PvpAMM: A Perpetual Market for Unbalanced Long-Short Positions

Perpetual contracts have emerged as the cornerstone of cryptocurrency derivatives trading, enabling leveraged speculation and hedging without requiring holders to roll over the underlying asset. Dominating digital asset markets with daily volumes exceeding $100 billion, they surpass traditional futures in both liquidity and accessibility [3]. However, their unique structure – continuous funding payments, no expiration, and 24/7 settlement – introduces critical challenges for maintaining balanced markets during extreme volatility or unilateral trading pressure.

Traditional implementations rely on centralized limit order books (CLOBs), where execution requires matching long and short order volumes. While straightforward, this mechanism proves fragile in skewed markets: liquidity providers face asymmetric risks that widen spreads, increase fees, and may cause market failure [5]. The zero-sum nature of derivatives exacerbates these issues, as directional exposure imbalances cannot be resolved without substantial trader costs.

We present a paradigm-shifting solution through the first complete automated market maker (AMM) framework specifically engineered for perpetual futures, which systematically addresses the persistent position imbalance problem while simultaneously enhancing capital efficiency and trader profitability. Unlike traditional AMM designs originally conceived for prediction markets [6] or simple spot trading [9, 12], our protocol introduces leveraged, state-aware liquidity provisioning that dynamically adjusts pricing curves and exposure limits in response to real-time market conditions. This innovative approach enables the autonomous absorption of unilateral order flow without dependence on external arbitrageurs or centralized liquidity pools, representing a fundamental advancement in decentralized finance infrastructure. Our solution finally bridges the critical architectural gap that has forced perpetual contracts to remain tethered to centralized order book models despite their inherent incompatibility with DeFi’s core principles.

Our research makes three foundational contributions to decentralized derivatives markets:

• $\mathrm{■}$

  A leveraged trading protocol resolving position imbalances, with profitability comparisons against traditional order books.

• $\mathrm{■}$

  Novel arbitrage mechanisms that sustain economic equilibrium, detailing liquidity providers’ role in maintaining protocol stability.

• $\mathrm{■}$

  Comprehensive agent-based simulations evaluating protocol efficacy through diverse participant interactions.

Perpetual contracts were first suggested by Shiller in [24] as a means of hedging illiquid assets, and they later gained widespread popularity as a method for taking leveraged positions in cryptocurrency markets [11]. The existing literature on perpetual futures primarily focuses on descriptive evidence. For instance, Alexander et al. [3] found that BitMEX derivatives lead the price discovery process across major Bitcoin spot exchanges. Hung et al. [23] identified significant pricing effects and breakpoints in market efficiency. Additionally, some researchers have addressed the challenges of perpetual trading by proposing derivatives with alternative return structures. For example, D. White proposed a perpetual contract with returns proportional to some power of the spot price in [27], leading to the creation of a decentralized perpetual protocol on the Ethereum blockchain called Squeeth [21]. However, this protocol splits total liquidity and does not resolve the issue of unequal positions.

AMMs have been utilized in various applications within the DeFi ecosystem, primarily popularized by token swap protocols [2, 4, 13]. They have also been adapted for several DeFi applications, including crypto options (e.g., Hegic [28]), rate swaps (e.g., Voltz [16]), and NFT exchanges (e.g., Caviar [10]). There is emerging research on the economic implications of AMMs; however, much of it focuses on the incentives for liquidity provision, particularly the relationship between transaction fees and impermanent loss [15, 17, 8].

In the context of derivatives, a virtual AMM (vAMM) model has been implemented, which differs slightly from the conventional AMM. In a vAMM, there is no proper asset pool that supports the counterparty risk, resulting in under-collateralization. Due to this under-collateralization, many vAMMs maintain an insurance pool to cover potential losses. However, this can lead to adverse selection by traders when liquidations do not occur in a timely manner. Some researchers have even argued that perpetual contracts designed through vAMMs resemble a Ponzi scheme that cannot be sustained [2]. In [14], GMX X4 introduced a peer vs. peer (pvp) mechanism for executing perpetual contract transactions via AMMs. LionDex later referenced this design in [18], but their analysis was restricted to narrow use cases – primarily liquidity pool initialization and single-price updates, without accounting for multi-period price dynamics. Their model also suffers from mathematical inconsistencies, particularly in modeling the effect of leverage on returns (see Section 2.3 for a detailed critique).

