Cross-Market Making

Updated 16 September 2025

Cross-market making is the practice of providing liquidity across multiple correlated assets and venues by balancing inventory and risk dynamically.
It employs advanced methods such as Bayesian updating, convex cost functions, and reinforcement learning to optimize quoting and manage multi-asset risks.
Applications span prediction markets, OTC, energy, and decentralized finance, where models adapt to non-Markovian microstructures and regime switches.

Cross-market making is the practice of providing liquidity, simultaneously and interactively, across multiple assets, venues, or interrelated markets. Unlike single-market making—where the market maker’s risk management and quoting are confined to a single asset or platform—cross-market making incorporates the effects of price correlations, liquidity transmission, co-integration, and dynamic risk balancing among multiple instruments or markets. This leads to increased complexity in market design, optimal quoting, inventory management, and strategic adaptation. Recent research addresses both the design of mechanisms (e.g., automated market makers, Bayesian or optimization-based algorithms) and concrete applications in prediction markets, OTC markets, energy/commodity markets, decentralized finance, and high-frequency multi-asset environments.

1. Foundational Mechanisms and Algorithmic Approaches

Early models of cross-market making build on inventory-based and information-based algorithms, notably the Logarithmic Market Scoring Rule (LMSR) and the Bayesian Market Maker (BMM) (Brahma et al., 2010). The LMSR sets prices based on the cumulative inventory, with the liquidity/adaptivity tradeoff governed by a single parameter $b$ . Its key feature is bounded loss:

LMSR Instantaneous spot price: $\rho(q_t) = \frac{e^{q_t/b}}{1 + e^{q_t/b}}$
Cost of trade: $C(Q; q_t) = b \ln[1 + e^{(q_t + Q)/b}] - b \ln[1 + e^{q_t/b}]$

By contrast, BMM explicitly tracks a belief distribution over the fundamental value, adapts its uncertainty through Bayesian updating, and performs belief resets via a sliding window "consistency index." These mechanisms allow BMM to dynamically increase risk appetite or adjust spreads after detecting market shocks but forfeit hard loss bounds enjoyed by LMSR.

Systematic theoretical frameworks generalize these approaches. Convex potential (cost function) based algorithms (Abernethy et al., 2010, Penna et al., 2011) show that any market mechanism obeying natural axioms—path independence, information incorporation, no arbitrage, expressiveness—must price securities via a convex, differentiable potential function $C(q)$ , with gradient mapping to the convex hull of payoffs. This yields tractable, modular, and bounded-loss market making even in combinatorially large or continuous outcome spaces.

In multi-market settings, modularity is achieved by mapping each market’s pricing to a separate (or joint) cost-function and using bandit learning (EXP3, CAB) to dynamically select profitable parameters (overround/incentive levels), with distribution-free regret bounds (Penna et al., 2011). These frameworks allow efficient, adaptable cross-market pricing, even when faced with nonstationarity or abrupt changes in trading regimes.

2. Structural Models for Multi-Asset and Factor-Reduced Quoting

Classical single-asset models (e.g., Avellaneda-Stoikov) are extended to multi-asset cross-market making by modeling asset prices as correlated stochastic processes (e.g., multi-dimensional Brownian motions) and expressing the market maker’s inventory dynamics and risk as a function of the full vector $q\in\mathbb{R}^d$ (Guéant, 2016). The value function reduction—e.g., writing $u(t, x, q, S) = -\exp(-\gamma(x + q\cdot S + \theta(t,q)))$ —enables the high-dimensional Hamilton–Jacobi–Bellman (HJB) equations governing the optimal controls to be reduced to manageable systems of ODEs or low-dimensional PDEs.

Closed-form approximations (the Guéant–Lehalle–Fernandez-Tapia formulas) allow practitioners to compute skewed, inventory-sensitive optimal quotes for each asset $i$ :

$\delta^{i,b}_{\mathrm{approx}}(q) \approx \tilde{\delta}_{i,\xi}\left( \sqrt{\frac{\gamma}{2}} \left( \Gamma^{ii} \frac{2q_i+\Delta_i}{2} + \sum_{j\neq i} \Gamma^{ij}q_j \right) \right)$

This captures both self-inventory and cross-asset impacts: quotes for asset $i$ adjust based on the inventory vector $q$ , incorporating risk offsets from correlated assets.

