# Delta-Neutral Examples

## Context

*Based on Arthur Hayes' example in his article: **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

1 *USDe *= $1 of *ETH *+ Short 1 Ethereum / USD Inverse Perpetual

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

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