test
curve v1 通过实现 StableSwap 恒等式的交易池,达到比恒定乘积(Uniswap)交易池更高效低滑点的效果。适合相互之间价格稳定的交易对,比如两个稳定币(DAI-USDC)或同一锚定标的的资产(ETH-sETH synthetic ETH)。
overview
curve v1 核心公式
将恒定和等式与恒定乘积等式合并,给予系数 $\chi$ (希腊字母) $\chi D^{n-1}\Sigma x_i+\Pi x_i=\chi D^n + (\frac{D}{n})^n$
令 $\chi$ 为如下表达
$\chi=\frac{A\Pi x_i}{(D/n)^n}$
代入可得最终的核心公式
$An^n\Sigma x_i + D = ADn^n + \frac{D^{n+1}}{n^n\Pi x_i}$
pool-templates
pool 交易池合约有 5 种,他们基本构造方法相同,但有不同风格。(5 种合约模板参见 /contracts/pool-templates
)
a
: 用于交易 a-Token 类型的合约(余额自增的 token)的交易池子 aToken.balanceOf()base
: 基础的交易池子eth
: 带 ETH 的交易池子meta
: 元池,基本资产与 curve LP token 的交易池子y
: 用于交易 yearn-Token 类型的合约(可内生息的 token)的交易池子
使用模板合约部署交易池合约,主要修改以下初始化参数
___USE_LENDING___
: 布尔值数组,交易资产中是否有从借贷协议中借出的资产___N_COINS___
: 池子中的资产种类数量___PRECISION_MUL___
: 整型数组,每个资产的精度调整到 1e18 所需的乘数___RATES___
: 整型数组,表示每个资产统一到 1e18 精度后的相对比例(价格)
Metapools 元池因为是某个资产和 base pool 类型的 lp token 组成交易对,所以还用了以下参数:
__BASE_N_COINS__
: base pool 内的 N__BASE_PRECISION_MUL__
: base pool 内的 precision__BASE_RATES__
: base pool 内的 rates
pooldata.json
是主要用于测试的辅助参数,与合约内保持一致
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{
"wrapped_contract": "yERC20", // 测试用的 wrapped token 合约
"base_pool_contract": "" // 元池中所对应的 base pool
"coins": [ // each list item represents 1 swappable coin within the pool
{
"decimals": 18, // number of decimal places for the underlying coin
"tethered": false, // 调用token的 transfer/approve 操作是否返回 `None` (true 是没有返回,标准ERC20 应该有返回布尔值,但usdt没有返回值)
"wrapped": true, // token 是否为 wrapped token
"wrapped_decimals": 18, // wrapped token 的精度,如果不是wrapped可以省略
},
]
"rate_calculator_address": "" // 计算兑换率的策略合约地址
}
base pool
基础的交易池子合约,例如 3pool
constans
合约中的不变量
- fee 以 1e10 为 100%
- 手续费率,A 系数等参数可以经过治理投票后修改
- A 系数的数值的修改是线性变化的
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# These constants must be set prior to compiling
# 部署时的初始化参数
N_COINS: constant(int128) = ___N_COINS___
PRECISION_MUL: constant(uint256[N_COINS]) = ___PRECISION_MUL___
RATES: constant(uint256[N_COINS]) = ___RATES___
# fixed constants
FEE_DENOMINATOR: constant(uint256) = 10 ** 10 # 计算手续费时的分母
PRECISION: constant(uint256) = 10 ** 18 # 计算资产数量时需要统一到的精度水平
MAX_ADMIN_FEE: constant(uint256) = 10 * 10 ** 9 # 管理员手续费率的最大值,设置费率不能超过此值 10%
MAX_FEE: constant(uint256) = 5 * 10 ** 9 # 手续费率的最大值,设置费率不能超过此值 5%
MAX_A: constant(uint256) = 10 ** 6 # A 系数的最大值,当前3pool A = 2000
MAX_A_CHANGE: constant(uint256) = 10 # 规定每次对A的调整不能超过原有值的倍数范围 即 1/10 * A <= A' <= 10*A
ADMIN_ACTIONS_DELAY: constant(uint256) = 3 * 86400 # 配置更改的延迟生效时间
MIN_RAMP_TIME: constant(uint256) = 86400 # 线性修改A系数的最小时间间隔
A_PRECISION: constant(uint256) = 100 # A 的计算精度
KILL_DEADLINE_DT: constant(uint256) = 2 * 30 * 86400 # 部署合约多久后才能调用 kill_me 方法
variables
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coins: public(address[N_COINS]) # 资产 token 地址数组
balances: public(uint256[N_COINS]) # 资产的数量
fee: public(uint256) # fee * 1e10
admin_fee: public(uint256) # admin_fee * 1e10
owner: public(address)
lp_token: public(address)
initial_A: public(uint256) # A 的初始值,或调整之前的值
future_A: public(uint256) # A 的当前值,或调整的目标值
initial_A_time: public(uint256) # A 的调整开始时间
future_A_time: public(uint256) # A 的调整完成时间
admin_actions_deadline: public(uint256) # 当调整commit提交后,管理员执行调整的 deadlin
transfer_ownership_deadline: public(uint256) # 当移交所有权commit提交后,管理员执行调整的 deadlin
future_fee: public(uint256) # 调整手续费率的目标值
future_admin_fee: public(uint256) # 调整管理员手续费率的目标值
future_owner: public(address) # 调整所有权的目标地址
is_killed: bool
kill_deadline: uint256
functions
constructor
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@external