Let $P_t$ denote the spot price of the underlying asset at time $t$. Contract positions are indexed by $i,j,k$. For position $j$, let $t_j$ denote its creation time and $T_j$ its clearing time, with the ordering $t_1\le t_2\le t_3\le \mathrm{\dots }$ assumed without loss of generality. The collateral for the position $j$ is $m_j>0$, and the leverage is $b_j$, where $b_j>0$ indicates a long position, a short position, and $b_j=0$ designates a liquidity provider(LP) role.

At time $t_j$, the user borrows an amount $m_j|b_j|$ to acquire $n_j=m_j{b}_j/P_{t_j}$ units of the underlying asset. Ignoring liquidations, the conventional value of position $j$ at time $t$, where , is given by:

Position $j$ will be liquidated if $w_j^{rough}(t)$ falls below a specified threshold. Specifically, the bust time for $j$ is defined as:

where $inf$ denotes the infimum, and $ϵ\in [0,1)$ is typically set to 0. When the set is empty, we adopt the convention $inf\mathrm{\varnothing }=\mathrm{\infty }$, which implies no liquidation occurs before $T_j$. Let $I(\cdot )$ be the indicator function, with $I(x)=1$ if $x$ is true and $I(x)=0$ otherwise. The conventional value of the $j$-th perpetual contract over $[t_j,T_j]$ is:

Note that ${\tau }_j$ depends only on $b_j$ and $\{P_s:s\ge t_j\}$, so the process $\{w_j(t):t\ge t_j\}$ is adapted to the filtration generated by $(t_j,m_j,b_j,\{P_s:s\le t\})$.

In an AMM-based protocol, a liquidity pool serves as the sole counterparty for each transaction, utilizing a conservation function to algorithmically price positions and restrict price movements to predefined trajectories [29]. A perpetual AMM protocol typically involves two types of interaction mechanisms: (1) the establishment, clearing, and liquidation of leveraged positions; (2) the provision and withdrawal of liquidity. These interactions must be specified in a way such that desired invariant properties are upheld.

To illustrate the problem addressed by the pvpAMM protocol, we consider the simplest scenario where all positions remain active post-creation. Similar analysis in more complex scenarios leads to analogous conclusions (see B.2). For time $t$ where:

the total assets in the liquidity pool are:

and the total value of all positions is:

The primary challenge for pvpAMM is to distribute the available collateral $m_{\text{pool}}(t)$ to individual position values $w_j^{\text{pvp}}(t)$ when $m_{\text{pool}}(t)\ne w_{\text{pool}}(t)$, preserving protocol solvency and economic incentives. A straightforward method is to distribute funds in proportion to their conventional worth:

The proofs of these propositions (Appendix A.1) demonstrate the insufficiency of this natural approach.

To rectify the cash-wealth imbalance, GMX introduced an intermediate token, LPT. Let ${\lambda }^{gmx}(t)$ represent the price of LPT, and ${ℬ}_j(t)$ the LPT balance for position $j$ at time $t$. The protocol functions as follows:

• $\mathrm{■}$

  Establishment: ${ℬ}_j(t_j)=m_j/{\lambda }^{gmx}(t_j)$, effectively purchasing LPT with $m_j$.

• $\mathrm{■}$

  Price Updates: for $\mathrm{\Delta }{P}_t=P_{t+1}-P_t$, ${ℬ}_j(t+1)={ℬ}_j(t)\cdot \left(1+b_j\mathrm{\Delta }{P}_t/P_t\right)$.

• $\mathrm{■}$

  Liquidation: position $j$ is liquidated if ${ℬ}_j(t)\le 0$ at any time $t_j\le t\le T_j$

• $\mathrm{■}$

  Clearing: at , the position holder receives ${ℬ}_j(T_j)\cdot {\lambda }^{gmx}(T_j)$ back.

Some specific numerical examples are given in B.1. Actually we have:

The proof is provided in Appendix A.2. In DeFi implementations, spot prices update with each on-chain operation. Consequently, frequent operations compound leverage’s effect on LPT balances exponentially, resulting in exponential position value growth.

The defining idea of our pvpAMM is “concentrated collateral and proportional worth": all collateral $m_j$ and worth $w_j(t)$ for leveraged positions is pooled together, and the actual worth ${\tilde{w}}_j(t)$ of each contract $j$ is realized in proportion to its conventional value. We incorporate a strategy from the GMX model, where $m_j$ is divided by the current (cash)/(position value) scaling factor upon entering the pool. For better understanding, we introduce an intermediate token named PLT(PvpAMM Liquidity Token), priced at ${\lambda }^{pvp}(t)$. When a position $j$ is established, an amount of $m_j/{\lambda }^{pvp}(t_j)$ PLT is recorded. As time progresses, the position’s PLT balance is adjusted to $w_j(t)/{\lambda }^{pvp}(t_j)$ at time $t$, and its value in the pvpAMM pool is calculated as:

