We study the dynamics of the Terra-Luna collapse occurred in May 2022 by creating a simulation environment that embodies both the free market buying-selling transactions and the Terra-Luna protocol exchange features. The parameters used during the simulation generate the conditions necessary for triggering the deviation from the peg of the stablecoin UST, along with the subsequent collapse of its value to almost zero. Then we present three proposals to increase the stability of algorithmic stablecoins and we employ the simulation environment to show how they could help stabilize the algorithmic stablecoin's peg.
The decentralized finance (DeFi) environment introduces a new paradigm in the finance sector. It enables
unbanked users with smartphones and Internet access to engage in financial activities like money
transfer, lending, borrowing, or speculating on synthetic stocks or commodities, on a 24/7/365 basis,
without the need for a bank or other financial institution’s involvement [
The volatility of these digital assets has forced the introduction of collateralized stablecoins (CS). These are cryptocurrencies backed by a fiat currency or a commodity – typically USD, but also EUR, CHF, JPY, RMB, KRW, or gold. These digital assets maintain dynamically the peg with the corresponding fiat currency through a seigniorage process, along with a certification that for each collateralized token there exists a corresponding amount of value of the fiat currency, deposited in a bank, that can be redeemed.
Nowadays, we count dozens of fiat-backed stablecoins traded on the main cryptocurrency exchanges, such as USDT, USDC, TUSD, and BUSD. Their capitalization and success are increasing over time. They are used as a safe haven of stability when traders want to exit the volatility turmoil of the market and control the keys of the owned tokens, avoiding the use of a centralized exchange, the only able to swap a cryptocurrency with a fiat currency.
The spirit of the DeFi philosophy is that of creating a new paradigm for a decentralized financial system, totally detached from (central) banks, financial institutions, and fiat currencies. Unfortunately, collateralized stablecoins do not comply with this philosophy, since by using them the DeFi sector remains bonded with the traditional financial market via the fiat currency used as collateral. It is at this point that algorithmic stablecoins (AS) come into play. They are coins or tokens whose price is anchored to fiat currency solely by using an algorithmic protocol. More precisely, there are two diferent kinds of pure algorithmic stablecoin pegging mechanisms: rebasing and seigniorage.
In the rebasing model, stablecoin’s total supply is not fixed and is modified adaptively on a regular
basis, directly on the wallets of all users. The general idea is that when the price of the AS is above the
parity with the adopted fiat currency – say the USD – it is necessary to increase the supply, since the
demand is high and the token is too scarce. On the contrary, when the price is below the parity, it is
necessary to decrease the supply, since the demand is low and the token is too abundant. This implies
that a user with, say, 1000 tokens today, could find a greater amount in her/his wallet the day after,
say 1100, if AS price > $1, without any action by the user. On the contrary, if AS price < $1, then the
wallet content could be smaller, say 900 tokens. The Ampleforth protocol (AMPL) [
The seigniorage model typically has two tokens: the AS token and the governance token (GT). The AS token is the algorithmic stablecoin, while the GT token is used to absorb the volatility of the AS token. This means that when AS price < $1, we can profitably burn 1 AS for $1 worth of GT inside the protocol, which we can sell at market, yielding the diference as a profit. This reduces the total supply of AS and stabilizes the price. On the contrary, when AS price > $1, burning $1 worth of GT can mint 1 AS, which we can sell at market, yielding the diference as a profit. This increases the total supply of AS and stabilizes the price.
The Terra-Luna protocol [
After recalling the basics of Terra-Luna (Section 2), we present the first contribution of this paper, which is a simulation environment for Terra-Luna based on Matlab® (Section 3). This environment mirrors the dynamics of the free market, allowing users to buy and sell UST and LUNA. Moreover, by changing some parameters, we can induce the market conditions that led to the collapse of the UST peg.
The second contribution of this work is to illustrate three proposals that aim to improve the stability of the original protocol (Section 4). Firstly we propose a redesign of the original algorithmic stabilization mechanism used in the Terra protocol, whose failure caused the collapse of the system. The second proposal involves creating a USDT reserve pool that the protocol can utilize to automatically restore the AS price. Finally, we propose a solution that aims to prevent hyperinflation of GT by introducing an automatic BTC pool. The paper concludes with all simulation results (Section 5) and a final discussion (Section 6).
