4.1 The StableSwap Invariant

The StableSwap invariant is a combination of the constant product invariant and the constant sum invariant, which are defined as

and

respectively, where C and D are positive real numbers, N is the number of tokens in the pool, and, for each j ∈ {1, 2, …, N}, b_j denotes the balance of token j in the pool. Observe that the product invariant is equivalent to Balancer’s geometric mean invariant with equally weighted tokens. On the other hand, note that the constant sum invariant gives a constant price of 1 for each pair of tokens, since if we deposit an amount b of token i and receive an amount a of token o, from the constant sum invariant, we obtain that

and hence, b_i + b_o = b_i + b + b_o − a, from where it follows that a = b, which implies that the effective price paid per unit of token o in terms of token i is . In particular, there is no difference between the spot price and the effective price paid by a trader; that is, there is no price impact.

As we previously mentioned, the StableSwap AMM invariant is obtained as a combination of the constant sum and constant product invariants. We will explain now how it is constructed. Let N be the number of tokens in the pool, and for j ∈ {1, 2, …, N}, let b_j be a variable that represents the balance of token j in the pool. Recall that the constant sum invariant is given by

for a certain positive number D and the constant product invariant is given by

for a certain positive number C. In order to combine these invariants, we will establish a relation between the numbers C and D. To this end, assume for a moment that all tokens are equally priced. Hence, for all j, k ∈ {1, 2, …, N}, the spot price of token j in terms of token k is 1, and then from Equation 3.​1, we obtain that b_j = b_k for all j, k ∈ {1, 2, …, N}. Thus, from the constant sum formula, we obtain that for all j ∈ {1, 2, …, N}, and substituting these expressions in the constant product formula, we obtain that . Thus, our constant product invariant will be

The first idea to combine both invariants is to take a convex combination of both formulae; that is, for any chosen t ∈ [0, 1], we consider the expression

Note that when t = 0, we obtain the constant product formula, and when t = 1, we obtain the constant sum formula.

Now we will modify the previous expression, replacing the parameter t with a new parameter x so that we obtain the constant product formula when x = 0 and the constant sum formula when x tends to infinity. To this end, we need to replace the parameter t with a function g(x) such that

Although many functions can do the trick, we consider the map g : [0, +∞) → ℝ given by . Note that this function g satisfies thedesired properties and that 0 ≤ g(x) ≤ 1 for all x ≥ 0. Replacing t = g(x) in the preceding expression, we obtain

or equivalently,

In addition, we want the parameter x to be dimensionless. To this end, we will substitute x = χD^N − 1 to obtain

Finally, we want the parameter χ to be dynamic rather than a constant chosen value. To this end, we substitute

(4.1)

where A is a positive number that is called amplification coefficient. In the actual code, there is a parameter labeled A, which is a positive integer and is equal to AN^N − 1. Therefore, from now on, we will assume that AN^N − 1 is a positive integer—and this will indeed be needed later.

The idea behind the substitution of Equation 4.1 is the following. Assuming that (which will not hold in general as the pool balances change), by the inequality of arithmetic and geometric means,³ we have that

(with equality holding if and only if b₁ = b₂ = … = b_N). Hence,

and thus, χ ≤ A. On the other hand, when the balances move away of the equilibrium point b₁ = b₂ = … = b_N, the parameter χ decreases, as we can see for the case N = 2 in Figure 4-1, where we plot the graph of the function f : [0, 2] × [0, 2] → ℝ given by

As we can see, the function has maximum value 1 when b₁ = b₂ and decreases when we move away from this line.

Making the aforementioned substitution of χ, we obtain

and multiplying both sides by , we get

(4.2)

which is the StableSwap invariant [11].

In Figure 4-2, we compare the StableSwap invariant for three different values of the amplification coefficient A with the constant product and constant sum invariants in a pool with two tokens. Observe that at the equilibrium point, the price of token X in terms of token Y is 1, since the slope of the curve is –1. In addition, near the equilibrium point, the curve looks like a constant sum invariant curve, and hence, the price impact is very low, and the price is approximately 1. But if the balances of the tokens deviate from the equilibrium point, the price changes, and the curve looks more like a constant product invariant one.