def __init__(
_owner: address,
_coins: address[N_COINS],
_pool_token: address,
_A: uint256,
_fee: uint256,
_admin_fee: uint256
):
"""
@notice Contract constructor
@param _owner Contract owner address
@param _coins Addresses of ERC20 conracts of coins
@param _pool_token Address of the token representing LP share
@param _A Amplification coefficient multiplied by n * (n - 1)
@param _fee Fee to charge for exchanges
@param _admin_fee Admin fee
"""
for i in range(N_COINS):
assert _coins[i] != ZERO_ADDRESS
self.coins = _coins
self.initial_A = _A * A_PRECISION
self.future_A = _A * A_PRECISION
self.fee = _fee
self.admin_fee = _admin_fee
self.owner = _owner
self.kill_deadline = block.timestamp + KILL_DEADLINE_DT
self.lp_token = _pool_token
A
主要逻辑在 _A()
,由于 A 系数不能突然发生较大变化,这样造成的内部价格突然变化,从而导致被外部套利者利用,套走过多资产。所以对 A 修改是线性变化的,而非立即变成目标值。其计算逻辑可改写为如下形式:
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A_current = A_initial + abs(A_future - A_initial) * (t_current - t_initial) / (t_future - t_initial)
观察核心公式可知,A 值越大,恒定和等式部分的项权重越大,其图形就越趋近于直线,A 值越小则趋近于恒定乘积的等式,图形更为弯曲(当 N=2 时,可以理解为更趋近于 Uniswap 的形状)。所以当 A 值变化时,图形的弯曲程度会发生改变,资产之间的价格也随之改变。如果突然有很大的变化,会导致池子内部价格与外部价格偏差过大,产生较大的套利空间。
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@view
@internal
def _A() -> uint256:
"""
Handle ramping A up or down
"""
t1: uint256 = self.future_A_time
A1: uint256 = self.future_A
if block.timestamp < t1:
A0: uint256 = self.initial_A
t0: uint256 = self.initial_A_time
# Expressions in uint256 cannot have negative numbers, thus "if"
# 因为合约中不能有负值,所以这里要判断大小,给出减法的顺序
if A1 > A0:
return A0 + (A1 - A0) * (block.timestamp - t0) / (t1 - t0)
else:
return A0 - (A0 - A1) * (block.timestamp - t0) / (t1 - t0)
else: # when t1 == 0 or block.timestamp >= t1
return A1
@view
@external
def A() -> uint256:
return self._A() / A_PRECISION
@view
@external
def A_precise() -> uint256:
return self._A()
xp
存放每个资产价值(balance * price)的数组
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'''返回当前池子的xp'''
@view
@internal
def _xp() -> uint256[N_COINS]:
result: uint256[N_COINS] = RATES
for i in range(N_COINS):
result[i] = result[i] * self.balances[i] / PRECISION
return result
'''指定balance数量计算xp'''
@pure
@internal
def _xp_mem(_balances: uint256[N_COINS]) -> uint256[N_COINS]:
result: uint256[N_COINS] = RATES
for i in range(N_COINS):
result[i] = result[i] * _balances[i] / PRECISION
return result
virtual_price
p_virtual = D / lp_totalSupply
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@view
@external
def get_virtual_price() -> uint256:
"""
@notice The current virtual price of the pool LP token
@dev Useful for calculating profits
@return LP token virtual price normalized to 1e18
"""
D: uint256 = self._get_D(self._xp(), self._A())
# D is in the units similar to DAI (e.g. converted to precision 1e18)
# When balanced, D = n * x_u - total virtual value of the portfolio
# D 的单位与 DAI 相似,精度 1e18
# 当价格处于平衡点,D = n * 每个资产以u计算的总价值
# 此时D就是整个池子资产组合的总(虚拟)价值
token_supply: uint256 = ERC20(self.lp_token).totalSupply()
return D * PRECISION / token_supply
calc_token_amount
计算调用 deposit 和 withdraw 后,lp token 的变化数量。该方法会考虑滑点的影响,但不包括手续费。主要用于防止抢跑攻击,而非用于交易的精确计算。
根据 D 的变化比例来同比计算 lp token 的变化。
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@view
@external
def calc_token_amount(_amounts: uint256[N_COINS], _is_deposit: bool) -> uint256:
"""
@notice Calculate addition or reduction in token supply from a deposit or withdrawal
@dev This calculation accounts for slippage, but not fees.
Needed to prevent front-running, not for precise calculations!