The price of PLT, ${\lambda }^{pvp}(t)$, is determined by the ratio of total collateral to total PLT balance:

${\tilde{w}}_j(t)$ measures the amount of numéraire that the $j$-th contract can retrieve if cleared at time . The PLT price ${\lambda }^{pvp}(t)$ acts as a normalization factor, ensuring that the total value of all contracts equals the remaining cash amount, the conservation function for our pvpAMM protocol can be written as:

The PLT mechanism operates similarly to a casino’s chip system. Just as a casino maintains a fixed cash reserve backing all chips in circulation, the protocol pools all collateral to back PLT tokens. When traders open positions, their collateral converts to PLTs at the current exchange rate, much like purchasing casino chips with cash. The PLT price fluctuates based on the aggregate performance of all open positions - when collective gains exceed losses, the PLT appreciates, and vice versa. Traders closing positions redeem their PLTs at the prevailing rate, ensuring the system remains fully collateralized at all times. This creates a self-balancing ecosystem where value flows naturally between participants while maintaining systemic solvency.

For computational convenience, we set $\psi (t)=1/{\lambda }^{pvp}(t)$, and get:

In this section, we clarify the state transitions of the defined pvpAMM market when operating with leveraged positions. Consider two trading positions, $i=1$ for Alice and $j=2$ for Bob. To demonstrate the system’s behavior across different stages, we specifically consider a scenario where . The trading activities in our protocol function as follows:

• $\mathrm{■}$

  Initialization. we initialize $\psi (t)=1$ for $t\le t_1$.

• $\mathrm{■}$

  Establishment. for ,

  $$\psi (t)=\frac{w_1(t)\psi (t_1)}{m_1}=\frac{w_1(t)}{m_1},{\tilde{w}}_1(t)=\frac{w_1(t)}{\psi (t)}\equiv m_1.$$

  Alice can close her position at any time during this period to reclaim her collateral $m_1$, it occurs without any profit or loss, as there is no counterparty involved.

  For :

  $$\psi (t)=\frac{w_1(t)\psi (t_1)+w_2(t)\psi (t_2)}{m_1+m_2}=\frac{w_1(t)+w_2(t)(w_1(t_2)/m_1)}{m_1+m_2}.$$

  $$\{\begin{array}{cc}{\tilde{w}}_1(t)=w_1(t)\psi (t_1)/\psi (t)=w_1(t)\frac{m_1+m_2}{w_1(t)+w_2(t)(w_1(t_2)/m_1)}\hfill & \\ {\tilde{w}}_2(t)=w_2(t)\psi (t_2)/\psi (t)=w_2(t)(w_1(t_2)/m_1)\frac{m_1+m_2}{w_1(t)+w_2(t)(w_1(t_2)/m_1)}\hfill & \end{array}$$

  where ${\tilde{w}}_1(t)$ and ${\tilde{w}}_2(t)$ represent the amounts each position would receive if closed at time $t$, ${\tilde{w}}_1(t)+{\tilde{w}}_2(t)\equiv m_1+m_2.$

• $\mathrm{■}$

  Clearing. Alice closed her position at time $T_1$, withdrew ${\tilde{w}}_1(T_1)=w_1(T_1)/\psi (T_1)$ from the pool. for :

  $$\psi (t)=\frac{w_2(t)\psi (t_2)}{m_1+m_2-w_1(T_1)\psi (t_1)/\psi (T_1)},{\tilde{w}}_2(t)=w_2(t)\psi (t_2)/\psi (t)\equiv m_1+m_2-{\tilde{w}}_1(T_1).$$

The above discussion illustrates how $\psi (\cdot )$ is updated and its role in the trading process. For any leveraged position, the presence of a counterparty is not strictly necessary. Even with only positions in the same direction, such as all long positions, trading can still proceed. In this case, the less leveraged positions effectively act as short positions(“relatively short”).

Numerical examples can be found in Appendix B.2.

For the interval , we have:

This suggests that, from the traders’ perspective, the ratio $\psi (t)/\psi (s)$ determines whether their profit(or loss) is greater or less than those in a conventional perpetual setup.

For notational simplicity, let

Observe that $\xi (t)$ is left-continuous, it represents the balance of long and short positions in the pool: . $\xi (t)>0$ means that the weighted long shares, adjusted by $\psi (t_j)$, are heavier than the weighted short shares, and vice versus.