4.2 Mathematical Preliminaries

In this subsection, we will analyze several mathematical arguments that are used within the StableSwap AMM. We will also prove some results that will be needed later in order to explain the mathematical foundations of this AMM.

Consider a StableSwap liquidity pool consisting of N tokens. For each j ∈ {1, 2, …, N}, let B_j be the balance of token j in the pool. Let D₀ be the value of the pool parameter D. Recall that D₀ > 0. Let o ∈ {1, 2, …, N}. We will prove that the balance of token o is uniquely determined by D₀ and the balances B_j, j ≠ o.

For simplicity, let y = B_o. From the invariant Equation 4.2, we know that

that is,

Multiplying both sides of the equation by we get

Let

With this notations, the previous equation can be written as

(4.3)

and the value of y can be obtained through solving the previous quadratic equation. Observe that if y₁ and y₂ are the solutions of the previous equation, then y₁y₂ =  − c, and since c > 0, we obtain that y₁y₂ < 0. This means that one of the solutions of the previous equation is positive and the other is negative. Thus, the value of y that we are looking for is well defined since it is the only positive solution to the previous equation. Therefore, we have proved that the balance of token o is uniquely determined by D₀ and the balances B_j, j ≠ 0.

The previous proof allows us to give the following definition.

Definition 4.1. Consider a StableSwap liquidity pool that has N tokens. Let o ∈ {1, 2, …, N}. For each j ∈ {1, 2, …, N} − {o}, let B_j be the balance of token j in the pool. Let D₀ be the value of the pool parameter D. We define

as the only positive number that satisfies Equation 4.3. That is, the number is the balance of token o if the balance of token j is B_j for all j ∈ {1, 2, …, N} − {o} and the value of the pool parameter D is D₀.

We will analyze now how changes when we modify the balances B_j, j ≠ o, or the value of the pool parameter D.

Proposition 4.1. Consider a StableSwap liquidity pool of N tokens. Let o ∈ {1, 2, …, N}. For each j ∈ {1, 2, …, N} − {o}, let B_j and be positive real numbers. Let

and let

Let D₀ and be positive real numbers. If and for all j ∈ {1, 2, …, N} − {o}, then

If, in addition, any of the inequalities , or is strict, then

Proof. Let

and let

Let

Recall that y² + (b − D₀)y − c = 0 and and that y > 0 and y^´ > 0. Hence,

Suppose that for all j ∈ {1, 2, …, N} − {o} and D₀≥ . Then S ≤ Ś and P ≤ P´. And since , we obtain that c ≥ c´. On the other hand, since AN^N − 1 is a positive integer and N ≥ 2, it follows that . Thus,

Suppose that y < y^ʹ. Then

which entails a contradiction. Thus, y ≥ y^ʹ.

In addition, if any of the inequalities , or D₀≥ is strict, then c > c^ʹ. Suppose that y ≤ y^ʹ. Then

which entails a contradiction. Thus, y > y^ʹ.        □

4.2.1 Finding the Parameter D

After each transaction, the pool parameter D is updated based on the current pool state. Given a liquidity pool with N tokens and balances B₁, B₂, …, B_N, the parameter D can be found by solving the following equation:

(4.4)

which follows from the invariant Equation 4.2.

Note that in order for the value D to be well defined, we need the previous equation to have a unique positive solution. We will prove now that this is indeed the case. Let g : ℝ → ℝ be defined by

Note that g is a polynomial of degree N + 1 (and hence a continuous function) and that AN^N − 1 > 0 since AN^N − 1 is a positive integer and N ≥ 2. On the other hand, the derivative of g is

Observe that g^ʹ(x) > 0 for all x ≥ 0 since N, B₁, B₂, …, B_N and AN^N − 1 are positive numbers. Thus, g is a strictly increasing function in the interval [0, +∞). It follows that g has at most one root in that interval. In addition, since

(because the leading coefficient of g is a positive number), we obtain that g has at least one root in the interval (0, +∞). Therefore, g has exactly one positive root, and thus, the parameter D is well defined.