@param _amounts Amount of each coin being deposited
@param _is_deposit set True for deposits, False for withdrawals
@return Expected amount of LP tokens received
"""
amp: uint256 = self._A()
balances: uint256[N_COINS] = self.balances
D0: uint256 = self._get_D_mem(balances, amp) # 缓存变化之前的D
for i in range(N_COINS): # 遍历改变每个资产的数量
if _is_deposit:
balances[i] += _amounts[i]
else:
balances[i] -= _amounts[i]
D1: uint256 = self._get_D_mem(balances, amp) # 计算变化之后的D
token_amount: uint256 = CurveToken(self.lp_token).totalSupply()
diff: uint256 = 0
if _is_deposit:
diff = D1 - D0
else:
diff = D0 - D1
# 根据D的变化比例计算lp token变化量
# (delta D / D0) * totalSupply
return diff * token_amount / D0
add_liquidity
添加流动性,向池子注入资产。传入资产数量数组和最小 lp token mint 数量,返回实际 mint 的数量。
- 根据传入 token 数量,计算 D 的增量 delta D1
- 不考虑手续费的影响
assert delta D1 > 0
- 考虑手续费的影响,计算 D 的增量 delta D2
- 先算出不考虑手续费时,每个资产的变化数量(difference_balance)
- 根据费率分别计算每个资产的手续费
fee_i = fee_rate * difference_balance
- 将资产数量减去手续费的数量,然后重新计算考虑手续费影响的 D 值,进而得到
delta D2
- 根据 delta D2 与 D 的比例,同比计算新增 lp token 数量
mint amount = (delta D2 / D) * totalSupply
- 如果 mint 数量 < 设定的最小数量,交易回滚
- 将资产 token 从用户转入本合约,为用户 mint lp token,最后返回实际 mint 数量
关于 lp token 流动性数量的含义:
- 池子首次添加流动性,不会收取手续费,且 mint 数量就是 D 值
- 如果不考虑手续费影响,lp token 数量与 D 值同步
- 如果添加流动性的资产比例,不会改变当前价格,其过程也不会发生交易,因此就不会产生手续费
admin_fee 是一个百分比,含义是协议将手续手续中的一定比例,作为协议费用。收取的协议费(admin fee) 的资产仍然会计入 balances 字段,但是不会计入 lp token 数量。
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@external
@nonreentrant('lock')
def add_liquidity(_amounts: uint256[N_COINS], _min_mint_amount: uint256) -> uint256:
"""
@notice Deposit coins into the pool
@param _amounts List of amounts of coins to deposit
@param _min_mint_amount Minimum amount of LP tokens to mint from the deposit
@return Amount of LP tokens received by depositing
"""
assert not self.is_killed # dev: is killed
amp: uint256 = self._A()
old_balances: uint256[N_COINS] = self.balances
# Initial invariant
# 计算变化前的D数值
D0: uint256 = self._get_D_mem(old_balances, amp)
lp_token: address = self.lp_token
token_supply: uint256 = CurveToken(lp_token).totalSupply()
new_balances: uint256[N_COINS] = old_balances
for i in range(N_COINS):
if token_supply == 0:
assert _amounts[i] > 0 # 初次添加流动性,每种资产的数量都必须大于0
# balances store amounts of c-tokens
# ?? 这里注释有疑问,base pool 应该不会有c-token ??
new_balances[i] += _amounts[i]
# Invariant after change
# 计算变化后的D值(不考虑手续费影响)
D1: uint256 = self._get_D_mem(new_balances, amp)
assert D1 > D0
# We need to recalculate the invariant accounting for fees
# to calculate fair user's share
# 我们需要公平的计算用户的份额(考虑手续费)
D2: uint256 = D1
fees: uint256[N_COINS] = empty(uint256[N_COINS])
mint_amount: uint256 = 0
if token_supply > 0:
# Only account for fees if we are not the first to deposit
# 初次添加流动性不会收取手续费,换言之,后续的添加都会收取
# ?? N_COINS / (4 * (N_COINS - 1)) ??
fee: uint256 = self.fee * N_COINS / (4 * (N_COINS - 1))
admin_fee: uint256 = self.admin_fee
for i in range(N_COINS):
# ideal_balance 每个资产理想的数量不考虑手续费
# difference 每个资产理想数值与现值的差量
# fee_i = difference * fee_rate
# 实际资产数量 = 理想数量 - 手续费
# new_balance_i = ideal_balance - fee_i
ideal_balance: uint256 = D1 * old_balances[i] / D0
difference: uint256 = 0
new_balance: uint256 = new_balances[i]
if ideal_balance > new_balance:
difference = ideal_balance - new_balance
else:
difference = new_balance - ideal_balance
fees[i] = fee * difference / FEE_DENOMINATOR
self.balances[i] = new_balance - (fees[i] * admin_fee / FEE_DENOMINATOR)
new_balances[i] -= fees[i]
# 根据扣除手续费之后的资产数量,计算新的D值
# 然后同比计算 lp token 的增量,即为 mint 数量
D2 = self._get_D_mem(new_balances, amp)
mint_amount = token_supply * (D2 - D0) / D0
else:
# 初次添加流动性,不收取手续费,池子资产数量即为本地添加的数量
# lp token 数量就是D,
self.balances = new_balances
mint_amount = D1 # Take the dust if there was any
assert mint_amount >= _min_mint_amount, "Slippage screwed you"
# Take coins from the sender
for i in range(N_COINS):
if _amounts[i] > 0:
# "safeTransferFrom" which works for ERC20s which return bool or not
_response: Bytes[32] = raw_call(
self.coins[i],
concat(
method_id("transferFrom(address,address,uint256)"),
convert(msg.sender, bytes32),
convert(self, bytes32),
convert(_amounts[i], bytes32),
),
max_outsize=32,
)
if len(_response) > 0:
assert convert(_response, bool) # dev: failed transfer
# end "safeTransferFrom"
# Mint pool tokens
CurveToken(lp_token).mint(msg.sender, mint_amount)
log AddLiquidity(msg.sender, _amounts, fees, D1, token_supply + mint_amount)
