op-amm

An interactive companion to Oracle-Parametrized Constant Function Market Makers: From Price Feeds to Pricing Rules by Hamed Amini and Zachary Feinstein (arXiv:2609.33799).

The app, Oracle Dial, turns the paper's results into instruments you can operate. One control, the oracle sensitivity λ, sets how far the pool's quote follows an external price oracle. At λ = 0 the pool is a plain constant-product AMM (CPMM). Every panel recomputes as the dial turns.

What's on the page

Panel	Paper	What it shows
Live market	§3, §4.2	A synthetic trading day in one-second steps. An oracle-priced pool and a CPMM face the same arbitrageurs, and their quote errors, trades and cumulative LP losses update live.
Weighted average	Theorem 2.8, Example 2.11	The quote sits between the oracle price π and the inventory price π*, a share λ of the way toward the oracle. A second chart shows depth against an equal-value CPMM as the oracle error grows.
Lag risk vs oracle risk	Corollary 3.4, §4.1, Figure 1	Residual variance split into market lag and oracle noise, the depth and loss path as λ runs from 0 to 0.99, and where the pool beats the CPMM on both.
Sandwich	Proposition 3.11, Remark 3.13, Lemma 3.14	An attacker front-runs the stale curve, the oracle refreshes, and the attacker back-runs the new curve. The panel shows the attacker's take and the fee that neutralizes it.
Backtest	§4.2, Figures 2 and 3	The paper's three oracle setups (L1, L2 and pull), rerun across a λ × fee grid in the browser.

Models and data

• Pricing rule. The paper's geometric-average design on a constant-product baseline: p(r, π) = λπ + (1 − λ) log r, where r = y/x is the reserve ratio and prices are logs.

• Market. Synthetic, not SPY data: geometric Brownian motion with no drift in price, 13% annual volatility, one-second steps over a 6.5-hour session, and a one-cent spread on a share near $420.

• Oracles. Updates follow the paper's rule (3.7). The oracle refreshes when its delayed signal moves more than ε from the posted price, or when the heartbeat h expires.

  Regime	Heartbeat h	Threshold ε	Delay δ
  Perfect	1 s	0	0
  Pull	1 s	0	1 s
  L2	1 h	5 bps	2 s
  L1	1 h	25 bps	60 s
  Noisy	1 s	0	1 s, plus a mean-reverting error with a 3 bps SD

• Arbitrage. Each second, arbitrageurs make the smallest trade that brings the pool's fee-adjusted quote back to the market bid or ask. The LP loss is their profit net of fees, which is the loss-versus-rebalancing (LVR) the paper measures.

Findings credited to the paper on the page are quoted from it. Everything else is computed from the paper's formulas.

A note on the formulas

The arXiv HTML rendering of the paper drops radical signs. Two formulas here use the square roots that the derivations require:

• The pool stays deeper than a CPMM while |log S − π| < ((1 − λ)/λ) · log((1 + √λ)/(1 − √λ)).
• In the same state, the pool's LVR is below the CPMM's while σ_η < √((1 − λ)/λ) · σ, which is λ < 1/(1 + α²) with α = σ_η/σ.

scripts/verify.js checks both. It also checks the worked sandwich example in Remark 3.13: a front-run cost of 2.175, an attacker take of 0.512 (25.6% of the pool), and a plain refresh loss of 0.054.

Run locally

There is no build step and there are no dependencies.

python3 -m http.server 8000

Then open http://localhost:8000.

Check the engine

node scripts/verify.js

Preview image

og.png is the 1200 × 630 card that link previews use. Its source is scripts/og-card.html, which draws an hour of the live panel's simulated market with the page's own fonts and colors. To regenerate it after a design change:

"/Applications/Google Chrome.app/Contents/MacOS/Google Chrome" --headless=new --use-mock-keychain --hide-scrollbars --force-device-scale-factor=1 --window-size=1200,630 --virtual-time-budget=10000 --user-data-dir="$(mktemp -d)" --screenshot="$PWD/og.png" "file://$PWD/scripts/og-card.html"

Chrome writes the file within a few seconds. If it doesn't exit afterwards, stop it with Ctrl+C.

Deploy on Vercel

The site is static. Import the repository in Vercel and choose the Other framework preset with no build command; Vercel serves the repository root. From the CLI:

vercel --prod

Layout

File	Role
index.html	Page markup
styles.css	Design tokens and layout for light and dark themes
engine.js	Pricing rules, liquidity, LVR, sandwich values, and the market and oracle simulation
core.js	Dial state, canvas helpers and number formatting
hero.js	Live market panel
plates.js	Weighted-average and trade-off panels
sandwich.js	Oracle-update sandwich panel
backtest.js	Backtest rerun
scripts/verify.js	Checks against the paper's numbers
og.png	Link-preview image
scripts/og-card.html	Source for og.png