# Deep Down Rabbit Hole #1: A Quantitative Analysis of Impermanent Loss by Asteria

Deep down the rabbit hole is our new series focused on using a quantitative analysis towards research and narrative building for our community.

First up in this series:

Now let’s quantitatively analyze the risk in an AMM using the ETH-USDT liquidity pool but under five different stimulations.

Let’s assume that an LP deposits x’0 amount of ETH and y’0 amount of USDT into the liquidity pool at the price of P’0. Here’ the total value of the liquidity pool will be x’n ETH and y’n USDT.

Now, let’s remember from our previous ramblings on AMMs that — constant product makers, including UniV2 required to satisfy the following equation whenever a trade is executed.

k= x ✕ y

Since LP is just providing USDT, the yield generation equation for stimulation 1 users will be as follows:

LP yield = y’n/y’0–1 = (P’n/P’0) ^ 0.5–1

The yield minus the income in LP fees for this pool of users looks as depicted in graph A:

Stimulation 1 users have the risk of ETH price dropping. Given the imbalance in the dollar value of the assets in this pool, arbitrageurs can sell ETH into the pool and buy USDT at a relative discount. Arbitrageurs will try to extract this profit and in the process will rebalance the pool.

Since LP is just providing ETH, the yield generation equation for stimulation 2 users will be as follows:

LP yield = y’n/y’0–1 = (P’n/P’0) ^ 0.5–1

The yield minus the income in LP fees for this pool of users looks as depicted in graph B:

Stimulation 2 users have the risk of ETH price rising.

Since the LP is providing both ETH and USDT, the yield generation equation for stimulation 3 users will be as follows:

LP yield =2 y’n/y’0–1 = 2 ×(P’n/P’0) ^ 0.5–1

Risk for stimulation 3 users?

Stimulation 3 users have the risk of both the price dropping and rising.

Sounds familiar? You know it as an impermanent loss.

Since the LP is providing both ETH and USDT, the yield generation equation for stimulation 4 users will be as follows:

LP yield =2 y’n/y’0–1 = 2 ×(P’n/P’0) ^ 0.5–1

Risk for stimulation 4 users?

Stimulation 4 users have the risk of both the price dropping and rising, which again translates to impermanent loss.

# Conclusion

• Impermanent loss occurs when the price of one asset in the pool rises or falls relative to the price of the other asset in the pool.
• If the price of both assets rises or falls perfectly simultaneously and in the same proportion (in percentage terms), then there will be no impermanent loss since the dollar value of both assets will remain constant.
• PnL in AMMs is closely related to the price of the asset.

In the coming article, we will explore different strategies to help reduce impermanent loss. For example: does oracle help reduce impermanent loss? Does Asteria’s time selection strategy reduce impermanent loss? etc!

We’re excited to embark on building a quant-based DeFi and exploring new narratives using quantitative analysis as our core fundamental.

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## More from Asteria Finance Lab

Asteria, committed to be one crucial component of DeFi infrastructure, defines decentralized protocol of option pricing, trading and hedging of AMM mechanism, and provides APIs and templates for structured option application developments.

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Asteria, committed to be DeFi infrastructure, defines decentralized protocol of option pricing, trading and hedging of AMM mechanism