return mint_amount
exchange
交易,输入单一资产 i
,输出另一个单一资产 j
,规定最小获得的输出数量 _min_dy
,返回实际交易输出数量 (dy)。
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@external
@nonreentrant('lock')
def exchange(i: int128, j: int128, _dx: uint256, _min_dy: uint256) -> uint256:
"""
@notice Perform an exchange between two coins
@dev Index values can be found via the `coins` public getter method
@param i Index value for the coin to send
@param j Index valie of the coin to recieve
@param _dx Amount of `i` being exchanged
@param _min_dy Minimum amount of `j` to receive
@return Actual amount of `j` received
"""
assert not self.is_killed # dev: is killed
# 根据当前资产数量计算现有每个资产的总价值 xp
old_balances: uint256[N_COINS] = self.balances
xp: uint256[N_COINS] = self._xp_mem(old_balances)
# 计算输入资产后,输出资产的价值,即y
rates: uint256[N_COINS] = RATES
x: uint256 = xp[i] + _dx * rates[i] / PRECISION
y: uint256 = self._get_y(i, j, x, xp)
# 计算输出资产的价值变化量,即 dy
# 防止计算精度引起的数值不准确,向下舍入 (-1)
# 计算手续费价值 dy_fee
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
# 将扣除手续费后的价值 (dy - dy_fee) , 并统一精度 PRECISION = 1e18
dy = (dy - dy_fee) * PRECISION / rates[j]
assert dy >= _min_dy, "Exchange resulted in fewer coins than expected"
# 从手续费价值中抽取一定比例的协议费,并转根据价格转换成实际的输出资产数量
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
# 将输入数量加到balances上,我们实际运算的是 ERC20 数量
self.balances[i] = old_balances[i] + _dx
# When rounding errors happen, we undercharge admin fee in favor of LP
# 当舍入错误发生,协议将不收取协议费来支持流动性的发展
# ?? 没有看到 undercharge 的逻辑 ??
# 猜测是 / FEE_DENOMINATOR 时,会将小于 FEE_DENOMINATOR 的数量舍入,所以认为这部分是少收了协议费
#
# 输出资产的数量 减去实际输出给用户的数量(fee会保留在合约中), 再扣除协议费数量
# 值得注意的是,这里 balance - dy 已经将交易手续费fee留给了 lp 提供者
# 然后协议又从 lp 提供者的fee份额中,抽取了协议费
self.balances[j] = old_balances[j] - dy - dy_admin_fee
# 转入输入资产
_response: Bytes[32] = raw_call(
self.coins[i],
concat(
method_id("transferFrom(address,address,uint256)"),
convert(msg.sender, bytes32),
convert(self, bytes32),
convert(_dx, bytes32),
),
max_outsize=32,
)
if len(_response) > 0:
assert convert(_response, bool)
# 转出输出资产
_response = raw_call(
self.coins[j],
concat(
method_id("transfer(address,uint256)"),
convert(msg.sender, bytes32),
convert(dy, bytes32),
),
max_outsize=32,
)
if len(_response) > 0:
assert convert(_response, bool)
log TokenExchange(msg.sender, i, _dx, j, dy)
return dy
remove_liquidity
移除流动性,传入要移除的 lp token 数量 _amount
,最小赎回资产数量数组 _min_amounts
,转给调用者的资产数量将按照池子当前的资产比分配,返回赎回的资产数量数组。
由于期间没有发生资产的转换(exchange),该方法不收取手续费。
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@external
@nonreentrant('lock')
def remove_liquidity(_amount: uint256, _min_amounts: uint256[N_COINS]) -> uint256[N_COINS]:
"""
@notice Withdraw coins from the pool
@dev Withdrawal amounts are based on current deposit ratios
@param _amount Quantity of LP tokens to burn in the withdrawal
@param _min_amounts Minimum amounts of underlying coins to receive
@return List of amounts of coins that were withdrawn
"""
lp_token: address = self.lp_token
total_supply: uint256 = CurveToken(lp_token).totalSupply()
amounts: uint256[N_COINS] = empty(uint256[N_COINS])
for i in range(N_COINS):
# 根据移除的 lp token 数量与总数的比例,同比计算balance的变化量,即为赎回的数量
# 这里没有收取手续费
old_balance: uint256 = self.balances[i]
value: uint256 = old_balance * _amount / total_supply
assert value >= _min_amounts[i], "Withdrawal resulted in fewer coins than expected"
self.balances[i] = old_balance - value
amounts[i] = value
_response: Bytes[32] = raw_call(
self.coins[i],
concat(
method_id("transfer(address,uint256)"),
convert(msg.sender, bytes32),
convert(value, bytes32),
),
max_outsize=32,
)
if len(_response) > 0:
assert convert(_response, bool)
CurveToken(lp_token).burnFrom(msg.sender, _amount) # dev: insufficient funds
log RemoveLiquidity(msg.sender, amounts, empty(uint256[N_COINS]), total_supply - _amount)
return amounts
remove_liquidity_imbalance
(不平衡的)移除流动性,传入要赎回的资产数量数组 _amounts
, 最大能销毁的 lp token 数量 _max_burn_amount
,转给调用者的资产数量将按照入参分配,返回实际销毁掉的 lp token 数量。
由于期间发生资产的转换(exchange),该方法要收取手续费。
- 根据传入 token 数量,计算 D 的减量 delta D1
- 不考虑手续费的影响
- 考虑手续费的影响,计算 D 的减量 delta D2
- 先算出不考虑手续费时,每个资产的变化数量(difference_balance)
- 根据费率分别计算每个资产的手续费
fee_i = fee_rate * difference_balance
- 将资产数量减去手续费的数量,然后重新计算考虑手续费影响的 D 值,进而得到
delta D2
- 根据 delta D2 与 D 的比例,同比计算减少 lp token 数量
burn_amount = (delta D2 / D) * totalSupply
assert burn_amount != 0
assert burn_amount <= _max_burn_amount
- 将资产按照入参的数量转给用户,销毁用户的 lp token,最后返回实际 burn 数量
为何不直接从 _amounts
中扣除 fee,再计算新的 D?