2.1. The Terra Stabilization Mechanism
The Terra algorithmic market module (TMM) played a central role in maintaining the price stability of
UST. This is the module that provides incentives for arbitrageurs2 to mint or burn UST in response to
price deviations from the peg [
When UST’s market price falls below the peg, e.g., $0.98, arbitrageurs can profitably burn 1 UST
obtaining automatically $1 worth of LUNA from the protocol, making a $0.02 profit per UST burnt.
Conversely, if UST’s price exceeds the peg, e.g., $1.02, they can burn $1 worth of LUNA and mint 1 UST,
again yielding a $0.02 profit. This mechanism operates via the protocol’s algorithmic market-maker,
the virtual liquidity pool (VLP), with LUNA’s price sourced from validator oracles, see Figure 1.
The VLP is implemented through a variant of the classical constant-product market-making algorithm
[
The TMM integrates the stabilization mechanism, with the parameter indicating the deviation of the UST amount into the VLP compared to its base size : = + , =
The dynamics of play a crucial role in adjusting the liquidity pool sizes in response to market activities. As swaps happen and the balance between UST and LUNA quantities shifts, changes to ensure that stays constant. A key aspect of the market module’s functionality is its ability to replenish the VLP, progressively bringing back towards zero. The rate of this replenishment is determined by the parameter. At the end of each block – with one block being produced approximately every 6 seconds – is updated using the following formula: := · ︂( 1
1
−
︂)
(2)
(3)
This formula governs the adjustment of , with influencing the pace of the
adjustment. This parameter was determined by the Terra community, and at the time of the de-pegging
event, its value was 36, meaning that a partial replenishment of the VLP occurs every 36· 6 = 216 seconds
if no transactions take place during this period [
This interpretation caused a final crash in the market, indicating the collapse of the Terra protocol.
These events highlight a cascade of failures within the Terra-Luna protocol, including an irreversible depeg of the UST stablecoin, the lost of about $54B of capitalization, and the collapse of the other two DeFi platforms involved in the Terra-Luna ecosystem. They are the lending platform Anchor and the exchange Mirror protocol, which allowed users to create and trade mirrored assets (mAssets) that mirror the price of stocks and/or commodities of the real world. 2.3. Literature Review Stablecoins have gained interest from researchers across disciplines and from financial institutions. The literature delves into their design, mechanisms, and impacts on financial stability, monetary policy, and regulation.
The ECB Crypto-Assets Task Force has tackled the issue of stablecoins in their report n. 247 [
As for the algorithmic stablecoins, Clements [
The Terra-Luna ecosystem has been the subject of several studies that explore its unique features and challenges, as well as the failure that occurred in May 2022.
Cho [
Briola et al. [
Liu et al. [
Kurovskiy and Rostova [
Uhlig [
Ferretti and Furini [
We present two distinct and independent simulations of the Terra stability mechanism. The first
one investigates the role of the mechanism in the collapse of Terra-Luna and is rooted in Cho’s
study [
The same simulation environment will be used in Section 4 to propose three enhancements to the
Terra protocol. The first one consists of a redesign of the stability mechanism, while the other two have
to do with limiting the LUNA supply growth.
3.1. Price Dynamics through an AMM
To achieve our goals efectively, we need an environment that can accurately reflect the price changes
in the free markets of the exchanges, for both the stablecoin and the governance coin, and the dynamics
of the Terra stabilizing protocol. This can be done by simulating an Automated Market Maker (AMM),
which operates within discrete time intervals called iterations or samples. AMMs serve as foundational
components in decentralized exchanges (DEX) [
In our simulations, two AMMs (or markets) are implemented: one that governs the buying and selling of the stablecoin, and the other that governs the buying and selling of the governance token. During each time interval, a swap occurs within each AMM, influencing simulated token prices. The impact of these swaps on prices depends on the volume of tokens involved: larger volumes have a greater efect on simulated prices, so inducing a slippage on it.