Now, in order to effectively compute the value of the parameter D, the smart contract applies the Newton’s Method⁴ to the function g. Explicitly, let d₀ > 0 be an initial approximate value for the parameter D. For example, we can take

where B₁, B₂, …, B_N are the balances of the tokens of the pool. Now let

Then, for each n ∈ ℕ, let

(4.5)

The process ends when the desired precision has been obtained, that is, when the difference between two consecutive terms, d_k and d_k + 1, is smaller than a certain number.

This process can be found in the get D function of the smart contract of the StableSwap AMM,⁵ as we will see now. For simplicity, let

Note that

This formula can be found in the code shown in Listing 4-1, where the translation between the variables of the code and our notation is the following:

Note also that d₀ = S.

def get_D (xp: uint256[N_COINS], amp: uint256) ->

↪ uint256:

    S: uint256 =0

    for _x in xp:

        S += _x

    if S == 0:

        return 0

    Dprev: uint256 = 0

    D: uint256 = S

    Ann: uint256 = amp * N_COINS

    for _i in range(255):

        D_P: uint256 = D

        for _x in xp:

            D_P = D_P * D / (_x * N_COINS) # If

            ↪  division by 0, this will be borked:

            ↪  only withdrawal will work. And that is

            ↪  good

        Dprev = D

        D = (Ann * S + D_P * N_COINS) * D / ((Ann - 1)

        ↪  * D + (N_COINS + 1) * D_P)

        # Equality with the precision of 1

        if D > Dprev:

            if D - Dprev <= 1:

                break

        else:

            if Dprev - D <= 1:

                break

    return D

Listing 4-1

get_D function of the smart contract of StableSwap

4.3 Trading Formulae

Consider a StableSwap liquidity pool of N tokens and no fees and let B₁, B₂, …, B_N be the balances of the tokens in the pool. Let D₀ be the value of the pool parameter D. Let i, o ∈ {1, 2, …, N}. As in the previous chapters, we want to compute the amount A_o of token o that a trader will receive when depositing a certain amount A_i of token i. For each j ∈ {1, 2, …, N}, let be the balance of token j after the trade. Clearly, and for all j ∈ {1, 2, …, N} − {i, o}. For simplicity, let . Note that A_o = B_o − y. We will compute y. From the previous section, we know that the value of y can be obtained by solving the quadratic Equation 4.3 and is given by following Definition 4.1.

4.3.1 Taking Fee into Consideration

Unlike Uniswap and Balancer, the pool fee for the StableSwap AMM is charged on the output token. In consequence, with the previous notations, after finding the balance y of token o using Equation 4.3, the amount A_o of token o that the trader receives is

where ϕ is the pool fee. Note that the trader is charged a fee of ϕ(B_o − y). In addition, a fraction ɷ of this charged fee is generally reserved for the protocol as an administrative fee and not deposited into the pool. And since, if there had been no fees, the balance of token o after the trade would have been equal to y, the balance of token o is updated in the following way:

Observe that

This last formula is the one that appears in the code of the StableSwap AMM, as we can see in Listing 4-3.

On the other hand, the updated balance of token i is clearly

def exchange(i: int128, j: int128, dx: uint256, min_dy:

↪  uint256):

    assert not self.is_killed # dev: is killed

    rates: uint256[N_COINS] = RATES

    old_balances: uint256[N_COINS] = self.balances

    xp: uint256[N_COINS] = self._xp_mem(old_balances)

    # Handling an unexpected charge of a fee on

    ↪  transfer (USDT, PAXG)

    dx_w_fee: uint256 = dx

    input_coin: address = self.coins[i]

    if i == FEE_INDEX:

        dx_w_fee = ERC20(input_coin).balanceOf(self)