- 用户的不平衡 remove 行为,让池子的资产比例发生变化,某些资产减少,某些资产增加,可以认为其过程中发生了交易(exchange)
_amounts
中并所有资产都参与了交易,必然有一部分没有参与交易,所以不能直接以_amounts
的数量来计算手续费
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@external
@nonreentrant('lock')
def remove_liquidity_imbalance(_amounts: uint256[N_COINS], _max_burn_amount: uint256) -> uint256:
"""
@notice Withdraw coins from the pool in an imbalanced amount
@param _amounts List of amounts of underlying coins to withdraw
@param _max_burn_amount Maximum amount of LP token to burn in the withdrawal
@return Actual amount of the LP token burned in the withdrawal
"""
assert not self.is_killed # dev: is killed
# 计算变化前的D值
amp: uint256 = self._A()
old_balances: uint256[N_COINS] = self.balances
D0: uint256 = self._get_D_mem(old_balances, amp)
new_balances: uint256[N_COINS] = old_balances
for i in range(N_COINS):
new_balances[i] -= _amounts[i]
# 计算移除指定资产数量后的 D 值
D1: uint256 = self._get_D_mem(new_balances, amp)
# ?? N_COINS / (4 * (N_COINS - 1)) ??
fee: uint256 = self.fee * N_COINS / (4 * (N_COINS - 1))
admin_fee: uint256 = self.admin_fee
fees: uint256[N_COINS] = empty(uint256[N_COINS])
for i in range(N_COINS):
new_balance: uint256 = new_balances[i]
ideal_balance: uint256 = D1 * old_balances[i] / D0
difference: uint256 = 0
if ideal_balance > new_balance:
difference = ideal_balance - new_balance
else:
difference = new_balance - ideal_balance
fees[i] = fee * difference / FEE_DENOMINATOR
self.balances[i] = new_balance - (fees[i] * admin_fee / FEE_DENOMINATOR)
new_balances[i] = new_balance - fees[i]
D2: uint256 = self._get_D_mem(new_balances, amp)
lp_token: address = self.lp_token
token_supply: uint256 = CurveToken(lp_token).totalSupply()
token_amount: uint256 = (D0 - D2) * token_supply / D0
assert token_amount != 0 # dev: zero tokens burned
token_amount += 1 # In case of rounding errors - make it unfavorable for the "attacker"
assert token_amount <= _max_burn_amount, "Slippage screwed you"
CurveToken(lp_token).burnFrom(msg.sender, token_amount) # dev: insufficient funds
for i in range(N_COINS):
if _amounts[i] != 0:
_response: Bytes[32] = raw_call(
self.coins[i],
concat(
method_id("transfer(address,uint256)"),
convert(msg.sender, bytes32),
convert(_amounts[i], bytes32),
),
max_outsize=32,
)
if len(_response) > 0:
assert convert(_response, bool)
log RemoveLiquidityImbalance(msg.sender, _amounts, fees, D1, token_supply - token_amount)
return token_amount
D
由于合约中不能直接解方程,curve 在合约中使用了牛顿法迭代求近似解。
$x_{n+1}=x_n-\frac{f(x_n)}{f’(x_n)}$
简单理解牛顿法就是利用上述公式,不断迭代 x_n+1
的值,使其越来越逼近真实的解
代码中使用的牛顿法的函数 f(D)
,是用核心公式推导得来,下面简述推导过程:
-
首先将核心公式变形成
f(D) = 0
的形式$f(D)=An^n\Sigma x_i + D-ADn^n-\frac{D^{n+1}}{n^n\Pi x_i}$
-
D_new = D - f(D)/f'(D)
$D_{new}=D-\frac{An^n\Sigma x_i + D-ADn^n-\frac{D^{n+1}}{n^n\Pi x_i}}{1-An^n}$
其分母省略了一项,便于节省gas,且对最终结果影响非常小
-
最终变成代码注释中的形态
$D_{new}=\frac{An^n\Sigma x_i-\frac{D^{n+1}}{n^n\Pi x_i}}{An^n-1}$
-
for 循环迭代求 D 的解,当迭代到 D 与上一次的差值 <= 1 时,停止迭代;
-
通常 4 轮迭代以内就会找到局部最优解;但如果迭代 255 次仍然没有找到合适的解,程序终止,交易回滚 (revert);用户可以通过
remove_liquidity
方法回收资产;
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@pure
@internal
def _get_D(_xp: uint256[N_COINS], _amp: uint256) -> uint256:
"""
D invariant calculation in non-overflowing integer operations
iteratively
A * sum(x_i) * n**n + D = A * D * n**n + D**(n+1) / (n**n * prod(x_i))
Converging solution:
D[j+1] = (A * n**n * sum(x_i) - D[j]**(n+1) / (n**n prod(x_i))) / (A * n**n - 1)
"""
S: uint256 = 0
Dprev: uint256 = 0
for _x in _xp:
S += _x
if S == 0:
return 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
# 只有当移除流动性时才有可能 _x 为0,这时程序会崩溃,这也是期望的结果
# 添加流动性,池内的xp值不可能为0
Dprev = D
D = (Ann * S / A_PRECISION + D_P * N_COINS) * D / ((Ann - A_PRECISION) * D / A_PRECISION + (N_COINS + 1) * D_P)
# Equality with the precision of 1
# 当迭代的新值和上次之差,小于等于 1,即0或1
# 认为找到了局部最优解
if D > Dprev:
if D - Dprev <= 1:
return D
else:
if Dprev - D <= 1:
