CONVEXSWAP

ConvexSwap started as a hackathon project addressing a simple problem: big trades on decentralized exchange pools like Uniswap move the price against you. A pool prices every swap off its own reserves, so the larger the order, the higher the slippage. ConvexSwap streams the trade out over time instead, in smaller chunks, letting other participants arbitrage the market price back to its starting point, saving the trader a part of their slippage costs.

uniswap/x·y=k — selling 40 ETH for USDC, same trade, same time

SELL 40 ETH

invariant	x · y = k	100 · 300,000 = 30,000,000
you receive	Δy = y · Δx / (x + Δx)	85,714 USDC
spot price	p₀ = y / x	3,000 USDC/ETH
execution price	p = Δy / Δx = y / (x + Δx)	2,143 USDC/ETH
slippage	1 − p / p₀ = Δx / (x + Δx)	28.6%
in 4 chunks	n · y · (Δx/n) / (x + Δx/n)	109,091 USDC · 9.1%

Ideal fill at p₀ is 120,000 USDC. 4 chunks keep 23,377 USDC more than one swap, if arbitrage restores the price between them. Fees ignored.

The interesting part is deciding how many chunks to use. Splitting a trade to many chunks cuts slippage costs but increases gas costs and exposure to volatility. This forms a convex optimization problem. The ConvexSwap algorithm finds the number of chunks executed at a set interval that minimize trade costs. Inputs are gas cost estimates, market liquidity depth, and recent volatility.

convexswap/objective — what the chunk count minimizes

minimizeC(b) = I(b) + G(b) + V(b)over whole chunk counts, 1 to 50

price impactI(b) = b · [ k / (y − R/b) − k / y ]b swaps of R/b each; falls as b grows

gasG(b) = ḡ · bḡ is the mean cost of one swap over recent blocks

drift riskV(b) = ½ · b(b − 1) · (γ_yσ_y² + γ_kσ_k²)σ is the volatility of y and k while the trade waits; γ weights it

answerb* = argmin C(b)one chunk every 500 blocks

convexswap/cost — total cost against chunk count

Illustrative numbers on a toy pool; the shape is the one the optimizer searches.

> cd ..