    # "safeTransferFrom" which works for ERC20s which

    ↪  return bool or not

    _response: Bytes[32] = raw_call(

        input_coin,

        concat(

            ↪  method_id("transferFrom(address,address,uint256)")

            convert(msg.sender, bytes32),

            convert(self, bytes32),

            convert(dx, bytes32),

        ),

    max_outsize=32,

) # dev: failed transfer

if len(_response) > 0:

    assert convert(_response, bool) # dev: failed

    ↪  transfer

if i == FEE_INDEX:

    dx_w_fee = ERC20(input_coin).balanceOf(self) -

    ↪  dx_w_fee

x: uint256 = xp[i] + dx_w_fee * rates[i] /

↪  PRECISION

y: uint256 = self.get_y(i, j, x, xp)

dy: uint256 = xp[j] - y - 1 # -1 just in case

↪  there were some rounding errors

dy_fee: uint256 = dy * self.fee / FEE_DENOMINATOR

# Convert all to real units

dy = (dy - dy_fee) * PRECISION / rates[j]

assert dy >= min_dy, "Exchange resulted in fewer

↪  coins than expected"

dy_admin_fee: uint256 = dy_fee * self.admin_fee /

↪  FEE_DENOMINATOR

dy_admin_fee = dy_admin_fee * PRECISION / rates[j]

# Change balances exactly in same way as we change

↪  actual ERC20 coin amounts

self.balances[i] = old_balances[i] + dx_w_fee

# When rounding errors happen, we undercharge admin

↪  fee in favor of LP

self.balances[j] = old_balances[j] - dy -

↪  dy_admin_fee

# "safeTransfer" which works for ERC20s which

↪  return bool or not

_response = raw_call(

    self.coins[j],

    concat(

        method_id("transfer(address,uint256)"),

        convert(msg.sender, bytes32),

        convert(dy, bytes32),

    ),

    max_outsize=32,

)  # dev: failed transfer

if len(_response) > 0:

    assert convert(_response, bool) # dev: failed

    ↪  transfer

log TokenExchange(msg.sender, i, dx, j, dy)

Listing 4-3

exchange function of the smart contract of StableSwap

Example 4.1. Consider a StableSwap pool having three tokens, USDC, DAI, and USDT, with the following balances:

	Token 1	Token 2	Token 3
	USDC	DAI	USDT
Bj	100,000	120,000	80,000

Suppose that the fee of the pool is 0.03%, that the pool administrative fee is 50%, and that the value of the amplification coefficient A is 500.

Applying the algorithm described by Equation 4.5, we obtain that the value of D is approximately 299,999. Observe that it is very close to the sum B₁ + B₂ + B₃ of the balances of each token, which is 100,000+120,000+80,000=300,000.

Suppose that a trader wants to obtain USDC depositing 10,000 USDT. In order to compute the amount of USDC that the trader receives, we first compute the value of y from Equation 4.3, which represents the balance of USDC in the pool after the deposit of 10,000 USDT, keeping constant the value of D and assuming that there are no fees. Solving the quadratic equation, we obtain that y ≈ 89,999.71. Hence, the amount A_o of USDC that the trader receives is

The updated balance of USDC in the pool is given by

Observe that the balance of USDC in the pool has decreased by approximately 100,000 - 90,001.21 = 9,998.79, and since just 9,997.29 USDC are given to the trader, we obtain that the difference between those amounts—1.5 USDC—has been charged as an administrative fee to the liquidity providers, since an additional amount of 1.5 USDC has been removed from the pool.

Note also that if there had been no fees, the trader would have received approximately 100,000 – 89,999.71 = 10,000.29 USDC (this is 3 USDC more than the amount they received when fees were considered), and thus, the balance of USDC in the pool would have been 100,000 - 10,000.29 = 89,999.71 (which is represented by y). Therefore, the pool now has 90,001.21 - 89,999.71 = 1.5 USDC more than it would have had if there were no fees. As we can see, the trader has been charged a fee of 3 USDC, and from this amount, 1.5 USDC goes into the pool, and the other 1.5 USDC is reserved separately as an administrative fee of the protocol.

Finally, note that the balances of the tokens in the pool after the trade are as follows:

	USDC	DAI	USDT
Bj	90,001.21	120,000	90,000

Also, after the trade, the parameter D is recalculated with the algorithm given by Equation 4.5, and we have that the new value for the parameter D is approximately 300,000.57.