return D
# convergence typically occurs in 4 rounds or less, this should be unreachable!
# if it does happen the pool is borked and LPs can withdraw via `remove_liquidity`
raise
@view
@internal
def _get_D_mem(_balances: uint256[N_COINS], _amp: uint256) -> uint256:
return self._get_D(self._xp_mem(_balances), _amp)
y
输入资产 i
,变化之后的数量 x
,变化之前的 _xp
,计算输出资产 j
变化后的数量。
与计算 D 一样,这里也适用了牛顿法计算 y, 即 x_j
,其 f(x_j)
推导过程如下:
$An^n\Sigma x_i + D = ADn^n + \frac{D^{n+1}}{n^n\Pi x_i}$
- 设
sum'
,prod'
分别为排除输出资产数量的累加和累乘结果sum' = sum(x) - x_j
prod' = prod(x) / x_j
-
核心公式除以
A*n**n
$x_j\Sigma x_i+\frac{D}{An^n}-D=\frac{D^{n+1}}{An^nn^n\Pi x_i}$
-
乘以
x_j
,并代入sum'
和prod'
$x_j(x_j+sum’)-x_j(An^n-1)\frac{D}{An^n}=\frac{D^{n+1}x_j}{An^{2n}(prod’*x_j)}$
-
展开多项式,即为牛顿法使用的
f(x_j)
$x_j^2+x_j(sum’-(An^n-1)\frac{D}{An^n})=\frac{D^{n+1}}{An^{2n}prod’}$
-
将其写为
x_j**2 + b*x_j = c
形式,代入牛顿法公式x = x - f(x) / f'(x)
x_j = (x_j**2 + c) / (2*x_j + b)
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@view
@internal
def _get_y(i: int128, j: int128, x: uint256, _xp: uint256[N_COINS]) -> uint256:
"""
Calculate x[j] if one makes x[i] = x
Done by solving quadratic equation iteratively.
x_1**2 + x_1 * (sum' - (A*n**n - 1) * D / (A * n**n)) = D ** (n + 1) / (n ** (2 * n) * prod' * A)
x_1**2 + b*x_1 = c
x_1 = (x_1**2 + c) / (2*x_1 + b)
"""
# x in the input is converted to the same price/precision
assert i != j # dev: same coin
assert j >= 0 # dev: j below zero
assert j < N_COINS # dev: j above N_COINS
# should be unreachable, but good for safety
assert i >= 0
assert i < N_COINS
A: uint256 = self._A()
D: uint256 = self._get_D(_xp, A)
Ann: uint256 = A * N_COINS
c: uint256 = D
S: uint256 = 0
_x: uint256 = 0
y_prev: uint256 = 0
for _i in range(N_COINS):
if _i == i:
_x = x
elif _i != j:
_x = _xp[_i]
else:
continue
S += _x
c = c * D / (_x * N_COINS)
c = c * D * A_PRECISION / (Ann * N_COINS)
b: uint256 = S + D * A_PRECISION / Ann # - D
y: uint256 = D
for _i in range(255):
y_prev = y
y = (y*y + c) / (2 * y + b - D)
# Equality with the precision of 1
if y > y_prev:
if y - y_prev <= 1:
return y
else:
if y_prev - y <= 1:
return y
raise
@view
@external
def get_dy(i: int128, j: int128, _dx: uint256) -> uint256:
xp: uint256[N_COINS] = self._xp()
rates: uint256[N_COINS] = RATES
x: uint256 = xp[i] + (_dx * rates[i] / PRECISION)
y: uint256 = self._get_y(i, j, x, xp)
dy: uint256 = xp[j] - y - 1
fee: uint256 = self.fee * dy / FEE_DENOMINATOR
return (dy - fee) * PRECISION / rates[j]
还有个 _get_y_D()
函数,与上述区别是可以自定义 D 的值来求 y
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@pure
@internal
def _get_y_D(A: uint256, i: int128, _xp: uint256[N_COINS], D: uint256) -> uint256:
calc_withdraw_one_coin
计算移除流动性只赎回单一资产。
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@view
@internal
def _calc_withdraw_one_coin(_token_amount: uint256, i: int128) -> (uint256, uint256, uint256):
# First, need to calculate
# * Get current D
# * Solve Eqn against y_i for D - _token_amount
# 首先需要先计算出移除流动性后,资产i的新数量
amp: uint256 = self._A()
xp: uint256[N_COINS] = self._xp()
D0: uint256 = self._get_D(xp, amp)
# 根据 lp token 变化量同比计算D的新值
# 牛顿法计算新的资产 i 的数量 new_y - 不考虑手续费
# xp_reduced 资产价值减少后的量
total_supply: uint256 = CurveToken(self.lp_token).totalSupply()
D1: uint256 = D0 - _token_amount * D0 / total_supply
new_y: uint256 = self._get_y_D(amp, i, xp, D1)
xp_reduced: uint256[N_COINS] = xp
fee: uint256 = self.fee * N_COINS / (4 * (N_COINS - 1))
for j in range(N_COINS):
# dx_expected 资产j价值的变量
# - 相对于按照比例移除流动性的数量
# - 不考虑手续费
# 资产i相对新的xp要少,其他则是相对于新的xp要多,因此需要注意正负号
dx_expected: uint256 = 0
if j == i:
dx_expected = xp[j] * D1 / D0 - new_y
else:
dx_expected = xp[j] - xp[j] * D1 / D0
# 将每种资产对应扣除手续费
xp_reduced[j] -= fee * dx_expected / FEE_DENOMINATOR
# 使用扣除手续费的xp计算 y,进而得出dy
dy: uint256 = xp_reduced[i] - self._get_y_D(amp, i, xp_reduced, D1)
