Pricing Challenge (Bounty)

Bounty (100'000 LQTY) for finding a fair price formula for bTokens

Problem statement

Chicken Bonds rely on the bTokens to trade at a premium above their redemption value (see Flywheel effect between bonding and investing). Having a good understanding of the expected market price is crucial for modeling the system, determining the optimal behavior of its users, and potentially guiding them to take the best decisions.
In particular, we are looking for a formula to calculate the fair price of the bTokens in function of the current system state as described by the following parameters:
• Size (`q_p`) and rate of return (`r_p`) of the Pending Bucket
• Size (`q_a`) and rate of return (`r_a`) of the Reserve Bucket
• Size (`q_d`) and rate of return (`r_d`) of the Permanent Bucket
• bToken supply (`s`)
• Accrual parameter `α` or target Chicken In time `T_t`(controller version)
For simplicity, we assume that the bucket sizes and returns are independent variables. In reality, the returns may depend on the amounts deployed to a given yield-generating venue. While the fair price may depend on the actual size and age distribution of the pending bonds, it seems inevitable to use simplifying assumptions in that regard.
Our team has performed multiple approaches with different simplifying assumptions to estimate the fair value of the bTokens. The task turned out to be an interesting challenge, leading to very different results that are hard to reconcile. The approaches described in section 4.3 of the whitepaper are:
• Naive approach (4.3.1)
• Recursive approach (4.3.2)
• Yield comparison approach (4.3.3)
• Conservative approach (4.3.4)
We invite the researcher community to help us by suggesting a formula or approach to determine a fair price that "makes sense" under economically reasonable assumptions. To that end, we are offering a total bounty of 100'000 LQTY. Note that we're not looking to establish the fair price as a model result, but rather need it to instantiate our economic model.

Submission

Please submit your solution to [email protected] before September 1st 00:00, 2022.

Reward criteria

Rewards will be awarded at the sole discretion of Liquity AG. Rewards are denominated and paid out in LQTY. If both parties agree, rewards can also be paid out in other crypto assets.
The reward can be paid out in total to one submission or split up among up to 3 submissions if several of them fulfil the criteria. Submissions that only partially fulfil the criteria (essential clues but no fully-fledged solution) may be awarded a reduced bounty, and we might not pay out the full reward amount if we don't receive satisfactory solutions to the problem. The quality of the writeup and the other deliverables can impact the reward.
An ideal submission should meet the following criteria
• All assumptions and simplifications regarding system configuration (e.g. size and age distribution of pending bonds) and user behaviour (e.g. risk-seeking vs. risk-averse, rational vs. irrational) are clearly laid out
• The full derivation of all formulas with economic rationale for each step that is not purely mathematical in nature
• Closed-form solution or concrete way to approximate the fair price given the input variables
• Validation of the formula through simulation (e.g. using our existing Python model) or at least Desmos, e.g. by plotting the fair price in function of the pending bucket size
• Explanation if and how it relates to our existing approaches in the whitepaper
• The approach should work with both the accrual parameter and target Chicken In time as an input
Deliverables
• Must-have: Writeup of the formula with step-by-step explanations and visuals (e.g. plots)
• Ideal: Simulation code and/or interactive plots (e.g. Desmos, Colab workbook)