4.4 All-Asset Deposit

In a similar way as with Uniswap and Balancer, liquidity providers can deposit assets into a StableSwap pool in order to earn trading fees. In exchange for their deposit, they receive StableSwap LP tokens, which represent ownership of the assets contained in the pool. During the pool trading activity, the fees from swaps accrue into the pool, resulting in added value for the StableSwap LP tokens. In this section, we will describe how the all-asset deposit works in the StableSwap AMM.

Consider a StableSwap AMM with N tokens, and suppose that a liquidity provider wants to deposit amounts x₁, x₂, …, x_N of tokens 1, 2, …, N, respectively. Let B₁, B₂, …, B_N be the pool balances before the deposit. Since the pool needs to remain balanced, the amounts x₁, x₂, …, x_N must satisfy that

for all i, j ∈ {1, 2, …, N}.

Let M be the amount of existent StableSwap LP tokens before the deposit and let be the share that the new liquidity provider adds to the pool. Hence, they will receive a share q of the amount of existent StableSwap LP tokens; that is, they will receive an amount qM of StableSwap LP tokens. Note also that for all j ∈ {1, 2, …, N},

and thus, x_j = qB_j for all j ∈ {1, 2, …, N}. Hence, in order to receive a share q of the existent StableSwap LP tokens, the liquidity provider has to deposit, for each j ∈ {1, 2, …, N}, an amount qB_j of token j.

We will prove now that the number of newly minted StableSwap LP tokens is exactly reflected in the change of the StableSwap pool parameter D. Concretely, assume that the new liquidity provider deposits a share q of each token. Let D₀ and D₁ be the values of the pool parameter D before and after the deposit, respectively. Let B₁, B₂, …, B_N be the balances of the tokens in the pool before the deposit, and let be the balances of the tokens in the pool after the deposit. Note that for all j ∈ {1, 2, …, N}. From Equation 4.4, we know that D₀ and D₁ satisfy

and

Hence,

Let g : ℝ → ℝ be defined by

Note that both D₀ and are positive solutions of the equation g(x) = 0, and hence, since the map g has exactly one positive root (see subsection 4.2.1). Thus, D₁ = (1 + q)D₀.

Suppose, in addition, that the current number of StableSwap LP tokens in circulation is 10,000.

If a liquidity provider wants to perform an all-asset deposit of 1% of the pool size, they will need to deposit the following amounts of each of the tokens of the pool:

In addition, they will receive an amount of 10,000.0.01 = 100 StableSwap LP tokens.

4.5 All-Asset Withdrawal

We will now analyze the case in which a liquidity provider redeems StableSwap LP tokens. Suppose that the liquidity provider owns an amount m of StableSwap LP tokens, and let M be the amount of StableSwap LP tokens in circulation. Let ; that is, σ is the share of the pool that the liquidity provider owns. Let B₁, B₂, …, B_N be the balances of the tokens in the pool. Clearly, in return for the amount m of StableSwap LP tokens, the liquidity provider will receive, for each j ∈ {1, 2, …, N}, an amount σB_j of token j.

Note also that for all j ∈ {1, 2, …, N}, the balance of token j in the pool after the all-asset withdrawal is B_j − σB_j = (1 − σ)B_j. As we did in the case of the all-asset deposit, we will prove that the parameter D changes in the same proportion as the balances of the tokens of the pool. That is, we will prove that if the value of the parameter D is D₀ before the all-asset withdrawal, then its value after the withdrawal is (1 − σ)D₀.

We proceed in a similar way as we did in the previous section. Let D₁ be the value of the parameter D after the withdrawal. From Equation 4.4, we know that D₀ and D₁ satisfy

and

Hence,

Let g : ℝ → ℝ be defined by

Note that both D₀ and are positive solutions of the equation g(x) = 0, and hence, since the function g has exactly one positive root (see subsection 4.2.1). Thus, D₁ = (1 − σ)D₀.

Observe that the parameter D plays the same role as the parameter V in Balancer (and as the parameter L in Uniswap v2), meaning that for liquidity deposits (or withdrawals), the amount of tokens minted (or burned) is reflected in the change of the parameter D.