precisions: uint256[N_COINS] = PRECISION_MUL
dy = (dy - 1) / precisions[i] # Withdraw less to account for rounding errors
dy_0: uint256 = (xp[i] - new_y) / precisions[i] # w/o fees
# 返回 赎回的资产数量,扣除的手续费(价值), lp token 总量(销毁之前)
return dy, dy_0 - dy, total_supply
@view
@external
def calc_withdraw_one_coin(_token_amount: uint256, i: int128) -> uint256:
"""
@notice Calculate the amount received when withdrawing a single coin
@param _token_amount Amount of LP tokens to burn in the withdrawal
@param i Index value of the coin to withdraw
@return Amount of coin received
"""
return self._calc_withdraw_one_coin(_token_amount, i)[0]
remove_liquidity_one_coin
移除流动性,赎回单一种类资产。可以认为这期间是将其他资产交易为了单一资产,需要手续手续费。
传入要销毁的 lp token 数量 _token_amount
,赎回资产序号 i
, 最小要收到的资产数量 _min_amount
, 返回实际赎回的资产数量。
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@external
@nonreentrant('lock')
def remove_liquidity_one_coin(_token_amount: uint256, i: int128, _min_amount: uint256) -> uint256:
"""
@notice Withdraw a single coin from the pool
@param _token_amount Amount of LP tokens to burn in the withdrawal
@param i Index value of the coin to withdraw
@param _min_amount Minimum amount of coin to receive
@return Amount of coin received
"""
assert not self.is_killed # dev: is killed
# 预先计算本次移除将赎回的资产数量 dy
# 如果 dy 小于设定的最小数量,交易回滚
dy: uint256 = 0
dy_fee: uint256 = 0
total_supply: uint256 = 0
dy, dy_fee, total_supply = self._calc_withdraw_one_coin(_token_amount, i)
assert dy >= _min_amount, "Not enough coins removed"
# 将 admin fee 从balances中扣除
# 销毁掉设定的 lp token 数量
#
# 注意 _calc_withdraw_one_coin() 返回的 dy 是已经扣除掉手续费的数量
# 所以这里需要单独从balances中扣除协议费
self.balances[i] -= (dy + dy_fee * self.admin_fee / FEE_DENOMINATOR)
CurveToken(self.lp_token).burnFrom(msg.sender, _token_amount) # dev: insufficient funds
# 将赎回token转给用户
_response: Bytes[32] = raw_call(
self.coins[i],
concat(
method_id("transfer(address,uint256)"),
convert(msg.sender, bytes32),
convert(dy, bytes32),
),
max_outsize=32,
)
if len(_response) > 0:
assert convert(_response, bool)
log RemoveLiquidityOne(msg.sender, _token_amount, dy, total_supply - _token_amount)
return dy
Admin functions
管理员调用的方法
A’s adjustment
A 参数的调整,传入目标值 _future_A
,预定调整结束时间 _future_time
。
当调用 ramp_A()
后,A 参数会随时间线性改变,直到 _future_time
完成调整,达到预设的目标值。这期间池子的计算涉及 A 值都会线性的计算当前值。
调整期间管理员可以调用 stop_ramp_A
终止调整进度,A 会保持当前的值。
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@external
def ramp_A(_future_A: uint256, _future_time: uint256):
assert msg.sender == self.owner # dev: only owner
assert block.timestamp >= self.initial_A_time + MIN_RAMP_TIME
assert _future_time >= block.timestamp + MIN_RAMP_TIME # dev: insufficient time
initial_A: uint256 = self._A()
future_A_p: uint256 = _future_A * A_PRECISION
assert _future_A > 0 and _future_A < MAX_A
if future_A_p < initial_A:
assert future_A_p * MAX_A_CHANGE >= initial_A
else:
assert future_A_p <= initial_A * MAX_A_CHANGE
self.initial_A = initial_A
self.future_A = future_A_p
self.initial_A_time = block.timestamp
self.future_A_time = _future_time
log RampA(initial_A, future_A_p, block.timestamp, _future_time)
@external
def stop_ramp_A():
assert msg.sender == self.owner # dev: only owner
current_A: uint256 = self._A()
self.initial_A = current_A
self.future_A = current_A
self.initial_A_time = block.timestamp
self.future_A_time = block.timestamp
# now (block.timestamp < t1) is always False, so we return saved A
log StopRampA(current_A, block.timestamp)
fee’s adjustment
fee 和 admin fee 的调整
- 管理员提交调整方案
commit_new_fee()
- 如果期间有其他呆执行的方案,将不能提交
assert self.admin_actions_deadline == 0
- 自动设置最早执行时间为
block.timestamp + ADMIN_ACTIONS_DELAY
- ADMIN_ACTIONS_DELAY = 3 * 86400 s 即等待期 3 天
- 在等待期过后,管理员执行
apply_new_fee()
让新的费率参数生效- 根据 admin_actions_deadline 是否为零判断当前是否有待执行方案
assert self.admin_actions_deadline != 0