Let us explore in detail how AMMs are implemented within the simulations. Each AMM is described by a LP Π , composed of two tokens and . At the discrete time instant , the state of a LP is defined by: • () and (), which represent the supplies of and , respectively, at iteration . • = ()· (), which is the invariant at time . (4) (5) In our simulations, we assume zero transaction fees and constant pool liquidity for simplicity, as these factors are deemed unimportant for our analysis goals. The liquidity pool state at a given time can be written as:
Π ,,((), ()) We can define a swap as a function that operates on a liquidity pool state. The swap function takes a token and a swapped quantity as input and produces a token and the related quantity as output. The token can either be or , and the token is consequently or . If the swap is performed at time + 1, the diference in supply at the output is equal to: () − ( + 1) =
() − () + where () and () are the pool quantities of tokens and respectively.
In an AMM, the token price is measured in terms of the other token in the liquidity pool. Therefore, it is crucial to establish a fixed reference for token pricing. Here, the reference is USD, serving as a stable benchmark for analyzing price fluctuations of the algorithmic stablecoin. As fiat currency cannot be used in DeFi, we need to make an assumption. Let’s designate the other token as , a fully collateralized stablecoin pegged to USD; for example, USDT or USDC. This allows token values to be expressed in USD, assuming remains at a constant external value of $1.
The concept of algorithmic stablecoin involves a stabilization mechanism that is usually based on
two tokens: the stablecoin (e.g. UST) and the governance token , which is volatile (e.g. LUNA).
Both simulations instantiate the AMM model above by employing two liquidity pools to emulate the
price of and . The first pool, denoted by Π , is designed to simulate the market operations on
and consists of and . The second pool, denoted by Π , replicates the market dynamics of and
consists of and . At each time instant , a swap occurs within Π and Π altering the price of
and , the state of the liquidity pool is given by Π , ,((), ()) where = or = , and
the price () of the token is determined by the ratio of the quantities of the two tokens inside the
pool at that specific moment:
() =
()
()
This is what Uniswap, the most used DEX with the highest TVL, refers to as the “mid price” [
= 0 + 1 + 2 + . . . + In the stochastic swap model presented in this study, a random walk is employed to determine the probability of a token to be swapped within the pool at a given iteration. Let’s consider the example of liquidity pool Π ,,. At iteration , = with probability (), and = with probability 1 − (). So, there are three possible scenarios: () = 12 () < 12 () > 12 =⇒ =⇒ =⇒ equilibrium condition experiences buying pressure experiences selling pressure At the beginning of each iteration, () is varied from its current value by an amount ∆ : ( + 1) = () + ∆ where ∆ is a random variable with a normal distribution Φ( Δ, Δ2) such that its mean value Δ is equal to zero, while its variance Δ2 is a simulation parameter initialized during the simulation setup that expresses the market volatility. At the beginning of the simulations, we suppose that the market is in a state of equilibrium, hence (0) = 12 . The value of () is subsequently updated based on equation (7), allowing for both positive or negative steps, that increases or reduces the probability of sale. 3.4. Inducing the Collapse We have now the problem of representing the conditions of a panic sell (FUD – Fear, Uncertainty, Doubt) or of an irrational buy (FOMO – Fear of Missing Out) that can aflict the free market. During the FUD phase, users who are in a state of panic tend to sell their cryptocurrencies with great intensity, triggering what is known as a “bank run”. Widespread panic can be triggered by any external event. In the case of the Terra-Luna ecosystem, it was the withdrawal of large amounts of UST from the Anchor protocol and the partial loss of the peg.
We can model this by introducing the concept of panic zone for the stablecoin. When the market enters the panic zone, a mechanism is triggered to represent the irrational behavior of users, whose decisions are driven more by emotions than rationality. Otherwise, we are in the healthy zone, where the users act as usual, in a normal condition of the market. It is understood that the stablecoin tends to maintain the peg inside the healthy zone, while tends to lose it outside, when entering the panic zone.