In traditional perpetual contract trading markets such as centralized exchanges, prices of perpetual contracts are determined by secondary market transactions, and quantitative analysis between the spot and perpetual markets can identify arbitrage-free prices [1]. In contrast, our pvpAMM model determines the value of a position primarily through the price of PLT, denoted as $\psi (t)$. This value influences the strategies for position entry and exit, and arbitrageurs effectively trade over$\psi (t)$. Unlike in CEX, $\psi (t)$ in pvpAMM is unaffected by secondary market transactions or market tendencies towards long or short positions; the determination of $\psi (t)$ follows exclusively from the stochastic differential equation defined in Section 5. The pvpAMM protocol introduces an arbitrage mechanism that attracts minority orders, automatically balancing the long-short ratio by driving $\xi (t)$, as defined in 3.3, toward zero.

Liquidity provision is a key mechanism for enhancing stability in decentralized finance (DeFi) systems – a critical factor in ensuring protocol security. In pvpAMM, liquidity providers act as counterweights to dominant pool positions, mirroring the stabilizing role of market makers in traditional perpetual markets.

By letting $b_j=0$ for the $j$-th contract, we get a liquidity position:

this position will profit from a decrease in $\psi (t)$. Under the condition that $\xi (t)>0$, a decline in $\psi (t)$ indicates a fall in $P_t$, meaning the liquidity position is opposite to the direction of the majority of the pool’s positions. This is precisely the role of the LP, as it partially acts as a counterparty to the pool majority. Let $b_1=0$, i.e. the first position be an LP position, actually, we may regard the first LP position establishment point $t_k$ as the process origin. We have:

Flash loan attacks – a prevalent threat in DeFi, as outlined in Proposition 2 – leverage rapid, uncollateralized borrowing to artificially distort market conditions. While earlier pvpAMM iterations were susceptible to such exploits, our protocol eliminates this vulnerability by preventing unanticipated profits from immediate position entry and exit, as formalized below.

In decentralized finance (DeFi), accurate and timely price updates are critical to the functionality of the protocol, particularly for perpetual decentralized exchanges such as GMX and Jupiter. These platforms depend on oracles to provide asset price data, which underpins position valuation and overall protocol stability. However, oracles introduce potential attack vectors, including off-chain data manipulation, update latency, and centralization risks, all of which may compromise system integrity [26].

To mitigate these risks, modern DeFi protocols employ advanced oracle mechanisms. For example, Jupiter Perp integrates the Signal Oracle, a collaborative solution with Jupiter that enables compute-efficient multi-asset price updates in a single transaction, ensuring low latency and high reliability. Additionally, it supplements security with the Pyth Oracle for redundancy [25].

Decentralized oracle networks like Chainlink further enhance robustness by aggregating data from multiple sources, reducing manipulation risks. These systems use cryptographic attestations to verify data providers and employ aggregation methods to derive consensus prices – often accompanied by confidence intervals to indicate reliability [7].

The underlying market price is updated at each time step (after all agents have completed their actions) using the formula:

where $X\sim 𝒩(0,1)$ is drawn from a normal distribution, and $\mu ,\sigma \in ℝ$ represent the mean returns and market volatility. In our simulation, we set $\mu =0$ and $\sigma =0.01$. The initial price $P_0$ is set to $1.0$, with the timestep defined as $\mathrm{d}t=\frac{1}{100000}$.

We analyze the performance of positions in a simple market with three different types of users, where the leverage factor $b_j$ and the collateral $m_j$ are defined as in Section 2.1. The users and their default values are set to be:

• $\mathrm{■}$

  Alice, who joins at $t_1=1$, uses no leverage, i.e. $b_1=0$, with a $m_1=1000$ collateral.

• $\mathrm{■}$

  Bob, who joins at $t_2=1000$, uses $3\times$ leverage long, i.e. $b_2=3$, with a $m_2=1500$ collateral.

• $\mathrm{■}$

  Chris, who joins at $t_3=2000$, uses $2\times$ leverage short, i.e. $b_3=-2$, with a $m_3=1000$ collateral.

We assume that price fluctuations are relatively small and users have a low leverage ratio, ensuring no position will be liquidated. The variables $\psi (t),w_j(t),{\tilde{w}}_j(t)$ for $j=1,2,3$ are updated through equations 4, 2.1, 3.1, respectively.

To investigate the property of $\psi (t)$, multiple experiments with different price series $P_t$ are conducted, we calculate $\psi (t)$ and take their mean as our final value. The results, shown in Figure 1, indicate that $\psi (t)$ consistently fluctuates around its expected value of 1.0, with lower volatility compared to $P_t$.

The unique properties of our proposed pvp model under various market conditions are experimentally verified as follows: