The Gyro Bonding Curve

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Background

In navigating the DeFi, several critical challenges emerged within traditional stablecoin systems. These challenges included concentrated risk, slow governance responses, scalability hurdles, fragility of Oracle systems, and liquidity dry-ups during market turmoil.

The de-pegging of stable coins which destabilized much of the stablecoin market.

• Risk concentration: Traditional stablecoin systems often suffered from concentrated risk due to reliance on a single collateral or insufficient isolation among multiple collaterals, leading to systemic failures.

  • The solution that Gyro implements is to use a reserve that diversifies risk to the greatest extent possible. A stratified reserve structure that limits exposure to different existential risks ahead-of-time.

• Slow governance: Decentralized governance, while effective for long-term strategic decisions, proves inadequate for swift crisis response, hindering the ability to react promptly to emergent issues.

  • Autonomy requirement: Recognizing the need for autonomous system reactions, Gyro aimed to delegate strategic decisions to governance while enabling the system to autonomously address crises.

• Scalability challenges: Complex and algorithmic stablecoin systems sometimes experience mismatches between supply and demand dynamics, resulting in imbalances.

  • A system that scales and adapts to market stress automatically and transparently. The Dynamic Stability Mechanism (DSM), also known as "PSM 2.0," autonomously manages reserves, employing programmatic risk control and diverse strategies to ensure stability. It incorporates contingency pricing to address fluctuations effectively. Governance primarily handles strategic decisions, which may face delays due to its slower nature

• Fragility of Oracle systems: Many stablecoin systems, including ours, relied on Oracle prices for underlying assets, often with a heavy reliance on a single provider like Chainlink, introducing fragility concerns.

  • Gyroscope relies on Chainlink to obtain prices, treating them as potential solutions to a system of equations defined by on-chain trading tools' interconnections. The system compares these prices to on-chain DEX pools, detecting discrepancies that may signal unusual market states or potential manipulation. However, the system lacks the ability to confirm manipulation definitively. Liquidity remains a significant concern in this process.

• Liquidity dry-ups: During market turmoil, liquidity providers faced challenges due to liquidity shortages, complicating risk management efforts.

  • A liquidity network: It allows the stablecoin GYD to serve as a central asset, enabling efficient swaps between GYD and other pairs. Separate pools, like 2-CLP, 3-CLP, and E-CLP, ensure that assets are isolated from each other, providing redundancy and maintaining liquidity. This setup ensures that depegs are isolated, keeping pools liquid and allowing stablecoin trades to compose them effectively. GYD's stability as a central asset enhances the network's efficiency.

Dynamic Stability Mechanism

Gyroscope's GYD stablecoin leverages a novel mechanism called the Dynamic Stability Mechanism (DSM), previously known as the Primary Automated Market Maker (P-AMM). This mechanism utilizes a bonding curve to maintain the peg of GYD to its underlying assets. The DSM adjusts its price quotes based on the reserve ratio and outflows. The dynamic system is parameterized by a virtual anchor point that captures the system's health and redemption pressure. This anchor point serves as a mathematical modeling tool to represent the current state of the system, although it is not directly observable at any given time. The system is governed by a few hyperparameters, which can be fine-tuned via governance. However, the desired properties are robust across the entire parameter space.

One of the critical issues plaguing decentralized finance (DeFi) platforms is the sluggishness of decentralized governance processes. Governance decisions often suffer from delays, and inefficiencies, hindering the platform's ability to adapt quickly to changing market conditions and user needs. To counter the challenge Gyroscope has implemented a solution that leverages DSM which is capable of scaling and adapting to market stress automatically and transparently. DSM ensures that the platform can swiftly respond to market fluctuations and user demands, all while maintaining a high level of transparency and accountability.

Design Deep dive

Simple Primary Market Model

For a decentralized reserve-backed stablecoin system, collateralization levels are a critical consideration. In the ideal case, the system maintains at least 100% collateralization with liquid assets backing every unit of the stablecoin in circulation. This allows for a straightforward minting and redemption process where users can deposit $1 worth of assets to mint 1 unit of the stablecoin, or redeem 1 unit to withdraw $1 worth of assets.

However, real-world scenarios introduce complexities. There may be instances where the system is overcollateralized overall, but some of the assets in the reserve are illiquid and cannot be easily disbursed during redemptions. An even more precarious situation arises if the system becomes undercollateralized due to factors like asset price volatility or excessive redemptions depleting the reserve.

In such cases of undercollateralized, the protocol needs robust mechanisms to autonomously stabilize and recapitalize itself without relying on centralized oversight. Careful system design is required to outline the specific actions the protocol will programmatically take to regain a collateralized state.

Redemption Curves

Redemption curve = price of redemption as fn. of total redemptions

A useful analytical framework for evaluating undercollateralization response mechanisms is the redemption curve. This curve models the relationship between the level of redemptions from the reserve $x\text{-axis}$ and the resulting redemption price $y\text{-axis}$. In an ideal fully collateralized scenario, the curve would be a flat horizontal line, with redemptions happening at the $1 peg regardless of the amount redeemed.

However, when undercollateralized, the curve must curve downwards as the reserve gets depleted and redemptions need to be haircut below $1. A steeper curve indicates redemption prices drop more rapidly as the level of redemptions increases. The shape and steepness of this curve is a key design decision that has major implications.

Potential redemption curve policies like redeeming at $1 until reserves are fully exhausted, then dropping to $0, are suboptimal due to price instability and unfair treatment of late redeemers. More nuanced approaches that gradually decrease redemption prices as collateral gets depleted can provide smoother and fairer price discovery.

The design criteria for the Redemption Curve within a single block include several key properties

• Firstly, the relative collateralization, or reserve ratio, of the protocol must remain above a certain lower bound, barring any exogenous shocks to reserve assets.

• Secondly, the P-AMM should typically maintain a region of open market operations where the stablecoin price hovers around $1.

• Thirdly, there should be a lower bound on the worst-case P-AMM redemption rate.

• Fourthly, the redemption curve of the P-AMM should be continuous and not excessively steep, unless this would conflict with the other desiderata.

• Lastly, there should be no incentive for redeemers to strategically subdivide their redemptions.

More broadly, across multiple blocks, the system should ensure that the reserve can only be depleted over an extended period. Additionally, if the stablecoin loses its peg, there should be a clear path for it to regain the peg. Finally, the P-AMM mechanism should be capable of efficient and on-chain implementation and computation.

Redemption Curve Design

Gyro uses a piecewise-linear redemption curve for its stablecoin. This curve defines how the redemption price of the stablecoin changes based on the total amount of stablecoin being redeemed. This design reflects a trade-off between:

• Short-Term Stability: Maintaining a $1 redemption price

• Long-Term Sustainability: ability to maintain its peg to the target asset (often a fiat currency like USD) over an extended period while ensuring the smooth functioning of its core mechanisms.

In Region 1, redemptions occur at $1.

Redemption curve design with piecewise-linear price decay

The curve prioritizes maintaining a $1 redemption price when redemption pressure is low. This means as long as there's minimal demand for redeeming the stablecoin, users can still get back $1 for each unit they redeem.

In Region 2, represented by ( $\chi_U$ to $\chi_L$), the redemption pricing decays linearly.

The system dynamically computes a point $\chi_U$ based on several parameters where the redemption price needs to decrease gradually from $\chi_U$ until another point $\chi_L$.

In Region 3, represented beyond ( $\chi_L$ ), redemptions occur at a new reserve ratio, ensuring full sustainability for the entire supply.

Once this point is reached, the system stabilizes, and the redemption price no longer needs to decrease. Instead, it can remain constant, ensuring stability moving forward.

The chosen points $\chi_U$ & $\chi_L$correspond to observing a collateralization ratio where the redemption amount is zero. This ratio is termed the anchor collateralization ratio.

$$r_a \in (\bar{\theta},1) \space\text{at} \space x=0$$

Three dynamic parameters describe the three regions of the P-AMM:

Redemption curve design with piecewise-linear price decay

$\bar{\theta}$: Lower bound on collateralization ratio which is the value $y$ or height of the line in region 3

$\bar{\alpha}$: Lower bound on the slope $\alpha$ of the linear segment in region 2

$\bar{\chi_U}$: Upper price on the unit price redemption level

where $\alpha$ is the slope of the line

$\chi_U$ is the point until which redemption occurs at $1

$\chi_L$ represents the point where the price becomes constant again

$r_L$ is the price at point $\chi_L$

The parameters of the curve fulfils the design criteria of efficient on-chain computation. The math & proofs for Gyro Bonding Curve are robustly discussed here

Dynamic Anchor Determination in Changing Market Conditions

The challenge lies in determining a starting anchor reserve ratio. Since the system must react to the current state rather than relying on a fixed anchor point or historical data, which may no longer be relevant due to changing market conditions, we need a dynamic approach. To address this, we observe the current state. From this current state, we can identify that there is only one consistent starting point for the anchor reserve ratio, assuming the model holds true. We compute this initial state and then use it to determine subsequent actions and outcomes.

Theorem: For any fixed redemption level $x$,the collateralization ratio $r$at $x$is strictly monotonic in $r_a$, as long as $r> \bar {\theta}$ Corollary: There is a unique $r_a$ that leads to a given pair $(x,r)$.

