Maximal Arbitrage Value (MAV): Theory and Applications

• MAV is a mathematically defined measure quantifying the maximum arbitrage profit attainable in financial markets, invariant to market transformations.
• Its computation employs drift null-space projection, robust statistical optimization, and superhedging duality across centralized and decentralized settings.
• Empirical studies show MAV’s critical role in portfolio design, market efficiency testing, and fair extraction mechanisms in MEV and decentralized finance.

Maximal Arbitrage Value (MAV) is a rigorously defined concept that quantifies the highest attainable arbitrage profit—under precise mathematical, geometric, and model-invariant criteria—in both traditional and decentralized financial markets. MAV is not restricted to a specific mechanics or asset type; it appears variously as a gauge-invariant residual drift in stochastic differential models, as the optimal solution to parabolic partial differential inequalities in Markovian and Knightian contexts, as a superhedging ratio, as the output of robust statistical optimization, as a payoff-class maximum over geometric portfolio constructions, or as realized profit in multi-market execution settings—determinable only after accounting for all market frictions, liquidity impacts, and relevant constraints. Across foundational literature, MAV is the structurally unavoidable, model-independent "core" of arbitrage, immune (by construction) to cosmetic changes in numéraire, equivalent probability measures, or—under certain circumstances—market protocol rules such as transaction ordering or block production frequency. Its theoretical, numerical, and empirical determinations, outlined below, elucidate its central role in financial mathematics, portfolio construction, decentralized finance architectures, and emerging methods on blockchains and rollups.

1. Fundamental Definition and Mathematical Structure

Across stochastic finance, MAV is the maximized, gauge-invariant arbitrage component of asset price drifts, or equivalently, the supremum of attainable arbitrage profit from admissible strategies—after exhausting all possible "degenerate" market transformations (e.g., changes of probability measure, change of numéraire). In the Ito process framework for asset price dynamics, the drift is decomposed as

$$\alpha_{(\mu)} = \alpha + \sum_a \beta^a \sigma_{(\mu)}^a + \sum_{A \in \mathcal{N}} \alpha^A J_{(\mu)}^A,$$

where the last term collects the gauge-invariant (residual) components, with the $\{J_{(\mu)}^A\}$ forming a basis for the null space $\mathcal{N}$ orthogonal to the "market risk" directions. The canonical measure of arbitrage, which is maximal in the sense of being irreducible by any admissible transformation, is

$\mathcal{A}^2 \equiv \sum_A (\alpha^A)^2.$

This arbitrage curvature is geometrically interpreted as the curvature of a gauge connection, with vanishing curvature corresponding to the absence of arbitrage (0908.3043).

In robust optimization and semimartingale models, the MAV is given by

$$U(T) = \sup \left\{c > 0 : \exists H \in \mathcal{A}_1 \text{ with } V_T^H \geq c \text{ a.s.}\right\},$$

where $\mathcal{A}_1$ is the set of admissible (nonnegative, suitably bounded) portfolios, and $V_T^H$ is the strategy’s terminal wealth. Canonically, $U(T) = 1 / SP^+(1)$, the reciprocal of the superhedging price for the unit payoff (Chau et al., 2013).

In decentralized market models—particularly automated market makers (AMMs)—MAV is a closed formula parameterized by on- and off-chain prices $P_A$, $P_C$, and liquidity $y$:

$\mathrm{MAV} = \frac{y (P_A - P_C)^2}{4P_A},$

where the quadratic dependence on price deviations reflects both the degree of misalignment and the limiting impact of pool liquidity (Gogol et al., 24 Mar 2024).

2. Invariance, Geometry, and Theoretical Underpinnings

MAV is invariant under two central classes of market transformation:

• Gauge invariance: Under a change of numéraire, prices are rescaled ($X_{(\mu)} \to \Lambda X_{(\mu)}$), but the arbitrage curvature $\mathcal{A}^2$ is unchanged; its geometric structure as a connection one-form $A$ renders arbitrage both path-dependent and physically meaningful only when $dA \ne 0$ (0908.3043).
• Equivalent probability transformation: Any risk-neutral or physically equivalent probability measure erases only the market risk drift, leaving the residual arbitrage component—and thus MAV—intact.