Assume that the number of StableSwap LP tokens in circulation is 100,000 and that a liquidity provider owns 100 StableSwap LP tokens. Hence, the corresponding share is ; that is, the liquidity provider owns 0.1% of the pool. If they decide to redeem their StableSwap LP tokens by performing an all-asset withdrawal, they will obtain the following amounts of each token:

4.6 Single-Asset Withdrawal

We will now tackle the single-asset withdrawal feature of Curve Finance. Suppose that we have a StableSwap pool that has N tokens with balances B₁, B₂, …, B_N and that the pool has a nonzero fee ϕ. Assume that a liquidity provider owns a share σ of all the StableSwap LP tokens and that instead of redeeming a proportion σ of each of the tokens of the pool, they want to obtain all their share in terms of just one token of the pool, say, token o. Let D₀ be the value of the parameter D before the withdrawal. If we were to perform an all-asset withdrawal, the value of the parameter D after the withdrawal would be (1 − σ)D₀, as we saw in the previous section. Thus, it makes sense to use this value of D to calculate the amount of token o that will be left after the single-asset withdrawal since the parameter D also reflects the liquidity of the pool in a certain sense.

Let y be the amount of token o after the withdrawal taking into account only the previous considerations. Since for each j∈ {1, 2, …, N} − {o} the amount of token j in the pool does not change after the single-asset withdrawal, applying the invariant Equation 4.2, we obtain that

Proceeding as in Section 4.2, we obtain that

Hence, we can obtain the value of y by solving the previous quadratic equation. Applying Definition 4.1, observe that

Therefore, if there were no fees, the liquidity provider would have to receive an amount B_o − y of token o (note that y < B_o by Proposition 4.1). However, since the liquidity provider receives only one token, the proportions of the tokens in the pool are modified, and thus, several spot prices also change, as occurred with the Balancer AMM. Thus, a fee has to be charged, in a similar way as when a trade is performed.

It will be reasonable to charge the fee on the difference between the balances of the tokens in both situations, since a direct single-asset withdrawal should be equivalent to an all-asset withdrawal followed by the corresponding trades, that is, trading, for each j∈ {1, 2, …, N} − {o}, the amount σB_j of token j received in the all-asset withdrawal so as to obtain token o. Hence, for each j∈ {1, 2, …, N} − {o}, the fee on token j should be ϕσB_j (note that here we are considering charging fees on the way in). Therefore, if we consider the situation of an all-asset withdrawal followed by the corresponding trades, for each j ∈ {1, 2, …, N} − {o}, the balance of token j in the pool after the trades will be (1 − σ)B_j + σB_j = B_j as expected. However, in order to obtain the amount of token o that the liquidity provider should receive, we have to compute the balance of token o with respect to the balances of all the other tokens considering that the liquidity provider deposits an amount σB_j − ϕσB_j of token j, due to the fees that are charged. In addition, the value of the parameter D to be considered is (1 − σ)D₀, that is, the same value as after the all-asset withdrawal, since in a trade, the value of D is preserved in order to compute the amount of tokens that should be given to the trader (as if there were no fees) and then when the fees are charged, the value of parameter D is updated. Observe that a similar situation occurs in Uniswap v2 and in Balancer’s AMMs.

where the amount y^ʹ has to be computed from the values of the balances of tokens j, with j ≠ o and the value of the parameter D by means of the quadratic Equation 4.3. Equivalently,

where for each .

In consequence, the amount of token o that has to be given to the liquidity provider should be B_o − y^ʹ. However, the StableSwap AMM also charges a fee on token o corresponding to the difference of balances of token o shown in the second table of this section. This fee is ϕ((1 − σ)B_o − y) (note that y < (1 − σ)B_o by Proposition 4.1). In conclusion, the amount of token o that is given to the liquidity provider is

(4.6)

This formula can be seen in the code of the StableSwap AMM that is shown in Listing 4-4, with the following translation between the variables of the code and our notation:

Suppose that the fee of the pool is 0.03%, the administrative pool fee is 50%, and the value of the amplification coefficient A is 500.

