Pilgrim: An Atomic Valuation Framework for Low Liquidity Assets

I. Introduction

• Market liquidity must be manually maintained. Even if illiquid assets are fractionalized into shares or shards, they are still derivatives of the underlying asset and therefore require liquidity in order for them to be traded. As liquidity requirements inherently force some shards to be locked indefinitely with corresponding liquid assets even though there are a limited number of shards available, this is very capital inefficient and difficult to manage in the long term.
• Reconstitution of underlying assets is difficult. Shard-based models often require either a certain percentage of shard owners to coordinate or participate in governance proposals to retrieve the underlying asset. Not only is this participation requirement rigorous and time-consuming, such an approach comes with optimistic coordination assumptions that may result in problems as described below. Reconstitution difficulties while shards are being traded also prohibit assets from being actively used with DApps that allocate certain features, such as with Axie Infinity or other NFT-based games.
• Coordination assumptions are often optimistic. Performing actions collectively under the underlying NFT under a shards model require coordination that, inevitably, result in multiple parties with conflicting interests. For instance, people that bought shards below a given reconstitution value will most likely agree to redeem their shards, but others that bought shards above this value will likely be against redemption. While approaches such as involving gradual inflation of shards given to active participants may be able to solve this particular issue, fundamentally it still stands true that fractionalization-based models result in other inefficiencies listed here.
• This still does not fix nonatomic liquidation properties of illiquid assets. Unlike fungible and liquid assets, illiquid assets require a time-consuming auction to liquidate and gain access to liquid assets in return. Fractionalization based approaches do not resolve this problem, as reconstitution must be triggered in order to liquidate assets (of which has coordination issues in itself).

• Liquidity should be generated and withdrawn programmatically in proportion to market demand. As there could be an infinite number of assets in existence, requiring every single one of them to manually manage market liquidity is extremely inefficient. Therefore, an automated liquidity control mechanism is required, whereby liquid assets provided with buy (or short) orders act as both collateral and liquidity for buy bids, implemented on top of a prediction market-based price curve as those of Uniswap.
• All features of the underlying asset should be accessible even when it is held by a custody contract for valuation. Fractionalization-based mechanisms effectively block access to the underlying asset, especially in cases when an asset is allocated certain features by an external application. Therefore a mechanism should exist to relay the concept of ownership to a staking derivative-like asset that could act as a functional pointer to its underlying asset while it is actively being traded.
• No coordination assumptions should be made outside of financial incentives. Coordination is difficult to resolve through non-incentivized governance processes that must be executed manually. Ideally, outside of financial incentives and free market principals, effective coordination should not be one of the assumptions for designing such a valuation system.
• Liquidity should be readily available when requested by the underlying asset's original owner. There should exist a mechanism where the owner of an asset may borrow a portion of locked liquidity generated from market demand. This would allow for partial liquidity exposure for owners of illiquid assets, even when an auction or a sale on the asset is not finalized.

II. The Pilgrim Trading Protocol

• market liquidity is generated in proportion to demand
• valuation of illiquid assets locked as collateral is recursively determined based on an AMM-like prediction market system
• locked market liquidity may be partially borrowed against illiquid collateral
• a staking derivative-like asset used for borrows, pool liquidations, and feature access, is minted on pool deployment

• Pool liquidity is dynamically adjusted per trade. Under the Uniswap model, the liquidity parameter $k$ only changes when liquidity is explicitly provided or withdrawn from the pool. However, this assumes that assets provided on both sides are liquid and fungible assets; such a model results in liquidity fragmentation for asset types that are not fungible. To compensate for this, we define the liquidity parameter $k$ to
• increase when there is more demand for the corresponding asset
• decrease when there is less demand for the corresponding asset

Defining $L$ as the base round liquidity parameter, and $p$ as the current exchange rate per round, this could be expressed as

$$L^2 \cdot p = z$$

with $L \cdot (p - p_{genesis})$ as the total amount of liquidity locked.

Simplifying this as

$$x \cdot y = z$$

with $z$ defined as a dynamic product and not a constant, buying $r$ units of LBP rounds therefore should then result in:

$$\frac{1}{x} \cdot (x + r)^2 \cdot y = z$$

therefore, required liquid assets to buy $r$ units of LBP rounds equate to

$$\Delta y = y' - \frac{y \cdot (x+r)}{x} = \frac{y \cdot r \cdot (x + r)}{x^2}$$

and vice versa. Buying LBPs with $n$ units of liquid assets equate to

$$n = \Delta y = \frac{y \cdot r \cdot (x + r)}{x^2}$$

therefore the amount of LBP rounds returned from this pool is

$$r = \frac{x \cdot (\sqrt{1 + \frac{4n}{y}} - 1)}{2}$$

and vice versa.