In functionally generated portfolio contexts, the only portfolios that consistently attain relative arbitrage (uniform outperformance under diversity and volatility) are those generated by strictly positive concave functions on the unit simplex, subject to multiplicative cyclical monotonicity (MCM). MCM is both necessary and sufficient for MAV maximization in equity market geometry (Pal et al., 2014).

The connection to Farkas’ Lemma and the Arbitrage Theorem underlines that, in finite-dimensional payout spaces, MAV emerges as the optimization of Av over all portfolios v, constrained to the intersection of the column space of the payoff matrix and the positive orthant. The geometry of hyperplane arrangements bounds the number of distinct arbitrage regimes, shaping the extremal value (Naiman et al., 2017).

3. Practical Determination, Empirical Detection, and Numerical Methods

MAV can be empirically estimated, or operationally computed, through several formal and applied methodologies:

• Drift null-space projection: By estimating the covariance of asset log-returns then extracting the (near-)null eigenspace, one can project realized returns to measure the $\alpha^A$ and compute sample paths of $\mathcal{A}^2$. Peaks at high frequency signal active arbitrage opportunities, while daily windows typically yield $\mathcal{A}^2\approx 0$ (0908.3043).
• Statistical arbitrage optimization: Portfolios are optimized to maximize in-band volatility via convex–concave procedures, yielding statistically significant mean-reverting structures with high MAV, robust to leverage constraints and multi-asset configurations (Johansson et al., 12 Feb 2024).
• Superhedging duality: The reciprocal of the superhedging price under the physical (arbitrage-admitting) measure vs. the risk-neutral reference measure anchors MAV; model construction using measure changes allows explicit MAV computation (Chau et al., 2013).
• Multi-market construction: In AMMs and DEX–CEX price misalignments, MAV is derived analytically as a function of price gaps, pool reserves, and protocol fees. For concentrated liquidity pools, MAV is summed over the relevant tick intervals (Gogol et al., 24 Mar 2024).
• Copula-based optimization: Arbitrage detection in multi-asset derivative markets is formulated as an optimization over copula functionals consistent with all observed marginal risk-neutral distributions. Inconsistency (no feasible copula) certifies the presence of multi-instrument MAV even when all individual derivatives satisfy no-arbitrage bounds (Papapantoleon et al., 2020).
• Algorithmic extraction in blockchains: Profitable sequences of swaps (cyclic arbitrages, sandwich attacks) are identified using token balance conversions and depth-first path enumeration, capturing only transactions with true positive net profit, i.e., nonzero MAV. Algorithmic advances over naïve heuristics yield markedly improved coverage (Chi et al., 28 May 2024).

4. Robustness, Fragility, and Model Uncertainty

MAV’s exact value and attainability can be robust or fragile depending on model or protocol specification:

• Fragility: Many diffusion model arbitrages, while mathematically possible, disappear under small transaction costs or under “ε-close” perturbations of the price process (fragility in the sense of arbitrage vanishing in the presence of microstructure noise). Non-fragile, or robust, arbitrages are constructed via jump models or stopping times not sensitive to infinitesimal modifications (Chau et al., 2013).
• Knightian uncertainty: In models with parameter (covariance, drift) uncertainty, MAV is given by the inverse of the minimal initial fraction that almost surely covers the market portfolio under all admissible scenarios. This is obtained as the minimal supersolution of a fully nonlinear Hamilton–Jacobi–Bellman PDE, or as the value function of a stochastic control/game problem (Fernholz et al., 2012).
• Protocol invariance: In deterministic protocols with path-independent, frictionless liquidity pools (e.g., constant product or concentrated liquidity AMMs), the total value of arbitrage available is invariant to transaction ordering or block subdivision—operations that merely redistribute rather than increase arbitrage profit (Guo, 2023).

