# ConstantProduct AMM Contract

Zoe v0.24.0. Last updated August 25, 2022.
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The Constant Product AMM is an automated market maker (AMM) that supports multiple liquidity pools and direct exchanges across pools. It's called the "Constant Product" AMM because it uses the constant product rule, which is one of a family of rules that enable a market maker to guarantee to be able to continue to make trades, regardless of how prices change. (The constant product rule does this by ensuring that the product of the contents of two pools of assets remains constant as trading takes place.)

Each liquidity pool maintains a price for exchanges between the central token and a secondary token. Secondary tokens can be exchanged with each other, but only through the central token. For example, if BLD and ATM are two token types and IST is the central currency, a swap giving ATM and wanting BLD would first use the pool (ATM, IST) then the pool (BLD, IST). There are no direct liquidity pools between two secondary tokens.

There should only need to be one instance of this contract, so liquidity can be shared as much as possible. Each secondary currency has a separate pool of liquidity.

When the contract is instantiated, the terms specify the central token. Invitations for adding and removing liquidity and for making trades are available by calling methods on the publicFacet. Other publicFacet operations support querying prices and the sizes of pools. Create new pools with addPool().

When making trades or requesting prices, the caller must specify that either the input price (swapIn, getInputPrice) or the output price (swapOut, getOutputPrice) is fixed. For swaps, the required keywords are In for the trader's give amount, and Out for the trader's want amount. getInputPrice() and getOutputPrice() each take two amounts. When getInputPrice() or swapIn() is called, the amountOut parameter indicated the desired amountOut; if amountIn is insufficient to provide that much, the result indicates that no trade will take place. (The returned amountIn and amountOut will both be empty amounts.) Similarly, when swapIn() or getOutputPrice() is called, amountIn is treated as a maximum. If it would take a greater amount to get the specified amountOut, the result indicates no trade.

When adding and removing liquidity, the keywords are Central, Secondary, and Liquidity. Adding liquidity uses Central and Secondary in the give section and Liquidity in the want section. Removing liquidity reverses the keywords: Liquidity in the give section, and Central and Secondary in the want section. If the proposal specifies amounts directly taken from a recent quote, and any trading has intervened, the trade is unlikely to be accepted. You can either specify limits on how far the price may have moved, or specify limits of zero and trust the contract to trade fairly.

Transactions that don't require an invitation include addPool() and the queries (getInputPrice(), getOutputPrice(), getPoolAllocation(), getLiquidityIssuer(), and getLiquiditySupply()).

# The ConstantProduct API

These examples use IST as the central token. BLD and ATM are secondary currencies.

# Trading with the ConstantProduct AMM

Once trading pools have been set up (see below), a new trader can interact with the market by asking for the current price, making an invitation, and making an offer. If Sara has ATM and needs 275 BLD for a deal she has negotiated, she can use getOutputPrice() to get a quote. (An empty amount indicates no limit on the amountOut of the result.)

const quote = E(publicFacet).getOutputPrice(
  AmountMath.make(BLDBrand, 275n),

Let's assume the quote says she needs to provide 216 ATM. Sara believes the price is somewhat volatile, and she doesn't want to make repeated calls, so she pads her offer. If the appropriate pools don't exist, she'll get an error (brands were not recognized). If someone sells a lot of ATM into the pool just ahead of her, the price will increase, and she'll have to decide whether to deposit more ATM or wait for the price to stabilize. If someone buys a lot of ATM just ahead of her order, she'll get the 275 BLD for less and will get some ATM back.

const saraProposal = harden({
  want: { Out: AmountMath.make(BLDBrand, 275n) },
  give: { In: AmountMath.make(atmBrand, 220n) },

const swapInvitation = await E(publicFacet).makeSwapOutInvitation();
const atmPayment =
  harden({ In: saraAtmPurse.withdraw(AmountMath.make(atmBrand, 220n)) });

const saraSeat = await E(zoe).offer(swapInvitation, saraProposal, atmPayment);
const saraResult = await saraSeat.getOfferResult();

If the result is Swap successfully completed., she got the BLD for 220 ATM or less (she'll want to deposit any refund). Otherwise the market price moved against her, and she'll have to check the price again and make another offer.

const BLDProceeds = await E(saraSeat).getPayout('In');
const atmRefund = await E(saraSeat).getPayout('Out');

const BLDProceedsAmount = E(saraBLDPurse).deposit(BLDProceeds);

# Creating a New Pool

When the contract is first instantiated, there won't be any pools ready for trading. addPool() adds a new currency, which can then be funded. (All currencies must be fungible.) When a pool is first funded, there's no other basis on which to decide how much liquidity to create, so the liquidity amount equals the amount of the central token in the offer.

const BLDLiquidityIssuer = await E(publicFacet).addPool(BLDIssuer, 'BLD');

Alice sees that the current rate in the external market is 2 BLD for each IST, so she deposits twice as many BLD as IST to fund the market.

const aliceProposal = harden({
  want: { Liquidity: BLDLiquidity(50n) },
  give: {
    Secondary: AmountMath.make(BLDBrand, 100n),
    Central: AmountMath.make(ISTBrand, 50n),
const alicePayments = {
  Secondary: aliceBLDPayment,
  Central: aliceISTPayment,

const aliceAddLiquidityInvitation = E(publicFacet).makeAddLiquidityInvitation();
const addLiquiditySeat = await E(zoe).offer(

# Adding Liquidity to an Existing Pool

When adding or removing liquidity to pools that have already been established, the amounts deposited must be in proportion to the current balances in the pool. The calculation is based on the amount of the Central asset. The Secondary assets must be added in proportion. If less Secondary is provided than required, the offer is exited with no trade. If more of the secondary is provided than is required, the excess is returned.

Bob calls getPoolAllocation() to find the relative levels. Let's say the answer is that the current ratio is 1234 BLD to 1718 IST.

const BLDPoolAlloc = E(publicFacet).getPoolAllocation(BLDBrand);
const ISTValue = BLDPoolAlloc.Central.value;
const BLDValue = BLDPoolAlloc.secondary.value;

Now he can add liquidity. The price ratio changes when anyone trades with the pool, so he should leave some flexibility in the proposal. The pool calculates the amount of secondary currency required based on the amount of central currency provided. Bob bumps up the amount of BLD he'll contribute by a little. If he was concerned about how much liquidity this would produce, he would calculate it and specify a rough figure, but there's no need in this case.

const bobProposal = harden({
  give: {
    Central: AmountMath.make(ISTBrand, 1800n),
    Secondary: AmountMath.make(BLDBrand, 1200n),
  want: { Liquidity: AmountMath.make(liquidityBrand, 0n) },
  exit: { onDemand: null },

const bobPayments = {
  Central: bobISTPayment,
  Secondary: bobBLDPayment,

const seat = await E(zoe).offer(addLiquidityInvite, bobProposal, bobPayments);

# Governance

The ConstantProduct AMM uses governance to manage changes to two parameters: PoolFee and ProtocolFee. The current values of the parameters and the history of governance voting to update their values is visible through the governance APIs.

An instance of the ConstantProduct AMM is managed by a contractGovernor, which controls the ability to change contract parameters and add new types of collateral. The contractGovernor adds these four methods to the contract's publicFacet:

  • getSubscription(): get a Subscription that updates when votes are called.
  • getContractGovernor(): returns the contractGovernor for verification.
  • getGovernedParamsValues(): returns a structure showing the current values of the two parameters.
  • getParamValue('PoolFee'): gets a description of the current value of either parameter. Note the initial capital letter.