# Rebalancer

<figure><img src="/files/z0yowD6lMfocndCvWyCM" alt=""><figcaption></figcaption></figure>

Assume the **AVAX-USDC** farm on Trader Joe has $22M liquidity and the price of **AVAX** is $18. That translates to $11M **USDC** and 611,111 (11M/18) **AVAX**. Assuming we have $1M worth of LP on Trader Joe which comes from a 3X LYF position on **Homora**, our debt in **USDC** is 1/6M and our debt in **AVAX** is 1/36M \~ **27.78K AVAX** or **$500K USD.**

Say after some time we accrue $10K in farm token rewards and $2K **USDC** plus $4000/18$ **AVAX** in interest. Therefore

<figure><img src="/files/GFeqQOuOUKoRCs9ol9pE" alt=""><figcaption></figcaption></figure>

Say the price of **AVAX** changes to $20. After the $10K in rewards is added to our **LP**, assume the farm’s liquidity becomes $23.20M so that it has N\_a=11.60M **USDC** and N\_b=580K **AVAX**. The amount of tokens in our **LP** is

<figure><img src="/files/3fGU7evGAsRiRZdPVG2O" alt=""><figcaption></figcaption></figure>

The current leverage becomes 3.29. In order to bring the leverage back to 3 and achieve delta-neutral, we need to remove some **LP tokens**, swap some **USDC** to **AVAX** and repay some of our debts.

The amount of **LP** we need to remove is proven to be

<figure><img src="/files/OFudJsKwI2YIoxcZ2DY8" alt=""><figcaption></figcaption></figure>

We need to remove 5.731% **LP** in our position during rebalance. After that the amount of tokens in our **LP** is

<figure><img src="/files/k0dFwYavW1MJGIoeKjvW" alt=""><figcaption></figcaption></figure>

The amount of tokens left in the pool is

<figure><img src="/files/7a1nvRJgp7U4WTC5aoci" alt=""><figcaption></figcaption></figure>

The optimal amount of **USDC** to be swapped to **AVAX** is proven to be

<figure><img src="/files/JQJFZ63OnwmXuRNYV2st" alt=""><figcaption></figcaption></figure>

which amounts to 3% of the **LP** position.

The swap satisfies

<figure><img src="/files/FoBQRch8MvUmmr6VvLrp" alt=""><figcaption></figcaption></figure>

Therefore

<figure><img src="/files/CPGPkZX8KaN25SMWOXEW" alt=""><figcaption></figcaption></figure>

The amount of **USDC** to repay Delta\_d\_a is the amount withdrawn minus the amount swapped

<figure><img src="/files/Y3t64tMOxT29VQDtyBtE" alt=""><figcaption></figcaption></figure>

Therefore the final amount of **USDC** in debt is

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

The final amount of **USDC** in the **LP** after the swap is proven to be

<figure><img src="/files/IjtSeBgUG0t4x6Z7RVJR" alt=""><figcaption></figcaption></figure>

And the amount of repaid **AVAX** is

<figure><img src="/files/Jgm3aIdSVM8Pe0XjtTPj" alt=""><figcaption></figcaption></figure>

The final amount of **AVAX** in debt is

<figure><img src="/files/fMMviQAUP3TyCSUL8SUC" alt=""><figcaption></figcaption></figure>

And the final amount of **AVAX** in the **LP** after the swap is

<figure><img src="/files/q96CpbuL2UdinyHPVJ7d" alt=""><figcaption></figcaption></figure>

which matches the debt amount(up to rounding errors) in **AVAX** to reach delta-neutral.

The swap price of **AVAX** in terms of **USDC** is

<figure><img src="/files/7oeweYYZ72qttKqjcOP6" alt=""><figcaption></figcaption></figure>

which indicates a slippage (including the swap fee) of 0.8%.

Also the pool price of **AVAX** in terms of **USDC** after the swap is

<figure><img src="/files/aFvCq2x0rKU2RS307KB8" alt=""><figcaption></figcaption></figure>

As a result the final leverage is

<figure><img src="/files/eCU50UBgf4ywqGFZVKtW" alt=""><figcaption></figcaption></figure>

The actual calculations are much more accurate and only have single digit errors (in Solidity `uint`). In a word our contract is able to rebalance the pseudo delta-neutral position by removing the optimal amount of **LP** and swapping the minimal amount of tokens, with less than [1,800,000 in gas](https://snowtrace.io/tx/0xe6212b62e78d5800f2f3e86ae043cc560d5a1618480d7487e9d168725f2fa38e).


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