5. Empirical Phenomena, Decentralized Finance, and MEV

Modern realizations of MAV, especially in decentralized finance (DeFi) and blockchain systems, are characterized by their operational, system-wide significance:

• Non-atomic arbitrage: The bulk of empirically observed MAV in leading DEXs (often >25% of volume) arises from non-atomic arbitrage, where the inter-market legs (on-chain and off-chain) cannot be executed in a single atomic transaction. The magnitude of MAV in this regime reaches $132B across five DEXs during the studied period, dominated by a small cadre of integrated searcher-builder entities (Heimbach et al., 3 Jan 2024).
• MEV (Maximal Extractable Value): MAV is a foundational component of total MEV, with arbitrage MEV being the largest segment. Its robust identification in actual blocks is improved when moving beyond sequential heuristics to path- and exchange-rate–based analysis, handling cases with out-of-order swaps and complex token flows (Chi et al., 28 May 2024).
• Rollup and cross-chain extensions: MAV on Layer-2 (L2) rollups is characterized by persistent price misalignments owing to rapid block times and fragmentary liquidity. Empirical MAV is measured as a fraction of trading volume (0.03–0.05% on major L2s, up to 0.25% on ZKsync), with persistent opportunities demanding metric adjustments (e.g., LVR modification) to prevent double counting (Gogol et al., 4 Jun 2024).
• Monetized arbitrage profit: In cyclic DEX arbitrage, MAV maximization requires considering all possible starting tokens in the arbitrage loop, with the overall profit measured in common fiat terms (token output × CEX price). Formal convex optimization strategies marginally outperform, but the simpler MaxMax strategy is empirically almost always optimal (Zhang et al., 24 Jun 2024).
• Sequencer and time-advantaged regimes: Auctioned time advantages (e.g., Arbitrum’s Timeboost) provide a privileged period in which an arbitrageur can extract a higher MAV than under FCFS or priority-gas–auction-based regimes. Dynamic programming shows that the optimal strategy is to wait for the end of the time window, substantially increasing mean realized arbitrage profits and providing grounds for protocol-level MEV redistribution, including AMMs capturing a share of the MAV via fee adaptation (Fritsch et al., 14 Oct 2024).

6. Implications for Portfolio Theory, Market Design, and Protocol Engineering

The comprehensive theoretical, statistical, and computational development of MAV has several crucial implications:

• The identification of gauge-invariant or model-invariant MAV provides a rigorous benchmark for market efficiency testing, portfolio design, and stress testing in both classical and digital markets.
• Fragility analysis informs both the resilience of arbitrage strategy design and the development of new financial instruments.
• Practical detection algorithms for MAV—for example, via copula methods or advanced swap cycle enumeration—are essential for market surveillance, risk management, and DeFi protocol auditing.
• The structural theorems demonstrate that efforts to boost on-chain trading opportunity by mere protocol rearrangements (e.g., block time reduction, transaction ordering) cannot increase but only redistribute MAV; only liquidity, price volatility, and structural market frictions set its upper bound.
• In decentralized systems, MAV-driven competition shapes the economics of block construction (e.g., Proposer-Builder Separation), necessitates mitigation of negative externalities (centralization, time-bandit attacks), and guides the design of both fair extraction mechanisms and protocol-level MEV sharing.

7. Open Problems and Future Directions

Despite significant advances, fundamental open challenges persist:

• The dynamical behavior of MAV under regime-switching volatility, non-stationary liquidity, and evolving on-chain/off-chain integration remains an active area of stochastic control and empirical analysis (Fernholz et al., 2012, Fernholz et al., 2016).
• The full characterization of MAV in high-dimensional, nonconvex, or adversarial market environments requires further research—especially for applications in multi-asset derivatives, multi-chain rollups, and cross-domain DeFi (Papapantoleon et al., 2020, Obadia et al., 2021).
• Theoretical and computational efficiency improvements in convex optimization methods for multi-hop/cyclic arbitrage loops continue to be refined, particularly given their computational complexity as the number of tokens increases (Zhang et al., 24 Jun 2024).
• Protocol-intrinsic mitigation of MAV centralization—especially in light of the steady market share consolidation among top searcher-builder pairs—warrants continued innovation in both incentive structures and MEV-redistributing design (Heimbach et al., 3 Jan 2024).

In summary, Maximal Arbitrage Value constitutes the mathematically precise, model-invariant, and operationally decisive quantifier of extractable arbitrage profit. Its geometric, analytic, statistical, and empirical structure now underpins both classical asset pricing theory and the newest developments in blockchain-enabled financial markets, and continues to guide the evolution of efficient, fair, and robust market architectures.