**Trigger Warning: Simple Mathematics**

**Trigger Warning: Simple Mathematics**

###### Welcome again to another Dopex Essentials Episode. In this episode, we are going to be taking an in-depth look into the Option Greek: Delta. In the previous article, we discussed Delta in a surface-level manner. If you have not read that article don’t worry, we have included a recap of what you need to know.

There is a glossary of terms at the bottom of the article so if you come across any terms you are not familiar with just scroll down to the bottom

###### We hope you enjoy this one!

# Beginning of Recap

## Delta

###### What is Delta?

## Delta is a measure of the change in an option’s price (the premium) resulting from a change in the underlying asset

###### The value of delta ranges from -1.00 to 0 for puts and 0 to 1.00 for calls. Puts generate a negative delta because they have an inversely proportional (negative) relationship with the underlying asset — that is, put premiums fall when the underlying asset rises, and vice versa.

###### On the other hand, call options have a directly proportional (positive) relationship with the price of the underlying asset. Provided that other variables such as implied volatility or time to expiry remain constant — the price of the underlying asset rises, and so does the call premium. Likewise, if the price of the underlying asset decreases, the call premium will also decrease, provided all other things remain constant.

## Analogy

###### Think of Delta like a race track: The tires = the delta. The gas pedal = the price of the underlying asset.

###### Options with low delta are like race cars with stock tires. When you rapidly accelerate, they won’t get a lot of traction. On the flip side, options with high delta are like drag racing tires. When you rapidly accelerate, they provide a lot of traction. Delta values closer to 1.00 or -1.00 provide the highest levels of traction.

## Example of Delta

###### Let’s suppose that there is Option X and Option Y, both have the same underlying asset.

###### Option X is an out-of-the-money option with a delta of 0.25

###### Option Y is an in-the-money option with a delta of 0.80

###### A $1 increase in the price of the underlying asset will lead to a $0.25 increase in Option X and a $0.80 increase in Option Y.

###### Options with high deltas are more likely to provide greater traction. Due to this, these options tend to be more expensive in terms of their cost basis since they’re likely to expire in the money.

###### Delta changes as options become more profitable or in-the-money. As the option gets further in the money, delta approaches 1.00 on a call and -1.00 on a put with the extremes eliciting a one-for-one relationship between changes in the option price and changes in the price of the underlying asset.

###### Basically what this means is that, at delta values of -1.00 and 1.00, the option behaves like the underlying asset in terms of price changes. This behavior occurs with little or no time value as most of the value of the option is intrinsic.

## Probability of Profitability

###### Delta is commonly used when determining the probability of an option being in the money at expiration. For instance, an out-of-the-money call option with a 0.30 delta approximately has a 30% chance of being in-the-money at expiration, whereas a deep-in-the-money call option with a 0.90 delta approximately has a 90% chance of being in-the-money at expiration.

###### The assumption made here is that the prices follow a lognormal distribution, like amounts of rainfall for example.

## Delta and Directional Risk

###### Delta is also used when determining directional risk. Positive deltas are long (buy) market assumptions, negative deltas are short (sell) market assumptions, and neutral deltas are neutral market assumptions.

###### Higher deltas may be ideal for more speculative, higher-risk, higher-reward strategies, whereas lower deltas may be more suitable for lower-risk strategies with higher win rates.

###### When you buy a call option, ideally you want it to have a positive delta since the price will increase along with the underlying asset’s price. Conversely, when you buy a put option, ideally you want it to have a negative delta since the price will decrease if the underlying asset’s price increases.

## Key Points:

- Delta is inclined to increase closer to expiration for near or at-the-money options.
- Delta is further evaluated by gamma, which is a measure of delta’s rate of change.
- Delta can also change in reaction to implied volatility changes.

# End Of Recap

###### Leading on from those key points, let’s keep a few more evaluations in mind:

- OTM options have a delta value between 0 and 0.5
- ATM options have a delta value of around 0.5
- ITM options have a delta between 0.5 and 1
- Call options have a positive delta
- Put options have a negative delta

# Let’s take a look at some scenarios:

**Asset:** “OhicToken”

**Price:** $8,312

**Option Type:** Call Option

**Option Name:** Option X

**Strike Price:** $8,400

**Asset:**

**Price:**

**Option Type:**

**Option Name:**

**Strike Price:**

###### I’m going to ask a question and put the answer right below, but for your own benefit, try to answer the question without looking at the answer.

**Question 1:**

###### What is the approximate delta value for the Option X when OhicToken is trading at $8,312?

**Answer:** In this case, Option X is OTM so the delta value should be between 0 and 0.5. Let us assume the delta is 0.4

**Question 2:**

###### Suppose OhicToken rallies from $8,312 to $8,400. What is the delta value of Option X now?

**Answer:** In this case, Option X is now ATM, so the delta value should be around 0.5

**Question 3:**

###### Now suppose, OhicToken rallies again from $8,400 to $8,500. What is the delta value of Option X now?

**Answer:** In this case, Option X is now ITM, so the delta value should be closer to 1. Let us say 0.8

**Question 4**

###### Assume, the market has a slight dip, and OhicToken drops from $8,500 back to $8,300. What happens to the delta value of Option X?