- 等待期间,管理员可调用
revert_new_parameters()
取消调整方案
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@external
def commit_new_fee(_new_fee: uint256, _new_admin_fee: uint256):
assert msg.sender == self.owner # dev: only owner
assert self.admin_actions_deadline == 0 # dev: active action
assert _new_fee <= MAX_FEE # dev: fee exceeds maximum
assert _new_admin_fee <= MAX_ADMIN_FEE # dev: admin fee exceeds maximum
deadline: uint256 = block.timestamp + ADMIN_ACTIONS_DELAY
self.admin_actions_deadline = deadline
self.future_fee = _new_fee
self.future_admin_fee = _new_admin_fee
log CommitNewFee(deadline, _new_fee, _new_admin_fee)
@external
def apply_new_fee():
assert msg.sender == self.owner # dev: only owner
assert block.timestamp >= self.admin_actions_deadline # dev: insufficient time
assert self.admin_actions_deadline != 0 # dev: no active action
self.admin_actions_deadline = 0
fee: uint256 = self.future_fee
admin_fee: uint256 = self.future_admin_fee
self.fee = fee
self.admin_fee = admin_fee
log NewFee(fee, admin_fee)
@external
def revert_new_parameters():
assert msg.sender == self.owner # dev: only owner
self.admin_actions_deadline = 0
ownership
移交合约 owner。与手续费调整类似,先提交方案,等待期过后,才能执行方案。
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@external
def commit_transfer_ownership(_owner: address):
assert msg.sender == self.owner # dev: only owner
assert self.transfer_ownership_deadline == 0 # dev: active transfer
deadline: uint256 = block.timestamp + ADMIN_ACTIONS_DELAY
self.transfer_ownership_deadline = deadline
self.future_owner = _owner
log CommitNewAdmin(deadline, _owner)
@external
def apply_transfer_ownership():
assert msg.sender == self.owner # dev: only owner
assert block.timestamp >= self.transfer_ownership_deadline # dev: insufficient time
assert self.transfer_ownership_deadline != 0 # dev: no active transfer
self.transfer_ownership_deadline = 0
owner: address = self.future_owner
self.owner = owner
log NewAdmin(owner)
@external
def revert_transfer_ownership():
assert msg.sender == self.owner # dev: only owner
self.transfer_ownership_deadline = 0
admin_fee withdraw and donate
提取和捐赠 admin_fee
-
withdraw_admin_fees()
提取协议费,将每种资产分别转给调用者(管理员)- admin_fee 的数量没有专门的字段存储,而是通过 合约持有的总量 - 流动性提供者持有的总量(balances) 获得
ERC20(coin).balanceOf(self) - self.balances[i]
-
donate_admin_fees()
将未提取的协议费全部捐赠给池子的流动性提供者self.balances[i] = ERC20(self.coins[i]).balanceOf(self)
- 上述操作实质上是将协议费的数量清零,全部转移到 balances 上
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@view
@external
def admin_balances(i: uint256) -> uint256:
return ERC20(self.coins[i]).balanceOf(self) - self.balances[i]
@external
def withdraw_admin_fees():
assert msg.sender == self.owner # dev: only owner
for i in range(N_COINS):
coin: address = self.coins[i]
value: uint256 = ERC20(coin).balanceOf(self) - self.balances[i]
if value > 0:
_response: Bytes[32] = raw_call(
coin,
concat(
method_id("transfer(address,uint256)"),
convert(msg.sender, bytes32),
convert(value, bytes32),
),
max_outsize=32,
) # dev: failed transfer
if len(_response) > 0:
assert convert(_response, bool)
@external
def donate_admin_fees():
assert msg.sender == self.owner # dev: only owner
for i in range(N_COINS):
self.balances[i] = ERC20(self.coins[i]).balanceOf(self)
kill_me
kill_me()
我杀我自己unkill_me()
诶,我又不杀了
当 is_killed
为true时,合约不能运行下列方法:
add_liquidity()
exchange()
remove_liquidity_imbalance()
remove_liquidity_one_coin()
注意:remove_liquidity
是可以运行的,允许流动性提供者撤走流动性,但期间不能发生交易行为,即必须按照当前价格移除流动性。
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@external
def kill_me():
assert msg.sender == self.owner # dev: only owner
assert self.kill_deadline > block.timestamp # dev: deadline has passed
self.is_killed = True
@external
def unkill_me():
assert msg.sender == self.owner # dev: only owner
self.is_killed = False