• Redemption level ($𝑥$): The amount of asset that has been redeemed.

• Collateralization ratio ($𝑟$): The ratio of collateral value to the debt or the amount of issued tokens.

• Initial collateralization ratio ($𝑟_a​$): The starting collateralization ratio before any redemptions.

• Threshold ($\bar{\theta}$): Lower bound on collateralization ratio. A certain level above which the theorem holds

By performing some mathematical calculations, including computing the derivative, you can verify that all the derivatives have the correct signs. This proof confirms that the function is strictly monotonic.

Anchor Point from the Current Market State

Therefore, given a current collateralization ratio $r_a$ and the observed amount redeemed $\text{x}$, you can use this monotonic function to determine the initial collateralization ratio $\text{r}$. This completes the process.

Scenarios:

1. Redemptions with collateral close to 100%

   • If the stablecoin becomes slightly undercollateralized and starts trading below par, the redemption curve begins near $1 to provide good liquidity around the peg.

2. Redemptions with collateral well below 100%

   • If redemptions continue and the collateralization falls significantly, the Bonding Curve enters the Circuit Breaker Phase (CBP).

   • During the Circuit Breaker Phase, the redemption rate decreases below the actual reserve ratio, ensuring redemptions are always sustainable. The bonding curve of the redemption market will provide decreasing redemption quotes.

   • The redemption price autonomously recovers back toward the peg as outflows equilibrate back to zero or the reserve recovers through yield. Recollateralization occurs through various methods, including the underlying reserve assets recovering in price, yield generation via reserve assets, sale of treasury assets, or protocol fees.

Peg Defenses:

1. First Line of Defense: All-Weather Reserve

   • The cornerstone of Gyroscope's stability mechanism is its all-weather reserve, designed to safeguard against fluctuations and mitigate risks inherent in decentralized finance (DeFi).

   • All issuance proceeds are stored within this reserve, which serves as a diversified safety net, spreading risks across various DeFi protocols and assets.

   Second Line of Defense: Autonomous Pricing Mechanism

   • In the event of a significant shock to the reserve, Gyroscope relies on its autonomous pricing mechanism to maintain stability.

   • When stablecoin units within the system become undercollateralized, the autonomous pricing mechanism intervenes. It utilizes the bonding curve of the redemption market, offering decreasing redemption quotes as a circuit breaker to prevent systemic instability and ensure the sustainability of the entire ecosystem.

   Tertiary Line of Defense: Recovery Mechanisms

   • Gyroscope implements additional recovery mechanisms to reinforce its peg defenses and ensure the resilience of its stability protocol.

   • These mechanisms enable the reserves to recover from adverse events or expand asset-backing as needed. For instance, in situations where the reserve faces depletion, it can be recapitalized through innovative strategies such as auctioning off governance tokens, thereby bolstering its financial strength and restoring stability to the system.

Risks of Using the DSM

The Dynamic System Mechanism (DSM) defines the system's monetary policy for utilizing reserve assets to maintain a stablecoin peg. The DSM aims to balance peg maintenance with preserving reserve assets, especially when these assets experience shocks, causing the system to become under-reserved. Due to the inherent design of the DSM, which operates transparently on-chain, there is no assurance that users can redeem GYD at or near the peg price at all times.

In addition to the drawdown risk from the volatility of reserve assets, the DSM can exacerbate reserve depletion. The DSM's monetary policy permits limited redemptions near the stablecoin peg even when the system is under-reserved, further reducing the reserve ratio available for future redemptions. This additional drawdown is influenced by the magnitude of inflows and outflows during periods of under-reservation and the specific calibration of the DSM parameters

Potential Next Steps

A critical area for further research is the optimal calibration of the P-AMM. This would benefit from formal game-theoretic models of speculative attacks, where an attacker might take an external short position to profit by causing the stablecoin to de-peg. Such models, integrating the P-AMM, secondary markets, and shorting markets. Existing models of speculative attacks on fiat currencies, which informed the P-AMM desiderata, provide a useful starting point

Exploring alternative P-AMM designs that might satisfy the desiderata is another promising direction. For instance, a P-AMM with a sigmoidal shape could offer smoother transitions. However, such designs might present computational challenges on-chain, including increased computational steps (i.e., gas requirements for on-chain execution) and the amplification of rounding errors from fixed-point arithmetic.

Conclusion

Gyro has designed a desirable P-AMM redemption curve based on an anchored state. This innovative design enables a stablecoin to adapt its monetary policy sustainably in response to crisis events without the need for external input. By doing so, the system can mitigate currency runs when the stablecoin is under-reserved. Furthermore, with some modifications to account for the liquid reserve ratio, it can address bankrun like risks associated with illiquid assets.