Deep dive into Loss-Versus-Rebalancing (LVR)

done

There is a prevalent notion that providing liquidity to an AMM over an extended period can result in losses. This concept was somewhat nebulous until two years ago when a seminal paper titled 'Automated Market Making and Loss-Versus-Rebalancing' clarified it and introduced the right metrics to explain how and when liquidity providers lose money.

Working of an AMM

Constant function automated market maker

A Constant Function Market Maker (CFMM) is a smart contract typically containing two assets, each with specific quantities. For instance, in the graph provided, we have an example of a CFMM with a certain amount of ETH (represented by $X$) and a certain amount of USDC (represented by $Y$ ). The CFMM is defined by an invariant function, depicted as the blue line on the graph, which determines when the CFMM is willing to trade.

The CFMM is always willing to accept a trade if, after the trade, the liquidity reserves change but remain on the invariant function. For example, if a trader approaches the CFMM wanting to buy a small amount of ETH ($x$) in exchange for a certain amount of USDC ($y$), and if this trade would still keep the reserves on the blue line, the CFMM will accept the trade. The price $p(x)$ of this trade can be visualized as the slope of the line connecting the initial liquidity reserves to the final reserves. This price is essentially the ratio of USDC $Y$ to ETH $X$ exchanged. Additionally, most CFMMs have fees, meaning the trader will need to pay slightly more than "y" due to these fees.

It's important to note that after a trade, the marginal price (the price of a very small trade) may change. If someone else wants to buy a very small amount of ETH after a trade of $X$ has occurred, they will need to pay the new marginal price, denoted as $p'(x)$ which may differ from the original price.

Therefore,

$p(x)$: Average price

$p'(x)$: Marginal price after trading $x$

$p'(x ) \neq p(x)$

Exploiting Liquidity Providers: The Impact of Rebalancing Events on AMMs

This specific design of the automated market maker (AMM) allows arbitrageurs to exploit liquidity providers whenever there is a rebalancing event. These events occur when there is a change in the equilibrium price of the assets being traded on the AMM. To illustrate, let's consider an example where an AMM allows the exchange of ETH for USDC.

Initially, let's assume that the price of ETH on the AMM is equal to the price $p'$ on a larger, more liquid trading venue like Binance, which is $3,000 . In this scenario, there is no incentive for traders to prefer one platform over the other because prices are in equilibrium.

Now, suppose for some external reason, the price of ETH on Binance suddenly jumps to p''=$3,300. This creates an opportunity for arbitrageurs to exploit the price discrepancy between the AMM and Binance. An arbitrageur can buy ETH at the lower price on the AMM and sell it at the higher price on Binance, thereby profiting from the price difference.

However, the problem arises because the price at which the arbitrageur buys ETH on the AMM (let's call it p(x)<p''=$3300 is below the new equilibrium price of $3300 . Essentially, the arbitrageur is purchasing ETH at a price lower than its actual market value, exploiting the outdated state of the AMM relative to the movement in the market.

This ability of the arbitrageur to profit comes at the expense of the liquidity providers on the AMM. These liquidity providers are effectively selling their assets at a price lower than the new market value, resulting in losses for them. Therefore, the liquidity providers are disadvantaged by the arbitrage activity, as they are forced to sell assets at prices that do not accurately reflect the current market conditions.

Volatility and Loss Versus Rebalancing

In the context of the Uniswap V2 system with unconcentrated liquidity, a formula to estimate $LVR$ (Loss Versus Rebalancing) is provided in the paper mentioned earlier by considering the volatility of asset prices. Increased price volatility leads to more frequent and larger rebalancing events, which directly impacts the potential losses experienced by liquidity providers (LPs). By establishing a connection between volatility and these losses, LPs can gain insights into the risks they face.

$$LVR_t=\int_{0}^{t} \ell(\sigma_{s},P_s)ds$$

$\forall t\ge 0$ where we define, for $P \ge 0$, the insatntaneous LVR by:

$$\ell(\sigma, P) \triangleq \frac{\sigma^2 P^2}{2} |x^\prime(P)| \geq 0$$

$\ell(\sigma,P)$ is always positive, so LVR is a non-negative, non-decreasing, and predictable process. Moreover, the cumulative profits of rebalancing arbitrageurs up to time t is equal to $LVR_t$

For instance, when examining the volatility $\sigma$ of assets such as ETH and BTC, it's evident that liquidity providers in ETH-stable and BTC-stable pairs lost approximately 5% to 7% of their capital to arbitrageurs due to LVR in 2023. Since other tokens typically exhibit higher volatility, the LVR effect is more pronounced. Additionally, LVR is exacerbated when liquidity is concentrated.

Daily realized volatility for the ETH-USDC pair, computed from Binance minutely closing prices, sampled at 60 minute intervals

Assuming a loss range of 7% to 11% due to LVR is deemed reasonable. Considering the total liquidity provided across various AMMs (excluding stable-to-stable pairs where volatility is minimal), which amounts to over $10 billion, an estimated loss of roughly $1 billion due to LVR can be derived. Thus a ballpark estimate suggests that liquidity providers (LPs) suffered around $1$Billion$ in losses due to arbitrage opportunities in 2023 alone. It's important to consider this figure as an approximation to provide a sense of the magnitude of the problem.

Arbitrageurs may not truly serve as bridges in this context, as only the first arbitrageur can profit from exploiting inefficiencies, and all arbitrageurs vie to be the first. Ultimately, this profit flows to validators as Maximal Extractable Value (MEV) because individuals must pay to prioritize their transactions in blocks. Consequently, the majority of this value becomes MEV.

This aspect presents an interesting problem often overlooked in discussions regarding AMM design's significant negative externalities. When considering both $LVR$ and sandwich attacks, another major issue in AMM design, the cumulative impact amounts to $1 billion or more annually. This sum essentially constitutes 90% to 95% of MEV. What remains primarily includes liquidation in lending protocols and other sporadic events, such as NFT transactions.

Understanding these dynamics sheds light on MEV's detrimental effects, contributing to assertions that Ethereum isn't fully decentralized. A significant portion of blocks is controlled by a few builders, giving them substantial influence over which transactions are included. This state of affairs is indirectly linked to AMM design's flaws. Improving this design to reduce MEV leakage could significantly enhance Ethereum's decentralization efforts.

Conclusion

LVR arises from the fact that AMMs always trade at off-market prices, leaving money to arbitrageurs trading the AMM against a CEX. LVR is greater when prices are more volatile, and when the AMM’s “marginal liquidity” is greater, that is, it trades more aggressively in response to price movements. The development and application of a delta-hedged AMM LP position, coupled with accurate predictions of LVR losses through quantitative modeling, offer promising avenues for enhancing the efficiency and stability of automated market makers in decentralized finance. By refining AMM designs to mitigate LVR risks, market participants stand to benefit from reduced trading fees and improved profitability.