Andrea Ciceri
fb378c4931
Start writing notes Notes on Uniswap V2's optimum Fix error in formula Test `computeAmountInt` using various deltas Add `concurrency` to the default configuration file Remove unused imports Correctly propagate error Allow dead code Make the priority queue a real FIFO Refactor: remove priority queue as stream and use channels Increase buffer size New `flashArbitrage` function Comment with some ideas Add pragma version Refactor: decrease the amount of calls Remove unused code Re-enable tests Remove comment Process known pairs when started Avoid re-allocating a new provider every time Ignore `nixos.qcow2` file created by the VM Add support for `aarch64-linux` Add NixOS module and VM configuration Add `itertools` Add arbitrage opportunity detection Implement `fallback` method for non standard callbacks Add more logs Fix sign error in optimum formula Add deployment scripts and `agenix-shell` secrets Bump cargo packages Fix typo Print out an error if processing a pair goes wrong Add `actionlint` to formatters Fix typo Add TODO comment Remove not relevant anymore comment Big refactor - process actions always in the correct order avoiding corner cases - avoid using semaphores New API key Add `age` to dev shell Used by Emacs' `agenix-mode` on my system Fix parametric deploy scripts Add `run-forge-tests` flake app Remove fork URL from Solidity source Remove `pairDir` argument Add link to `ArbitrageManager`'s ABI WIP
86 lines
2.2 KiB
XML
86 lines
2.2 KiB
XML
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= Notes
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Miscellaneous notes about *arbi*.
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== Uniswap V2's optimal input amount
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We consider two Uniswap V2-like pairs $A$ and $B$ both relative to the same two tokens.
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Let $X_A$ and $Y_A$ the reserves of the two tokens on the pair $A$ and $Y_A$ and $Y_B$ the reserves on the pair $B$ and assume that we want to perform 2 chained this way.
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$
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... ->^y^* A ->^(x_"out") B ->^(y_"out") ...
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$
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with $y^*$ the optimum amount to swap in order to maximize the gain function $G(y) = y_"out" - y^*$
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Let $0 <= f <= 1$ be the fee ($.03$ by deault on Uniswap V2), we know#footnote[https://www.youtube.com/watch?v=9EKksG-fF1k] that the optimum is one of the roots of the following second-grade equation:
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$
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k^2y^2 + 2k Y_A X_B y + (Y_A X_B)^2 - (1-f)^2 X_A Y_B Y_A X_B = 0
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$
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where
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$
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k = (1-f)X_B + (1-f)^2 X_A
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$
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In the Uniswap V2 implementation we have that $1-f = phi/1000$ (with $phi = 997$).
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Then we can rewrite:
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$
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k^2y^2 + 2k Y_A X_B y + (Y_A X_B)^2 - (phi/1000)^2 X_A Y_B Y_A X_B = 0
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$
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and
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$
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k = phi/1000 X_B + phi^2/1000^2 X_A
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$
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Let $a$, $b$ and $c$ be the three second-grade equation coefficients.
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$
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a = k^2
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$
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$
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b = 2k Y_A X_B
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$
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$
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c = (Y_A X_B)^2 - (phi/1000)^2 X_A Y_B Y_A X_B
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$
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Since $b$ is even we can find the roots with
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$
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y_i = (-b/2 plus.minus sqrt((b^2-4a c)/4))/a
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$
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Replacing our values:
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$
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(- k Y_A X_B y plus.minus sqrt(k^2 (Y_A X_B) ^2 ((Y_A X_B) ^2 -phi^2/1000^2 X_A Y_B X_B Y_A)))/k^2
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$
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$
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= -(Y_A X_B)/k plus.minus 1/k^2 sqrt(k^2 ((Y_A X_B)^2) -(Y_A X_B)^2 + phi^2/1000^2X_A Y_B X_B Y_A)
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$
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$
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= -(Y_A X_B)/k plus.minus 1/k sqrt((phi^2 X_B Y_B X_B Y_A )/1000^2)
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$
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Which, since the square root is positive, can be positive only considering $+$. In conclusion we get the following formula for the optimal amount of token $Y$:
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$
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y^* = 1/k (sqrt((phi^2 X_A Y_B X_B Y_A) / 1000^2) - Y_A X_B)
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$
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=== Solidity implementation details
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- Integer square roots can be effectively and cheaply computed using the Babylonian method #footnote[https://ethereum.stackexchange.com/a/97540/66173]
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- The square root can lead to overflow, in that case it can be convenient splitting it into something like
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$
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sqrt(phi times X_A div 1000 times Y_B) sqrt(phi times X_B div 1000 times Y_A)
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$
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