The following is a simple example of the logic illustrated above:

	$x$	$y$	$z$
initial condition	100	200	20,000
liquidity added	110	220	24,200
swapped	100	242	24,200

Note that $x$ stays constant, being the base round liquidity parameter $L$; while $y$ increases, reflecting added liquidity within this pool. Therefore the spot price per round has increased from 2 to 2.42, while the execution price stays at 2.

• Slippage should be counter-uniform across the entire pool. This to protect against slippage delta-based attacks; i.e. trading $nK$ units of liquid assets (assuming $n \in N$ & $K \in N$) should be equal to executing $K$ unit worth of trading operations $n$ times. This operation could be generalized as
• minting new rounds: let required liquidity in equal to $f_{nM}(y)$
$$f_{nM}(y) = \frac{(x + k)^{2n + 1} - x^{2n}(x + k)}{(2x + k) x^{2n}} y$$
• burning new rounds: let required liquidity out equal to $f_{nB}(y)$
$$f_{nB}(y) = \frac{(x + k)^{2n} - x^{2n}}{(2x + k) (x + k)^{2n - 1}} y$$

The above is proven with a separate Arbitrary Rounds Trading paper.

III. metaNFTs and Pool Ownership

• relay features of its underlying NFT to a third-party application that supports such,
• accept a full pool buyout bid to liquidate the entire pool and its corresponding LBP for reclaiming its underlying NFT,
• collect trading fees paid by round traders,
• and potentially compose them with third-party mechanisms based on its dynamic price feed enabled by the Pilgrim trading protocol, such as borrowing liquid assets within a particular liquidity pool with metaNFTs as collateral, allowing for dynamic liquidation implementations on illiquid-to-liquid money markets.

IV. Pool Buyouts and Liquidations

V. The Pilgrim Token & Financial Coordination

• Protocol fees generated from Pilgrim pools are used to buy PIL from Uniswap, subsidized with additional PIL from the Protocol Treasury at a predefined ratio based on relative pool TVL against the protocol’s total liquidity, and are distributed to both metaNFT holders and round holders.
• Total protocol rewards for a particular Pilgrim pool is defined as REWARD_CONSTANT * TOTAL_VALUE_LOCKED * TRADING_VOLUME.
• Protocol fees generated from Pilgrim pools with PIL as its base token has a higher PIL fee subsidization ratio than (i).
• Protocol fees generated from Pilgrim pools with PIL as its base token and paired with a PIL - ETH Uniswap v3 liquidity position has a higher PIL fee subsidization ratio than (ii).
• Protocol fees generated from Pilgrim pools with PIL as its base token and paired with a PIL - xPIL Uniswap v3 liquidity position has a higher PIL fee subsidization ratio than (iii).
• A portion of all Pilgrim protocol fees are distributed to xPIL holders, which are locked PIL positions committed for a predefined period. xPIL holders also receive additional PIL fee subsidizations from the Protocol Treasury, of which cannot be claimed until lockup expiry.

• to receive higher protocol fees and therefore higher PIL subsidizations,
• metaNFT holders must lock as much liquidity as possible from round traders. They also should set their base token to PIL for receiving additional protocol incentives.
• round holders must either hold round positions for significant periods of time to gain further exposure to this particular Pilgrim pool’s TVL growth, or actively rebalance their positions to be aligned with Pilgrim pools that they consider undervalued.
• this creates financial gain opportunities for active traders on both sides;
• pool buyout bids are inherently affected by current round prices, and vice versa; round prices are a direct outcome of TVL competition coordinated by multiple groups of metaNFT holders and round position holders.
• however, for buyout bidders, pool TVL should be lower for potential gains. as metaNFT holders are the only ones that may approve their bids, and only the bid that are closest to the current NFT valuation may be approved, buyout bidders should actively rebalance their bid positions. This also affects metaNFT and round token valuation, and therefore the amount of PIL rewards they receive.
• not only this encourages community-level coordination and competition to gain more exposure for subsidized protocol rewards, position rebalancing is also encouraged due to constantly changing buyout bids. This increases overall protocol fees, increasing PIL buybacks and fees being distributed to both protocol users and xPIL holders.

VI. Potential Applications & Conclusion

• Atomic valuation of illiquid financial positions, including Uniswap v3 liquidity positions, Maker vaults and Compound borrow positions.
• Leverage and hedging through combined atomic valuation of multiple illiquid positions:

• decentralized, community-coordinated valuation of corporate and DAO shares for funding.
• credit valuation and historical logs for both individuals and DAOs for under-collateralized borrows.

• shorting mechanisms for traders to allow NFT valuation to drop below its listed price, while still incurring positive trading fees and TVL
• native collateralized borrows against protocol liquidity utilizing metaNFTs.

Revision History