**Answer:** Due to the dip in the price of the underlying asset (OhicToken), Option X has changed from ITM back to OTM. Thus the delta also falls from 0.8 to say 0.35

**Final Question:**

###### What can we deduce from the above scenarios?

**Answer:** Well, it just shows that when the price of the underlying asset changes, the moneyness of the option changes, and therefore the delta value of the option also changes.

# Delta vs Underlying Asset Price

###### From the quiz we just went through, we can come to the conclusion that delta is not a fixed entity but a variable and the value is likely to change with the change in the value of the underlying. Let’s look at a graph and really visualize what this looks like.

###### The chart below shows the movement of delta versus the price of the underlying asset.

- The blue line shows the movement of a call option’s delta
- The red line shows the movement of a put option’s delta

For now, let us ignore the red line and focus on the blue line (delta of a call option)

###### Allow me to unpack the information the chart gives us:

- As you can see by looking at the X-axis from the left to the right, the moneyness of the option increases as the price of the underlying asset moves from OTM to ATM to ITM.
- Now observe the blue line that represents the delta of a call option. We can see that when the price of the underlying asset increases, so does the delta.
- As you can see, at OTM the delta is near 0. This also means that regardless of how much the price of the underlying asset drops ( going from OTM to deep OTM) the option’s delta will remain at 0
- When the price of the underlying asset moves from OTM to ATM, the delta value also starts to increase.
- Notice how the delta lies within the 0 to 0.5 range for options that are less than ATM. And then at ATM, the delta hits a value of 0.5
- When the price of the underlying asset moves from ATM towards ITM the delta starts to move beyond the 0.5 mark.
- However, notice the delta starts to flatten out when it hits a value of 1. This also implies that when the delta moves beyond ITM to deeper ITM the delta value does not change. It remains at its maximum value of 1

You can notice similar characteristics for the Put Option’s delta (red line)

# Delta Acceleration

###### There are four delta stages mentioned in the graph, let’s unpack each one.

# Predevelopment

## At this stage, the option is OTM or deep OTM. The delta will remain close to 0 even when the option moves from deep OTM to OTM

**For example:** When the price of the underlying asset is $8,400, an $8,700 Call Option is Deep OTM, which is likely to have a delta of 0.05. Even if the price of the underlying asset moves from $8,400 to say $8,500, the delta of the $8,700 Call option will not move significantly as it is still an OTM option. The delta will remain as a small non-zero number.

**Consider this:**

**Asset:** OhicToken

**Price:** $8,400

**Option Type:** Call Option

**Strike price:** $8,700

**Option Premium:** $12

**Option delta:** 0.05

**Moneyness:** Deep OTM

**Asset:**

**Price:**

**Option Type:**

**Strike price:**

**Option Premium:**

**Option delta:**

**Moneyness:**

###### Suppose the price of OhicToken moves from $8,400 to $8,500 (100 points), what happens to the premium?

###### Let’s calculate:

**Change in underlying asset price:** 100

**Change in underlying asset price:**

**Option Delta:** 0.05

**Option Delta:**

**Premium change:** 100 * 0.05 = 5 points

**Premium change:**

**New premium:** $12 + $5 = $17

**New premium:**

**Percentage change:** 41.6%

**Percentage change:**

**Conclusion**

###### Deep OTM options are inclined to offer a significant percentage return only if the price of the underlying asset moves significantly.

**My thoughts**

###### I would avoid buying deep OTM options because the deltas are really small and the underlying has to move quite significantly for the option to be beneficial.

# Take off & Acceleration:

## In this stage, the option transitions from OTM to ATM. This is where you can make the most money and therefore this is where the most risk lies.

**Consider this:**

**Asset:** OhicToken

**Price:** $8,400

**Option Type:** Call Option

**Strike price:** $8,500

**Option Premium:** $20

**Option delta:** 0.25

**Moneyness:** Slightly OTM

**Asset:**

**Price:**

**Option Type:**

**Strike price:**

**Option Premium:**

**Option delta:**

**Moneyness:**

###### Suppose the price of OhicToken moves from $8,400 to $8,500 (100 points), what happens to the premium?

###### Let’s calculate:

**Change in underlying asset price:** 100

**Change in underlying asset price:**

**Option Delta:** 0.25

**Option Delta:**

**Premium change**: 100 * 0.25 = 25

**Premium change**

**New premium:** $20 + $25 = $45

**New premium:**

**Percentage change:** 125%

**Percentage change:**

###### As you can see, for the same 100 point move, slightly OTM options behave very differently to deep OTM options.

**Conclusion**

###### Slightly OTM options are more sensitive to changes in the underlying asset’s price. Many options traders make a lot of money by purchasing slightly OTM options when they expect big moves in the underlying. Although keep in mind that many other factors have to be considered.

