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Review Summary: Liquidity Provider Returns in Geometric Mean Markets

Published onOct 22, 2021
Review Summary: Liquidity Provider Returns in Geometric Mean Markets


Geometric mean market makers (G3Ms), such as Uniswap and Balancer, comprise a popular class of automated market makers (AMMs) defined by the following rule: the reserves of the AMM before and after each trade must have the same (weighted) geometric mean. This paper extends several results known for constant-weight G3Ms to the general case of G3Ms with time-varying and potentially stochastic weights. These results include the returns and no-arbitrage prices of liquidity pool (LP) shares that investors receive for supplying liquidity to G3Ms. Using these expressions, we show how to create G3Ms whose LP shares replicate the payoffs of financial derivatives. The resulting hedges are model-independent and exact for derivative contracts whose payoff functions satisfy an elasticity constraint. These strategies allow LP shares to replicate various trading strategies and financial contracts, including standard options. G3Ms are thus shown to be capable of recreating a variety of active trading strategies through passive positions in LP shares.

Paper Summaries

“The paper applies option pricing theory to Geometric Mean market makers (G3M), defined as an investment in a fixed set of assets in which the feasible trade set is determined by the geometric mean of its reserves, which in turn is determined by a weighting of the assets in the set. The paper derives several results that characterize the value of the liquidity pool (LP) that underlies the G3M market maker. The key results are as follows:

  • Section 2 characterizes the elements of LP pricing and its movement over time, as a function of the weightings, the price of the assets and the LP trading frictions.

  • Section 3 shows that volatility in the prices of the assets in the presence of trading frictions in the G3M LP (causing delays in the rebalancing of the LP assets) results in the price of the LP falling below the price of a frictionless, continuously rebalancing portfolio.

  • Section 4 characterizes the pricing when the weighting scheme is allowed to vary over time; discretely and continuously.

  • Section 5 describes how to design a continuously time varying weight function so as to replicate any derivative contract, when the underlying assets are the risk free ass and a risky asset.”

“This paper generalizes the theory of geometric means market makers, called G3Ms, by allowing a variety of weighting schemes… The financial aspect of G3Ms can be described as follows: create a mechanism that keeps a fund with predetermined weights without any exogenous intervention by a human or a machines that rebalances the fund.

The paper extends the theory of G3Ms to many different weighting schemes in addition to constant weights. It assumes a geometric Brownian motion to represent asset prices in the equivalent martingale measure. Importantly, it shows that the payoff of investing in the fund replicates an option.”

“The paper studies the liquidity pool (LP) share returns for geometric mean market makers (G3Ms). Liquidity providers provide liquidity to G3Ms in exchange for shares. That is, providers deposit assets to the market maker in exchange for part of its valuation. The paper analyses the share values under market volatility… This appears to be the first work to explore the risks of providing liquidity to G3Ms which fills a gap in previous work.

The paper considers a stochastic model with risky assets and a money (risk free) asset to study the expected change in share value. Risky asset price changes are distributed according to Brownian motions with arbitrary correlations between assets… The paper quantitatively shows how LP value decreases depending on the volatility and correlation between risky assets. It further extends the model to consider dynamic G3Ms allowing contracts with a lower risk for liquidity providers.”

“The paper provides an analytical extension of geometric mean market makers to a more general context (time varying, stochastic). this is an important and useful extension that provides an analytic basis for understanding a more general class of AMM.”

Paper Strengths

“A very interesting result is that 'LP share payoffs of G3Ms that do not charge fees are supermartingales under the risk-neutral probability measure, due to having higher rebalancing costs than constant-mix portfolios'.”

“The paper extends the theory of G3Ms to many different weighting schemes in addition to constant weights. It shows that the payoff of investing in the fund replicates an option.”

“The paper provides a robust model to study LP value volatility and provide interesting quantitative results. Likely to motivate further work.”

Paper Critiques and Author Responses

“My questioning has to do with establishing:

(a) the relevance of the work to applications in crypto. Beyond mentioning Uniswap and Balancer, I am not sure if these type of environments are unique to crypto. Someone involved in DeFi would know. But others, like me, do not. Some more background would be helpful; and

(b) There is no discussion of the place of this work in the finance literature, e.g., is the contribution unique - perhaps because the G3M environment is unique?”

“I don’t know if the results, which are very abstract, comprise a new contribution or are a restatement of previously established results in the finance literature.”

“I wonder about the security implications of adaptive weights. Is it evident that liquidity providers cannot attempt to exploit these contracts to make unexpected profits?”

“Other than mentioning two market maker smart contract environments that operate on ETH (Uniswap and Balancer) which enable a user to customized design a automated market maker, the authors do not explain why their work is specially relevant to the crypto world. It might be the case that it is...and someone familiar with the current state of DeFi might recognize it a being so. But I do not. So, I think it needs to be addressed. Moreover, the key result of Section 5 assumes a continuously updating weight function. But the authors acknowledge that ETH SC's operate in discrete time. So, they have not proven the possibility of constructing a G3M that can replicate any derivative contract in the underlying assets for any actual cryptocurrency or distributed ledger.”

“To be honest, I understand that a Brownian motion is well suited for computing option prices. However, to my opinion, a geometric Brownian motion is probably not a good model for cryptocurrencies driven by speculative motivations.”

“There are a number of points that need discussion. First, what is the use of this mechanism? In a classical financial context it could be used to manage an index, for example a fixed weight index. An index provider might decide to let the market rebalance its index using the above mechanism. There are examples of funds managed in this way. However, it is difficult to believe that a classical market maker will find this mechanism more attractive than classical strategies. Second. A classical passive fund might make money or lose money in function of market events. But what happens if the market is not growing but is losing money? Under what conditions will investors find arbitrage opportunities in a prolonged market recession ?”


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