Concerning the pool Π , we have implemented two diferent definitions of panic zones, used in the simulations to represent the behavior of the buying/selling probability, when hovering over the boundaries separating the healthy zone from the panic zone of the market. The two approaches respectively use the price of the stablecoin and its probability of sale to trigger the risk of a collapse when entering the panic zone. 3.4.1. The Variable Mean Approach Based on Price Suppose that at iteration = 0 the price is set to (0) = 1. As the simulation progresses, various buy and sell transactions of will occur, leading to fluctuations in its probability and price on the basis of equations (6) and (7). At iteration , is in the healthy zone if: |1 − ()| < with < 1 In our simulation we have set = 0.05; this implies that the healthy zone of covers the interval $0.95 < () < $1.05. When the price of the algorithmic stablecoin exits this healthy zone, a price collapse mechanism for is triggered. In this scenario, the goal is to create selling or buying pressure that acts proportionally to the deviation of the price from $1. To implement this selling or buying pressure, we adjust the mean value Δ, initially set to 0, on the basis of the following formula: Δ = (1 − ()) · Δ So, when |1 − ()| ≥ we are in the panic zone, and the following market conditions hold: 1− () > 0 1− () < 0 =⇒ Δ > 0 =⇒ Δ < 0 =⇒ =⇒ selling pressure buying pressure Note that the slippage of the mean is proportional to the slippage of the price. 3.4.2. The Variable Probability Approach In the second approach the sell probability () freely fluctuates as described in equations (6) and (7), when we are in a healthy zone. The only diference is that now the healthy zone itself is defined in terms of probabilities, more precisely as the interval 0.3 ≤ () ≤ 0.7. Outside this interval, we are in the panic zone, and the user behavior is modeled through a completely diferent function representing her/his irrational mindset, which is an exponential-type function.
Define now the “ ℬ zone”, or buy zone, as the market condition in which () < 0.3: in this scenario, the demand for buying cryptocurrency is particularly high for various reasons. On the contrary, the “ zone”, or sell zone, represents the most critical phase of the market, in which () > 0.7 and the stablecoin risks the collapse. The functions ℬ() and () compute the sell probability within the panic zone: ℬ() =
1 1+e + ℬ () = 1+e− 1 + Here is the distance from the healthy zone, is a parameter determining the slope of the function, and ℬ and need to be computed based on the parameters lowerBound = 0.3 and upperBound = 0.7 ifxing the limits of the healthy zone, so as to align the values of inside and outside the interval. More precisely we have: ℬ = if
1 = lowerBound 1+e = if
1 1+e− = upperBound In our case ℬ = because 0.5 − lowerBound = upperBound − 0.5.
Inside the Terra-Luna protocol, the simulation keeps track of the number of tokens in circulation for both cryptocurrencies and models the principle of scarcity in the following way. When a new quantity of a token is minted, a positive step is taken during the random walk of the market in which it is traded, increasing its probability of sale. Conversely, when a certain quantity of a token is burnt, a negative step is taken during the random walk of the market in which it is traded, increasing its probability of purchase. The length of the step taken is proportional to the quantity minted or burnt.
The Terra-Luna protocol is not used during each time unit, but its probability of use ( ) increases with the deviation from the peg value. This is because a larger variation in the price of UST ofers greater profit opportunities for arbitrageurs, who tend to use the protocol more frequently. The function ( ) is defined as follows: 1–15 (8) (9) (10) (11) with and suitable constants. In our case = 0.55, = 5, and = |1 − UST.price| · 10. Equation (11) assumes diferent interpretations depending on whether the price of UST is above or below the parity. In the former case, the function returns the probability of minting UST; in the latter case, the function gives the probability of burning UST.
The scheme shown in Figure 1, describing the arbitrageurs’ opportunity when using the Terra-Luna protocol, is not used in practice when the market approaches zone . Normally, when the price of UST is less than 1, this would imply the purchase of discounted UST to make profit. However, if the price undergoes a significant drop triggering a bank run, this is no more true. In a panic phase users simply want to get rid of all the UST already in their possession, without buying new ones to make profit, also because in these cases the value of LUNA received in change is plummeting.
We now propose three mechanisms to improve the stability of the Terra-Luna protocol. They aim to
enhance the system’s redemption capacity during periods of crisis and high volatility.
4.1. Modifying the TerraPool Mechanism
At the core of Terra’s stabilization mechanism is the concept of virtual liquidity pool (VLP) [
(12) () () Each swap executed within the pool dynamically alters the value of ().
In the original implementation of the pool replenishing mechanism of equation (3), () goes down to zero only when →
∞. So, we suggest a diferent method that acts similarly, but can bring () to zero after exactly ( ) blocks. The idea is very simple. Let’s consider the scenario where the initial swap within the pool at time = 0 involves an amount of stable tokens . Then the replenishment of tokens in the VLP is spread among a list of / chunks, which will supply in the subsequent time instants of the simulation. More formally, let be a vector of length initially empty; at the time instant = 0 it is filled with the chunks: = [︁(︁ )︁ (︁ ,
)︁ , ..., ︁(
)︁]︁
|| =
as a consequence of the swap. The first element 1 of will determine the value of (1) as follows:
(1) = (0) − 1
Simultaneously, during this iteration, a swap operation takes place within the virtual pool denoted as
Consequently, the update of vector takes place according to the following procedure:
(, ), where ∈ {, }. Consider = [/ , / , ..., / ], where || = .
