Flip the problem upside down: LP returns are a function of how close the weights w (@BalancerLabs portfolio weights) are to the 'optimum' weight w*
Arbitrageurs can be viewed as a stochastic control mechanism that moves w around w*
Can you control |w-w*| as a function of fees?
Trad. Optimal Control: Robot/program has 100% deterministic control over the intervention (e.g. moving robot arm)
LP vs. Arbs: Stochastic control plus fees add a wrinkle — the fee interval. Arbs can never exactly get to w* because of fees, yet Martin/Dave show LPs are still 🤑🤑
Instead of looking at profits as a function of fees, drift, vol, we look at weights (controlled by arbitrageurs)
This gives you a Bellman equation (like those in reinforcement learning!) that only depends on your utility function indirectly (via ϕ)
2-assets: Solved w/ Wronskian
This paper is short and sweet, but gets to the heart of the debate between Dan & SBF: Why does utility matter?
Results intimate:
- When weights are very volatility you prefer linear
- When weights are static, you prefer logarithmic
Volatile weights ~ MMs frequently quoting
In Uniswap, you can replicate the volatile weights by adding and removing liquidity around orders (like sandwich attacks), which is _effectively_ the same as quoting a price schedule in an order book
By the way, @MartinTassy's discovery of the '0 fee phase transition' looks a lot more like conventional transitions in 1-dimension from the optimal control lens
October: @alexhevans had a genius idea of viewing LP returns from the perspective of a stochastic process on weights
November: @MartinTassy / @_Dave__White_ result out, we were confused as to why the proof was so specific to Uniswap
December: Rough proof that the zero-fee phase transition from Martin/Dave is real, but is better viewed via the control theory lens
January: Rewrite proof in Reinforcement Learning language — describe Bellman equation for LP returns w/ fees
tl;dr: Given the immense interest in optimal control in DeFi, we found a way to rephrase the 'do you need fees? how much?' problem for AMMs in terms of a RL-like Bellman equation
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!