lp-share-pricing.md

description
Methodology and implementation of pricing LP share tokens

LP share pricing

To price LP tokens, it is not enough to simply add the values of all assets in the pool as this is easily manipulated. Instead, a more robust procedure is required to calculate LP token values.

{% hint style="info" %} The precise instructions on how to calculate LP shares of the 2-CLP, 3-CLP, or E-CLP are available in section 5 of this technical paper. {% endhint %}

Example: Constant Product Pools

This example demonstrates the principles of robust LP share pricing applied to constant product pools.

For a given constant product Balancer pool containing assets 1, ..., n, define the following:

$$ w_i = \text{weight of asset } i \\ r_i = \text{amount (in # tokens) of asset } i \\ p_i = \text{price of asset } i \\ S = \text{total # LP tokens} $$

The constant product invariant of the pool is

$$ L = \prod_{i=1}^{n} r_i^{w_i} $$

Note that the amounts $$r_i$$are easily manipulatable through swaps, but the product $$L$$is not. And, as we require asset pricing oracles elsewhere, we can presume that the prices $$p_i$$ are also not easily manipulatable (controls to assure against this will be discussed elsewhere).

To calculate a manipulation-resistant LP token price, it will be enough to express the pricing of LP tokens solely in terms of manipulation-resistant variables $$w_i, p_i, L / S$$. Note that while $$L$$ and $$S$$ are individually manipulatable by adding or removing liquidity, $$L/S$$ is not easily manipulatable as long as proper accounting methods are in place for handling adding and removing of liquidity (see Section 5.5 in the technical paper for more details).

The portfolio value of the entire pool can be calculated as

$$ \text{Pool value} = L \prod_{i=1}^n \left( \frac{p_i}{w_i} \right)^{w_i} $$

In turn, the LP token price can be calculated in terms of manipulation-resistant variables as

$$ p_{\text{LP token}} = \frac{\text{Pool value}}{S} = \frac{L}{S} \prod_{i=1}^n \left( \frac{p_i}{w_i} \right)^{w_i} $$

Pricing LP tokens for 2-CLPs, 3-CLPs, and E-CLPs

LP share pricing for the CLPs follows the same general principles and is described in full detail in Section 5 of the following technical paper.

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