**My thoughts**

###### Although slightly OTM options are more expensive than deep OTM options, you stand a chance to make amazing returns if you play your cards right.

**Let’s take a look at how an ATM option would behave with the same 100 point rally**

**Asset:** OhicToken

**Price:** $8,400

**Option Type:** Call Option

**Strike price:** $8,400

**Option Premium:** $60

**Option delta:** 0.5

**Moneyness:** ATM

**Asset:**

**Price:**

**Option Type:**

**Strike price:**

**Option Premium:**

**Option delta:**

**Moneyness:**

**Premium change:** 100 * 0.5 = 50

**Premium change:**

**New premium:** $60 + $50 = $110

**New premium:**

**Percentage change:** 83%

**Percentage change:**

**Conclusion**

###### ATM options are more sensitive to changes in the price of the underlying asset when compared to OTM options. Due to the fact that an ATM option’s delta is very high, the underlying asset does not need to move significantly; even if the underlying moves by a tiny margin, the option premium changes. That said, purchasing ATM options is more expensive compared to buying OTM options.

**My thoughts**

###### I would buy ATM options when I’m looking to play safe. The ATM option will move even if the underlying asset does not move by a large value.

# Stabilization:

## When the option transitions from ATM to ITM and Deep ITM the delta starts to stabilize at 1

###### When the delta moves ITM the delta moves to 1. This means that an option can be ITM or deep ITM but the delta will remain fixed to 1 and will not change in value.

# Let us look at two examples

## Example 1: Option X

**Asset:** OhicToken

**Price:** $8,400

**Option Type:** Call Option

**Strike price:** $8,300

**Option Premium:** $105

**Option delta:** 0.8

**Moneyness:** ITM

**Asset:**

**Price:**

**Option Type:**

**Strike price:**

**Option Premium:**

**Option delta:**

**Moneyness:**

**Change in underlying asset price** = 100

**Change in underlying asset price**

###### Lets see how Option X will behave:

**Change in premium:** 100 * 0.8 = 80

**Change in premium:**

**New Premium:** $105 + $80 = $185

**New Premium:**

**Percentage Change:** 80/105 = 76.19%

**Percentage Change:**

## Example 2: Option Y

**Asset:** OhicToken

**Price:** $8,400

**Option Type:** Call Option

**Strike price:** $8,200

**Option Premium:** $210

**Option delta:** 1.0

**Moneyness:** Deep ITM

**Asset:**

**Price:**

**Option Type:**

**Strike price:**

**Option Premium:**

**Option delta:**

**Moneyness:**

**Change in underlying asset price**: 100

**Change in underlying asset price**

###### Lets see how Option Y reacts:

**Change in premium:** 100 * 1 = 100

**Change in premium:**

**New Premium:** $210 + $100 = $310

**New Premium:**

**Percentage Change:** 100/210 = 47.6%

**Percentage Change:**

**Conclusion**

###### In terms of the absolute change in the number of points, the deep ITM option scores over the slight ITM option. However, in terms of percentage change, it is the other way round. We can deduce that ITM options are more sensitive to the changes in the price of the underlying asset but they are more expensive.

###### Most importantly notice the change in the deep ITM option (Option Y) for a change of 100 points in the underlying there is a change of 100 points in the option premium. This means to say when you buy a deep ITM option it is as good as buying the underlying itself. This is because whatever is the change in the underlying, the deep ITM option will experience the same change.

**My thoughts**

###### I would buy ITM options when I want to play very safely. ITM options have a high delta, which means they are most sensitive to changes in the underlying.

# Final Conclusion

## Key Points:

- The delta value changes along with changes in the price of the underlying asset.
- As the option transitions from OTM to ATM to ITM, so does the delta.
- The delta value for ATM options is 0.5
**Delta predevelopment**is when the option transitions from deep OTM to OTM.**Delta Takeoff and acceleration**is when the option transitions from OTM to ATM.**Delta stabilization**is when the option transitions from ATM to ITM to Deep ITM.- Buying options in the take-off stage tend to give high percentage returns.
- Buying deep ITM options is as good as buying the underlying asset.

###### If you have any further questions, do not hesitate to reach out to us on the Dopex Discord server or via Twitter.

## Glossary of terms

**Underlying asset** — The underlying asset, the price of which is being speculated on, for example, Bitcoin.

**Underlying asset**

**Expiry date** — The date the option will expire and be exercised, after this date, the contract is no longer valid.

**Expiry date**

**Strike price** — The price at which the buyer has the right to buy or sell the underlying asset at expiry.

**Strike price**

**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.

**Option price (premium)**

**In the money (ITM):**

**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):**

**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):**

**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**

## About Dopex

###### 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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