# Delta-Zero

As per stated on [the-straddle](https://artichoke.gitbook.io/abstract/single-sided-liquidity-strategies/the-straddle "mention"), we can obtain a **delta-zero strategy** over Uniswap's LPs using a **long straddle** option strategy. Assuming a bidirectional hedging, this strategy profits if the price of the underlying asset (in this case, the tokens in the Uniswap liquidity pool) moves significantly in either direction. 

By holding both the underlying tokens and the call and put options, you can create a delta-neutral portfolio. This means that your portfolio's delta (i.e., sensitivity to changes in the price of the underlying tokens) is close to zero.

## Mathematical Approach

Let's assume we want to create a delta-neutral portfolio using a Uniswap LP with two tokens, token $$X$$ and token $$Y$$.

Let's say we currently hold an amount of token $$X$$ in the liquidity pool, and we want to create a delta-neutral portfolio by buying options with token $$Y$$. To do this, we can borrow an equivalent amount of token $$Y$$ from the **Lendor Contract,** using token $$X$$as collateral. 

Let's call the amount of token $$X$$we currently hold in the LP $$X\_0$$. In that sense, we can borrow an equivalent amount of token $$Y$$, which we'll call $$Y\_0$$, from the **Lendor Contract**. This creates a new portfolio with $$X\_0$$ amount of token $$X$$in the liquidity pool, and $$Y\_0$$amount of token $$Y$$ that we've borrowed from the lending protocol.

<figure><img src="https://2475965272-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2FtEBPtrAa4dj1cAaGGsPA%2Fuploads%2FYEmwLVXwdgFr8YgzZjkJ%2FCaptura%20de%20pantalla%202023-06-26%20a%20la(s)%2019.10.53.png?alt=media&#x26;token=9457f92e-7312-4e0f-a41a-a5380ad8b468" alt=""><figcaption></figcaption></figure>

To create a delta-neutral position, let's assume **User A** has 10k USD of token $$x$$. In order to provide liquidity using **Artichoke**, he has to deposit $$x$$ as collateral on the **Lendor SC.** After this, the smart contract will automatically borrow the user 10k USD of token $$y$$, and will deposit both assets on the $$(x,y)LP$$ for a time $$t$$.

After **User A** wants to get out the position, he must interact with the **Lendor SC** and close it. The smart contract will return $$\Delta x$$ assets to **User  A** plus the fees accured during the time $$t$$. On the other hand, $$\Delta y$$ tokens will be returned to the vault and compensate both $$(x, y)$$ for price movements.&#x20;

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