# Delta-Neutral Examples

## Context

*Based on Arthur Hayes' example in his article:* [*Dust on Crust*](https://blog.bitmex.com/dust-on-crust/)

*Here, we have used an inverse perpetual to explain the different payoff scenarios of a delta-neutral strategy. Ethena will utilize both inverse and linear perpetuals. The payoff outcomes on an inverse perpetual are more intricate and as a result, we focus on explaining those scenarios below.*

An Ethereum inverse perpetual which is worth $1 of Ethereum paid out in Ethereum has the following payoff function:

*$1 / Ethereum Price in USD*

If Ethereum is worth $1, then the Ethereum value of the perpetual is 1 *ETH*, $1 / $1.

If Ethereum is worth $0.5, then the Ethereum value of the perpetual is 2 *ETH*, $1 / $0.5.

If Ethereum is worth $2, then the Ethereum value of the perpetual is 0.5 *ETH*, $1 / $2.

***

## Worked Examples

{% hint style="info" %}
1 *USDe* = $1 of *ETH* + Short 1 Ethereum / USD Inverse Perpetual
{% endhint %}

To create 1 *USDe*, Ethena needs to delegate 1 *ETH* as margin with a derivatives exchange (via "Off-Exchange Settlement" solution) and short 1 ETHUSD perpetual.

#### Rapid ETH Price Decrease

* Now the Ethereum price falls from $1 to $0.1.
* The value of ETHUSD in ETH = $1 / $0.1 = 10 *ETH*
* The PNL of ETHUSD Position = 10 *ETH* (current value) – 1 *ETH* (initial value) = +9 *ETH* 
* We have 1 *ETH* delegated as margin with the exchange.
* Ethena's total equity balance with the exchange is 1 *ETH* (our initial margin) + 9 *ETH (*profit from the ETHUSD position), and the total balance is now 10 ETH. 
* The Ethereum price is now $0.1, but the system has 10 *ETH*, and therefore the USD value of the total portfolio is **unchanged** at $1, $0.1 \* 10 *ETH*.

#### Rapid ETH Price Increase

* Now the Ethereum price rises from $1 to $100.
* The value of ETHUSD in *ETH* = $1 / $100 = 0.01 *ETH*
* The PNL of ETHUSD Position = 0.01 *ETH* (current value) – 1 *ETH* (initial value) = -0.99 *ETH*
* Ethena's total equity balance with the exchange is 1 *ETH* (initial margin) – 0.99 *ETH* (loss from ETHUSD position), and total balance is now 0.01 *ETH*. 
* The Ethereum price is now $100, but Ethena has 0.01 *ETH*, and therefore the USD value of the total portfolio is **unchanged** at $1, $100 \* 0.01 *ETH*.

Delta-neutral strategies aim to ensure the portfolio value in synthetic USD terms is **unchanged** despite changes in value of the underlying collateral. In certain conditions and market environments this may not hold, as is described in more detail in the Risks section.

***

## Further Worked Examples

<figure><img src="/files/lvvhSeFuO4M5FP8qETjo" alt=""><figcaption></figcaption></figure>

<figure><img src="/files/jFYJs1ezaeq5sfkk4g1Q" alt=""><figcaption></figcaption></figure>


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