{︃() = ( − 1) +
() = ( − 1) −
if =
if =
(13)
After this, () is updated as follows, where now 1 is the first element of ():
() = ( − 1) − 1
Finally, the elements of the vector are shifted left and the last element is set to zero:
() = [1, 2, ..., →]−− −
ℎ
( + 1) = [2, 3, ..., , 0]
This process guarantees that, if no swaps occur, the virtual pool is fully replenished after exactly
blocks. Based on this setup, the improvement of the stability we are going to propose pertains to a
variant of the VLP replenishing mechanism suggested above. If the variation of exceeds ·
(where = 0.05 in our simulation), the array () is treated as if it had a length of /2. Essentially,
in a crisis scenario, the virtual pool’s redemption capacity doubles. Given that the variation of
correlates directly with the price () of (as shown in [
√︁ 0.905.9+54 − 1
⎩̃︀() = 2
4.3. Implementing a BTC Reserve Pool against LUNA Hyperinflation
This approach is an automatization of the attempt made by the Luna Foundation Guard to support the
value of UST, when during the agitated phases of the collapse they sold at the market about 80 000 BTC
[
When the value of the stablecoin reaches a critical level – e.g., $0.95 – a queue system is automatically activated, which involves a reserve of BTC. At this point, a user has three options available: 1. sell her/his own directly on the market; 2. use the classic stabilization protocol, burning 1 to obtain $1 worth of ; 3. burn 1 to obtain $1 worth of BTC.
It is necessary to clarify some aspects. For the correct functioning of the system, the presence of a reliable oracle, which provides the real-time price of BTC, is essential. Moreover, the users will not receive real BTC, as the latter is a native cryptocurrency of another blockchain. Instead, they will receive $1 worth of wrapped BTC (wrBTC), which is a synthetic token that represents the ownership of a BTC on a blockchain diferent from the original one. Obviously, it is not possible to prevent users from selling their own on the market. However, it is possible to induce them to undertake option 2 or option 3 through the creation of a double-queue system: queue collects transactions of users eager to exchange tokens for BTC, while queue contains transactions of those who prefer to normally use the protocol, obtaining $1 worth of for each inserted.
Both options ofer the benefit of reducing the circulating supply of , but queue has the additional advantage of avoiding the minting of . A probability function can govern the mechanism by selecting,
Collapses Δ Δ
Δ 10− 4 5 · 10− 4 10− 3 (14) at each time interval, which queue to activate. Both queues follow the FIFO (First In First Out) principle. The probability function can be a simple logistic function, which essentially depends on two parameters: • the deviation of the price () from the parity; • the filling percentage of the BTC reserve.
In our simulation, the function () returns the probability that queue is activated, and is defined as follows: () =
1 1 + − (− ) where determines the slope of the curve and establishes the point on the -axis where the graph has an inflection point. These parameters can be manipulated to obtain a function that conforms to one’s preference. The independent variable could be set as a combination of the filling percentage of the BTC reserve and the deviation from parity, for example the product of the two. To reproduce the market context before the collapse of the Terra-Luna system, the 100% of the reserve could be considered equal to $1B in BTC. When the reserve is empty, the probability of queue being selected is set to zero.
We carried out two kinds of simulations. The first one evaluates the performance of the original stabilization mechanism of the Terra protocol under normal market conditions, characterized by constant volatility Δ. In the second one, we simulated a crisis scenario, marked by escalating volatility, to induce a collapse and to test the eficacy of the three proposed improvements discussed in the preceding sections.
Firstly, we conducted a series of 10 simulations, each with diferent constant values of parameters Δ, , and , resulting in a comprehensive total of 270 simulation runs. The parameter Δ plays a pivotal role in determining market volatility, while and determine the redemption capacity of the stability mechanism. In each simulation, starts at $1, with a total supply of 107, while Π and Π initially consist of 2 · 106 units of and , respectively. The price of the volatile token is initially set to (0) = $100. Results are shown in Table 1. We considered collapsed when its price falls below $0.50 and the system is not able to recover. Figure 2 shows the behavior of the price of during a collapse.
