2 May, 14 tweets, 6 min read
Wow, y'all really aped into this paper, thanks for all the memes (@0xtuba) & thoughtful comments

I figured I'd go a little deeper into the connection between lending protocols and options selling but first, don't forget to follow @htkao, @GuilleAngeris, and @alexhevans ☺️
As @cuckqueeen points out, numerical methods for simulating barrier options are hard to execute correctly (one of the reasons we run all such simulations @gauntletnetwork against the contract execution state in EVM, otherwise you will mess up liq barrier)
The paper explicitly connects the option that a liquidator holds to a down-and-in knock-in option
- a: liquidation incentive
- l: ratio of liq threshold to LTV
- p(t0): loan entry price

But you might be asking: how the hell did we get an exact, analytical formula for this?
1st: What's a down-and-in option?

DOIC(K, S, H) with strike K, asset price S, barrier H pays off max(S-K, 0) if S < H

This is less than a normal call (S<H condition) & depends on price path (did S[t] ever hit H?)

If American options can't be explicitly priced, how can this?
Recall from stats 101: Two paths from a symmetric random walk that are reflections of each other have the same probability (see figure)

For these paths, we can look at hitting times t = {S < H} vs. t = {S > H} and know that they're equal via symmetry — a path-dependent result!
But for something like Black-Scholes, this isn't true — Geometric Brownian Motion + Lévy processes aren't symmetric [they're non-negative!]

BUT: they can be symmetric in a logarithmic sense, where a certain relation holds for all bounded, measurable g 👉 Put-Call Symmetry
What is Put-Call symmetry? If we own a call on Euro-denominated options on an asset, that's equal, in some sense, to owning a set of USD-denominated puts on an asset

What's the ratio calls to puts? The Put-Call symmetry formula above delineates this quantity for many processes
Put-Call symmetry lets you reason about the time for a price process S to hit a barrier H and you get *exact* formulas for replications — this is the magic of why in no-arb/no gas cost, we're able to exactly price the value of an @AaveAave or @compoundfinance liquidator's option
While Put-Call symmetry is amazingly useful in theory, it is natural to ask what its limitations / assumptions are. As shown by Carr & Lee, Put-Call symmetry implies that for stochastic volatility models, price and volatility must be independent (SABR/Heston fine, others bad)
This is why simulating the risk of default in Aave/Compound/Maker is quite difficult — you need to understand correlations induced by EVM execution, user behavior to get precise probabilities (what we do @gauntletnetwork)

OTOH, theoretical tools help provide 1st order intuition!
Other cool fact buried in the paper: if the idealized model of lending protocols as covered options sellers holds, you can replicate the protocol's exposure with a CFMM!

This means a synthetic asset representing the Compound protocol's exposure can be constructed *statically*
Why would a replication be useful?

If one were underwriting insurance to a lending protocol, you could compute the exposure of the protocol by summing up each outstanding borrow's option exposure 👉 construct a CFMM to securitize exposure

👉 Short CFMM to hedge your insurance
We believe these results open the door, for those quantitative enough, to construct ways for protocols to hedge and insure themselves in a more meaningful way than exists right now

But you're going to have to do more work than that of the idealized model of the paper 🙃
*positive Lévy processes (anything that could be used as a price)

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More from @tarunchitra

30 Apr
One mechanism for improving capital efficiency in CFMM trading is borrowing against LP shares (e.g. @AaveAave, @AlphaFinanceLab, @MakerDAO, @SushiSwap)

How safe is it compared to normal lending?

New 📝💧 from @htkao, @GuilleAngeris, @alexhevans y moi

stanford.edu/~guillean/pape…
We first show how to compare Loan-to-Value (LTV) / collateral factors between borrowing A with B as a collateral vs. borrowing A against an A/B LP pool share

Turns out, with dynamically adjusted LTVs you can make LP share lending *more* safe than lending of the underlying
👉🏾 Lending + CFMM protocols like @SushiSwap’s Kashi can provide way better efficiency if they dynamically adjust LTVs in response to price changes and fee accruals

I found this insanely counterintuitive until I wrote the equations — LP shares can be amazing collateral w/ care
9 Apr

One thing I will say is that most of the fair methodologies have a downsides themselves:
2. Lack of guarantees about economic price ordering
3. Extremely unproven in production (similar to ZKPs in 2012)

Theoretical (@vegaprotocol’s Wendy) and practical protocols (@valardragon) add a >= 1 block commit-reveal from validators OR added rounds of BFT-style message passing. Griefing vectors (DDoS-esque) are abundant + provable models have weak synchrony guarantees
Recent papers from @algo_class and Joachim Neu show lower bounds on these latencies and it is very unclear if the practical implementations even come close to saturating these bounds (Kelkar, et. al get to weaker bounds in their paper)
18 Jan

Remember the Twitter argument between @danrobinson and @SBF_Alameda?

Recall how it hinged on logarithmic vs. linear utility functions?

Using optimal control, we show utilities are a red herring

joint w/ @alexhevans @GuilleAngeris

stanford.edu/~guillean/pape…
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 🤑🤑
30 Nov 20

What does part of Paul Milgrom's 2020 Nobel Memorial Prize have to do with 🦍-ing into pool 2?

2nd part of our series on CFMM shape looks at:

💹 How do you compare LP return from different pools?
🤼 Quantifying adverse selection in CFMMs

medium.com/gauntlet-netwo…
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
24 Nov 20
What does Gauss's Theorema Egregium (1827) have to do with getting rug pulled in Uniswap?

This post (1 out of 3) introduces (only w/ pictures!) new work on understanding constant function MMs (CFMMs) as the primary market for an asset

Part I: Curvature

Why curvature?

@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
2 Nov 20
As much as I love the Penrose tiling and long-range order (I used to do glass research!), this seems like a terrible idea

1. Diverge correlation times/lengths means time to verify of a single transaction’s validity could take way longer than block propagation

😬😬😬
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)