I was looking through the latest literature on these topics and thought it might be worth getting feedback on because this is an important issue as we grow Pangolin.

The ever-marvellous Tarun Chitra, Guillermo Angeris and Alex Evans published a short paper last year discussing this very topic so I’m surprised it hasn’t received more attention. Or maybe it has but just not much in public. There are some important insights from this paper that are very relevant to us as we grow Pangolin and I’ve seen a lot of debate about how to incentivise pools that seems random and emotional when what it really needs is better analysis and expert advice.

I hope I can generate some interest in looking more closely at the implications offered by this research.

The paper can be found here: https://arxiv.org/pdf/2012.08040.pdf

The plain language version can be found here: Can you hear the shape of a CFMM? (part 3) | by Tarun Chitra | Gauntlet | Medium

The question the researchers posed themselves was very simple: what is the lower bound required to guarantee that the portfolio value of an LP, after arbitrage and subsidy, is nonnegative? And a corollary to that, what would be required to entice LPs from other platforms to a competing platform?

Well, these 3 brilliant minds have come up with an elegant formula to set that out. First, they start with some assumptions:

A. They define the initial price of the external and secondary markets as and

B. They assume the external market is **κ-liquid** if it satisfies, for ∆≥0, f(∆)−f(0) ≥ κ∆.

C. They assume that the price impact functiong for some market is μ-stable whenever a (nonnegative) trade of size ∆ does not change the market’s price by more than μ∆.

D. As before, they assume that the secondary market, with continuous, non-decreasing price impact functiong, is μ-stable with the definition given above.

E. They define

From this, they show that there is a simple expression for the amount of subsidy (R - in our case, PNG) that is sufficient to compensate an LP in the traded asset is:

I think that’s an incredibly elegant result.

In particular, they note that **more** subsidy has to be provided when:

(i) h becomes small (i.e., the token is subject to price changes with large drift) or

(ii) when μ/κ is large (i.e. the secondary market, for which the LPs are providing liquidity for, is very illiquid compared to the external market).

As I understand this, it means that how much subsidy we need to provide to LPs depends not just on the drift of the asset over time (such as that seen on tokens with low liquidity and new tokens), but also the relative curvature of the two markets - something that is generally negligible on established stablecoin pairs and more severe on stablecoin <=> non-stablecoin pairs.

The other important insight from this paper is how LPs can hedge for impermanent loss by selling a basket of put options on the traded assets as the LP leaves the pool. It’s a bit hard to do anything with this until we see an established derivatives market develop so I won’t dig into that much further here other than to note that derivatives are an important tool and should be given priority if we want Pangolin to become a market leader.