculate your delta and do mental maths.
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
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.
How to delta hedge a put option & earn money
We buy 1 “At the money” Tesla “put option” contract with a strike price of $300. Remember 1 option gives us the right to trade 100 shares. In this case, sell a 100 shares
Our option shares is our “delta” multiplied by the multiplier (100)
-0.5 * 100 = -50 sharesNow, this “put option” contract leaves us with an exposure a delta of 0.5Δ.
We need to neutralize our delta to zero. How do we do that? This time we need to buy the number of shares that the put option because it gave us a “net-short” position of 50 shares
So for our magic trick, we buy 50 shares this time.
If the price moves up to $350, our option exposes us to a delta of -0.3Δ
Our delta hedge was to buy 50 shares, so we have 50 shares on our book. Since the put option exposes us to 30 shares short
Let’s look at our book
+50 (buy shares)
______
+50 shares longThe put option is delta is 0.3Δ which is equal to 30 shares short50 - 30 = 20 shares long or delta 0.2Δ
Now we have a positive delta exposure of 0.2Δ and to get back to delta neutral and bring our delta back down to 0Δ, we need to sell 20 shares
Now, if the share price of Tesla moves down to $300, our option exposes us to a delta of -0.5Δ again.
We had that initial hedge by buying 50 shares and selling another 20 shares,
50 - 20 = 30 sharesNow we are back “At the money” at $300 which leaves us with an exposure of -50 shares short
30 shares and 50 shares short exposure which leaves us to -20 shares short exposureSo in total, we are “short” -20 shares which is a delta of -0.2Δ
Let’s look at our book
+50 (buy shares)
-20 (sell shares)
___
+30 shares longThe put option is delta is 0.5Δ which is equal to 50 shares short50 - 30 = 20 shares long or delta 0.2Δ
We need to fix that since we are now net short 20 shares, and we can do that by rebuying back 20 shares.
If Tesla shares drop to $250, our “put option” contract swings to “in the money”, and the delta exposure of the contract now has a delta of -0.7Δ
Our delta hedge is 50 shares right now because we repurchased 20 shares
+50 (buy shares)
-20 (sell shares)
+20 (buy shares)
____
50The put option is delta is 0.7Δ which is equal to 70 shares short50 - 70 = 20 shares short or delta -0.2Δ
Since our current delta is -0.2 we can get back to delta neutral by buying 20 shares again.
Now if Tesla's share price falls back to $300, back to “At the money” our “put option” delta is -0.5Δ
We have a 50-share short exposure and our previous trade, which leaves us with a delta of 0.2Δ
Since we have a positive delta of 0.2Δ, we need to get back to delta neutral and we can do that by selling 20 shares
We are back at square one, and you may wonder.
What was the point of doing this?
The price has returned. However, don’t forget that we executed trades to maintain delta neutrality. Our total profit is $2000 by continual hedging to maintain a delta neutral position.
I know this looks pretty complicated, but on an exchange platform such as Deribit, it’s easier to understand because they show you your net delta since they show that.
If your option gives you a negative delta exposure, you know that you have long BTC-perp to neutralize your delta. You won’t need a pen and paper to write down a table to recalculate your new delta of the option.
If your net delta is positive, you can delta hedge by shorting BTC-perp to the equal amount
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
Now you know how to delta hedge your position. It sounds like that because it is! It may sound too good and too easy to be true.
There’s another risk consideration, and that is Gamma or curvature risk.
But that’s for part 3 of the options trading article.
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
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.
Delta exchange
If you’re seeking to signup and want a 10% discount
You can use my referral link
I haven’t fully tested out their exchange yet, but they provide a lot of options for different altcoins, move contracts, and some exciting derivates contracts (which I need to study more)
Conclusion
Trading an option and buying/selling the underlying trades are made.
That’s what a lot of funds/dealers/market makers do. Keep trading to adjust their delta. If you buy a call option, you fill the sell order for that call of a market maker. The market maker is now stuck with a “short call option.” He must hedge that position by constantly delta hedging by buying/selling the underlying.
That’s what drives a lot of price action!
The options market keeps expanding, and maybe the options market drives a lot of price action.
The tail that wags the dog
You can continue reading options trading part 3: Gamma/Curvature risk
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