BPT Oracle

To price LP tokens, it is not enough to simply add the values of all assets in the pool as this is easily manipulated. We use a more robust procedure for calculating Balancer LP token values.
For a given Balancer pool containing assets 1, ..., n, define the following:
wi=weight of asset iri=amount (in # tokens) of asset ipi=price of asset iS=total # BPT tokensw_i = \text{weight of asset } i \\
r_i = \text{amount (in \# tokens) of asset } i \\
p_i = \text{price of asset } i \\
S = \text{total \# BPT tokens}

wi​=weight of asset iri​=amount (in # tokens) of asset ipi​=price of asset iS=total # BPT tokens
The constant product of the Balancer pool is
k=∏i=1nriwik = \prod_{i=1}^{n} r_i^{w_i}k=i=1∏n​riwi​​
Note that the amounts rir_iri​
are easily manipulatable, but the product kkk
is not. And, as we require asset pricing oracles elsewhere, we can presume that the prices pip_ipi​
are also not easily manipulatable (controls to assure against this will be discussed elsewhere).
To make the manipulation-resistant BPT oracles, it will be enough to express the pricing of BPT tokens in terms of solely manipulation-resistant variables wi,pi,k,Sw_i, p_i, k, Swi​,pi​,k,S
.
The portfolio value of the Balancer pool can be calculated as
BPT total value=k∏i=1n(piwi)wi\text{BPT total value} = k \prod_{i=1}^n \left( \frac{p_i}{w_i} \right)^{w_i}BPT total value=ki=1∏n​(wi​pi​​)wi​
Then, in turn, the BPT price can be calculated as
pBPT=BPT total valueS=k∏i=1n(piwi)wiSp_{BPT} = \frac{\text{BPT total value}}{S} = \frac{k \prod_{i=1}^n \left( \frac{p_i}{w_i} \right)^{w_i}
}{S}pBPT​=SBPT total value​=Sk∏i=1n​(wi​pi​​)wi​​

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