# Impermanent Loss in Extreme Market Conditions

It’s hardly a secret that automated market maker (AMM) liquidity providers (LPs) incur impermanent loss (IL). As an AMM, Lyra is no exception. We believe in transparency and want to give prospective LPs as much information as possible about how our system performs in adverse environments, along with the risks involved with being a Lyra LP. It is in this spirit that we have published the ‘Impermanent Loss’ paper available **here**.

In the paper, we simulate how the system would perform in a scenario similar to the May 2021 crash, where implied volatility (IV) for shorter dated (<1 month) ETH expiries spiked from 100% to ~300%. The simulations were performed with the harshest possible assumptions, to map out a reasonable ‘worst case’ volatility jump scenario for LPs. These assumptions are:

- 0 fees for LPs
- Arbitrageurs exploit any opportunity using optimal execution
- 0 two-way flow (i.e. IV immediately spikes to 300% with only buyers trading with the AMM)
- The true market value for IV really is 300% (these results do not apply if a large call buyer marks the internal Lyra IV to 300%).

In practice, some of these assumptions are too strict:

- Fees will reduce the IL shown in this report (particularly due to the fee component which scales according to the AMM’s net vega exposure).
- It is nearly impossible to execute as optimally as the attacker in the report, due to the need to split trades into infinitesimal size. It’s worth noting that market pressures (i.e. multiple arbitrageurs) may lead to an exploit that is considerably smaller than presented in this report, but determining this requires further research.
- The scenario in which volatility simply ‘spikes’ from 110% to 300% with only buyers showing up is unrealistic.

Nevertheless, the assumptions imposed offer a clear upper bound for IL and make the simulations easier to compute. These simulations were performed using an asset with a spot price of $2,000, listed strikes of (1800, 2000, 2100, 2300, 2500), and a 28 day expiry. Let’s get into some of the key results. **Result 1: Standard Size has a large impact on IL**

We begin by focusing on a single strike/expiry combination. This figure shows the IL incurred by the AMM for the 2100 strike option in the 28 day expiry, assuming volatility spikes from 100 to 300. Standard sizes of 10, 20, and 30 are shown via the pink, blue and red lines respectively, holding the skew ratio impact parameter equal to 0.0125. For a standard size of 20, the pool loses $304,000 to IL, increasing to $456,000 for a standard size of 30, and decreasing to $152,000 for 10 (all rounded to the nearest thousand dollars).

**Result 2: Quantifying the total IL across an expiry with 28 days to go**

This figure shows the IL incurred by the AMM for the total 28 day expiry, with the same colour scheme as the figure shown in result 1. For a standard size of 20, the total IL suffered for the expiry is $577,000, increasing to about $866,000 for a standard size of 30, and decreasing to $288,000 for 10. **Result 3: The sensitivity of IL to the skew ratio impact parameter**

The above figure shows the IL suffered by LPs assuming a standard size of 20, and varying the skew ratio impact parameter between 0.0075, 0.0125, and 0.0175 in the red, blue and pink lines respectively. For a volatility spike from 1 to 3, these ILs are $653,000, $577,000 and $521,000 respectively (to the nearest thousand dollars). Note that varying the skew ratio impact parameter has a less dramatic effect on IL than standard size, but still a significant one. This is because standard size impacts all strikes within an expiry, amplifying its effect relative to the skew ratio impact parameter. **Final Remarks**

It’s worth noting that the core Lyra mechanism (not including fees) is **independent of the liquidity in the pool**. This means that the projected ILs are equivalent for any pool, whether it has $10m or $50m in locked value. The larger the pool, the smaller relative IL (in percentage terms) this represents. This is worth considering when determining a suitable standard size - a $500,000 loss is much more palatable for a $50m pool than a $2m one, and the slippage/IL tradeoff is more manageable for the former.

It is also worth reiterating that this analysis was performed assuming **0 fees**, an assumption which, when relaxed, will result in less IL for LPs. We hope to publish more results elaborating on this in the coming weeks.

Hopefully this post has shed some light on the parameters within the Lyra system, and given some perspective on the risks associated with providing liquidity to the system. The Mathematica code for the simulations has been released **here**, for interested readers to construct their own scenarios and simulations. We’re looking forward to hearing your feedback!