In the second series of tests, we conducted 30 simulations with the goal of inducing the system to collapse by gradually increasing the volatility Δ from Δ = 0.0001, with a 0.00002 step every 1000 iterations, with a total of 100 000 iterations. Figure 3 illustrates the behavior of under these conditions. The results are shown in Table 2, and make it evident that the original implementation of Terra is unable to withstand such scenarios, with the peg being lost in 28 out of 30 simulations.
Better results are obtained with the modification of the TerraPool replenishing mechanism and the UST Reserve Pool. As for the BTC Reserve Pool improvement, we performed two separate sets of tests. The first one involved the use of the Terra standard stabilization protocol, without any queue system. The second one incorporated the queue system described in Section 4.3. In both scenarios, we conducted 100 runs of the simulation program, monitoring and recording the circulating supply of (LUNA). The results are shown in Figure 4.
Simulation
Total collapses Mean collapse time
The first conclusion we can draw is the structural weakness of the Terra-Luna protocol, since Table 2 shows an astonishingly high number of collapses as a consequence of an increasing volatility Δ of the market (28 out of 30). Also, the huge number of circulating supply of LUNA, of the order of 1012 and described in Figure 4a, is a clear warning of this structural weakness. Even if a perfect algorithmic stablecoin should resist all kinds of market destabilization forces, with only "0" in the second and third columns of Table 1, the improvements we obtain with the three proposals shed light on the mechanism one could implement to increase the strength of an algorithmic stablecoin protocol.
The TerraPool method indicates that it is possible to work on the replenishment protocol with a mild rate of success (24 collapses out of 30 instead of 28). Significantly better results are obtained with (a) Circulating supply of LUNA in the original protocol (b) Circulating supply of LUNA with the queue system implementing the BTC Reserve Pool the use of a UST Reserve Pool (7 collapses out of 30). Also, the method of limiting the LUNA supply with the queue system implementing the BTC Reserve Pool shows a good performance, reducing to 109 the original supply of LUNA of order 1012. Note that this huge supply was the main cause of the crash in the LUNA value and of the Terra-Luna collapse. Note, moreover, that this last method guarantees an almost constant value of supply for all simulations. These evaluations underscore the critical importance of incorporating reserve pools and refining virtual pool replenishing mechanisms to improve the stability of algorithmic stablecoin systems in the face of market volatility. The utilization of a reserve pool provides a significant cushion against adverse market movements, acting as a stabilizing force during periods of heightened volatility, and the obtained data serve as a clear illustration of the robustness exhibited by this hybrid collateralized-algorithmic stablecoin. It efectively demonstrates the coin’s ability to resist challenges posed by a highly volatile market. In general, our solutions extend beyond Terra-Luna, adapting to diverse seigniorage stablecoin frameworks.
As far as the limitations of our approach are concerned, while the introduction of a reserve pool shows promising results, it moves away from the notion of a pure algorithmic stablecoin, transitioning towards a partially collateralized stablecoin model. Future research should address these trade-ofs and explore avenues for optimizing stability while maintaining algorithmic integrity. Another limitation is related to the design choices we made. They are an inevitable approximation of the human behavior in ifnancial markets. Anyway, we tried to be as general as possible in designing these phenomena, since our simulations are initializable with diferent parameters, allowing for flexibility in capturing various market conditions and behaviors. A critical aspect for future exploration lies in refining the parameters and assumptions underlying our simulation model. This could be achieved by incorporating real-world data and historical market trends.
Moving forward, our research suggests several interesting directions. Further investigation into alternative replenishment protocols and reserve pool mechanisms could yield insights into improving the stability of this type of system. Our simulation-based approach could help researchers to efectively design and evaluate new algorithmic stablecoin protocols, ofering valuable information about their performance and robustness in various market conditions.
Acknowledgments. This research has been supported by the PRIN 2020 project NiRvAna – Noninterference and Reversibility Analysis in Private Blockchains.
Code Repository. The Matlab code implementing our simulations is publicly available at https://github.com/FedericoCalandra/Algorithmic-Stablecoin-Collapse-Simulation.