We now apply the algorithm given by Equation 4.5 to compute the value of the parameter D at the state of the pool given by the previous table, obtaining D ≈ 3,999,947.93. This value will be called D₀.

or in other words,

and

Solving the quadratic Equation 4.3, we obtain that y ≈ 1,759,997.36 and y^ʹ ≈ 1,760,003.96. Thus,

and the new value of the parameter D is 0.99D₀ ≈ 3,959,948.45.

Now, the liquidity provider trades 14,000 DAI for USDC. The amount of USDC that they receive is

where the value of y is obtained from Equation 4.3, or more precisely,

Hence,

that is, the liquidity provider receives 13,996.54 USDC more. On the other hand, the updated balance of USDC in the pool is given by

and the updated balances of DAI and USDT are 1,400,000 and 792,000, respectively. Also, the new value of the parameter D, which is obtained by applying the algorithm given by Equation 4.5, is D₁ ≈ 3,959,950.55.

Finally, the liquidity provider trades 8,000 USDT for USDC. The amount of USDC that they receive is

where . Thus,

that is, the liquidity provider receives 7,999.49 USDC more.

Therefore, the liquidity provider has received a total amount of approximately

which is a bit more than the 39,989.44 USDC they obtain performing a single-asset withdrawal.

4.7 StableSwap LP Token Swap

As we did in the previous chapters, consider two different (Uniswap, Balancer, or StableSwap) pools, P₁ and P₂, respectively. Let LP₁ and LP₂ be the LP tokens of pools P₁ and P₂, respectively. We want to compute the fair swap price between LP₁ and LP₂. To this end, we will calculate the total value of each of the pools in terms of a chosen token Z, and then we will compute the value of tokens LP₁ and LP₂ in terms of token Z so as to find the fair swap price.

Firstly, we need to make the following observation regarding StableSwap liquidity pools. Let P be a StableSwap liquidity pool that has N tokens Y₁, Y₂, …, Y_N with balances B₁, B₂, …, B_N, respectively. Let Z be any token, and for each j ∈ {1, 2, …, N}, let P_j be the price of token Y_j in terms of token Z.⁶ Then, the value of the whole pool P in terms of token Z is

Hence, if the circulating supply of StableSwap LP tokens of pool P is M, the value of each token is

Now, for j ∈ {1, 2}, let V_j(Z) be the value of each LP token of pool P_j in terms of token Z, which can be computed as in the previous paragraph for StableSwap pools and as in the previous chapters for Uniswap v2 or Balancer pools. Let P be the price of one LP₁ token in terms of LP₂ tokens. Hence, V₁(Z) = P ⋅ V₂(Z), and thus,

We will now give two examples where we will compute the fair swap price between StableSwap LP tokens and LP tokens of different pools.

We want to compute the fair swap price between Alice’s and Bob’s LP tokens. To this end, we will use USDC as the token Z of the previous argument. Suppose that the market prices of ETH and USDT in terms of USDC are 4,000 USDC/ETH and 0.9995 USDC/USDT. Then, the value of each Balancer BPT token is

and the value of each StableSwap LP token is

Thus, the price of each of Alice’s tokens in terms of Bob’s tokens is , which means that if Alice gives Bob 1 of her tokens from the Balancer pool, she will get 6.5 of Bob’s tokens from the StableSwap pool in return.

Again, we want to compute the fair swap price between Alice’s and Bob’s tokens. As in the previous example, we will use USDC as token Z, and we will assume that the market prices of ETH and USDT in terms of USDC are 4,000 USDC/ETH and 0.9995 USDC/USDT. Then, the value of each Uniswap v2 LP token is

and the value of each StableSwap LP token is

Thus, the price of each of Alice’s tokens in terms of Bob’s tokens is ; that is, if Alice gives Bob 1 of her Uniswap LP tokens, she will get 2.6 of Bob’s StableSwap LP tokens in return.

4.8 Summary

In this chapter, we explained how the invariant formula for the StableSwap AMM is obtained and how trades are performed. We also showed how liquidity can be deposited and withdrawn and gave a detailed description of the single-asset withdrawal feature.

In the next chapter, we will study the Uniswap v3 AMM, which introduces concentrated liquidity. As we shall see, the design of the Uniswap v3 AMM is very different from those of the AMMs that we have described before.