@CurveFinance made it clear that some assets perform better on 'flatter' CFMMs and others on 'sharper' CFMMs
But what does it mean to be 'better'?
Our paper studies what happens when traders arbitrage btw. two CFMMs and look at the max their prices differ by
When we dug into this a little more, it became clear that Gaussian curvature controls a lot of facets of CFMMs:
1. Price synchronization between two CFMMs 2. Adverse Selection for LP returns 3. Price stability 4. Optimal incentives for yield farming
There's a trade-off surface:
- Low curvature: large trade size for traders 📈, adverse selection for LPs 📉
- High curvature: large impact cost for traders 📉, LPs are profitable 📈
Prior thoughts on CFMMs aimed to try to optimize fees alone but..
We show (and will illustrate in the following posts) that curvature, fees, and adverse selection are tightly related and you can't change one without affecting the others
BTW: the name is in honor of the famous Marc Kac paper, "Can one hear the shape of a drum?" that tries to recover the metric on a manifold from the eigenvalues of its Laplacian
Can we recover the optimal time-dependent CFMM given a price process? 🧐
Post 1 tl;dr: Curvature controls pool price stability
Post 2: Curvature *directly* controls:
- LP profits when asset pairs are mean reverting
- ∃ a magic formula relating LP profit to adverse selection (probability α of LP realizing IL), curvature, and fees for *any* CFMM!
These results generalize Glosten & Milgrom (1984), Kyle (1985) to arbitrary CFMMs
This seminal work shows the shape of the order book represents the amount of adverse selection a market maker feels, leading to strategies where they remove liquidity to avoid adverse selection
2. If you want to anonymize a transaction graph by using a lattice with dense spectra (like the Penrose tiling) to define a DAG, note that you aren’t guaranteed that there isn’t *any* local structure that an adversary can find — only that no tx ordering will be unique
2. (cont.) It is possible that prefixes of tx ordering overlap an arbitrary amount, so there isn’t as much transaction ordering entropy as there is from cryptographic graph traversals (e.g. expander graph walks in supersingular isogeny signatures, lattice based crypto)
This effectively looks at a mean-field, agent-based model of: 1. Noise traders 2. Informed traders 3. Strategic LPs
It shows that as the # of LPs goes to ♾, ∃ a sharp phase transition in LP profits as a function of the number of informed traders (defined via simple signals)
There’s also a kind of curious stability result that is vaguely reminiscent of “rugpull” dynamics: there’s only a stable equilibrium when there are < 4 LPs, if there’s more you have sharp edge equilibria that you can oscillate between (akin to the “last LP holds the bag”)
The VC vs. trader “war” of crypto is reminiscent of the previous talent “war” between HFT and online ads: All of these boil down to latency vs. bandwidth trade-offs where "event-driven" investing depends on the condition number of a participants' value function
Trader: need max and min eigenval. of value fn. to be "close" (low condition number) because of regret minimization between your worst and best case outcomes
If your value function is smooth, this gives uniform bounds on the max/min eigenval. of hessian of your val. function
VC: need max eigenval. of value function to optimized
Things like the Tracy-Widom law force you to chase fat tails, terrible Sharpe, and anomalous portfolio construction
The number of traditional finance chads (e.g. @arbitragegoth) asking me questions about DeFi LP staking is 📈📈 📈
Here's what it is: 1. @synthetix_io / @kaiynne pioneered paying users for liquidity by staking CFMM LP shares 2. CFMM LP shares replicate options portfolios
👇🏾
∴ LP staking is equivalent to collateralizing a leg of an interest rate swap with future expected cash flows from an options portfolio
This is actually *really* hard to execute in normal finance — especially because the CFMM replication is a continuous combination of strikes
Traditional finance has focused on swaps as
a. In-kind (e.g. interest for interest)
b. Purely Synthetic (e.g. variance swaps, VIX)
DeFi let's you combine the two — in-kind on one side in exchange for synthetic on the other
Impossible to do this without non-custodial assets!
tl;dr:
- Synthetic levered assets in PoS and DeFi are MBSs.
- Improvements over meatspace/2008 MBS:
- Used to reduce inequality
- Avoid lending competition in PoS
- Numerical, probabilistic methods are key to correct design of these systems
The post motivates and provides background for our paper which just hit arXiv