The curse of dimensionality in $d$ -asset settings is addressed via factor models (Bergault et al., 2019). By projecting the risk matrix as $\Sigma = \beta V \beta' + R$ , the HJB control problem is reduced to low-dimensional factor space. The value function and optimal quotes then depend only on the projection $f = \beta' q$ , enabling practical computation for $d \gg 1$ and efficient cross-market calibration even with diverse asset classes or illiquid OTC baskets.

3. Mechanism Design, Optimal Transport, and Bundling

Recent advances interpret optimal multi-market (multi-good) market making as a dual to an optimal transport problem (Curry et al., 14 Feb 2024). The mechanism’s predetermined trading schedule—a convex trader utility function $u(x)$ —specifies all possible allocations and prices in advance, ensuring strategyproofness and robustness to adverse selection.

For $n$ goods, the menu includes bundles and acceptance of payment "in kind"—offering bundles at prices more attractive than the sum of individually quoted items. Under adverse selection (modeled by belief updates such as $\pi(c, x) = \lambda c + (1-\lambda)x$ ), profit optimization requires expanding the no-trade region and explicit bundling to mitigate information rents. The dual transport formulation characterizes the problem as minimizing the cost to move signed measure mass between "profit" and "loss" regions, constrained by geometric properties (integral stochastic orders). Solutions are constructed via differentiable economics: neural architectures (e.g., RochetNet) parametrize candidate utility functions, with dual certificates confirming optimality and revealing rich menu structures unseen in one-dimensional settings.

Empirically, mixed bundling yields profit uplifts (10%–11%) relative to independent pricing, and cross-market AMMs display intricate geometric boundaries—e.g., octagonal no-trade regions, intricate partitioning for beta-distributed types—demonstrating the complexity of optimal cross-market design.

4. Non-Markovian Microstructure and Cross-Venue Execution

Traditional limit order book (LOB) models (Markovian queue-reactive) do not accurately reproduce real-world liquidity dynamics, limiting their effectiveness in cross-market making (Lu et al., 2018). Empirical evidence shows order size heterogeneity (geometric, mixture, and "Dirac" distributions) and strong path dependence due to limit-removal events, which influence subsequent order flow and queue rebuilds.

Incorporating these non-Markovian features into Markov decision process (MDP) frameworks allows optimal strategies to better balance risk and avoid adverse selection, especially when cross-venue order flows are correlated and venues differ in execution characteristics. Simulations and backtests demonstrate that empirical, event-driven (non-Markovian) models yield profits with lower turnover and better inventory control than naive or Markovian models. Importantly, cancellation rules and queue-priority effects become critical in multi-venue arbitrage and liquidity provision.

These results indicate that cross-market makers should model and dynamically respond to the coupled microstructure state—across markets and venues—rather than naively mirroring orders or relying solely on instantaneous LOB metrics.

5. Cross-Market Dynamics: Capacity Constraints and Regime Switching

Cross-border and cross-market trading is fundamentally shaped by physical or regulatory capacity limits. In integrated models for coupled markets (e.g., electricity trading between countries), cross-market matching is regulated by transmission capacity bounds (Milbradt et al., 2022). Under heavy-traffic scaling, the order flow converges to a reflected multidimensional Brownian motion, with regime switching between coupled (active) and decoupled (inactive) states.

Key properties:

Active Regime: Coupled order books, shared bid prices, four-dimensional SRBM with oblique reflection.
Inactive Regime: Markets operate independently, reconnecting only when price processes converge or capacity is restored.
Capacity Process: Given by linear combinations of local times at the boundaries, representing accumulated cross-border volume.

Analytical tractability is preserved, allowing computation of quantities such as survival probabilities, price change likelihoods, and joint queue exit distributions. Optimal cross-market strategies must account for the risk of regime switches, liquidity fragmentation, and the dynamic allocation of liquidity between (and within) venues.

6. Control, Adaptivity, and Learning in Cross-Market Making

Modern advances incorporate machine learning and adaptive control:

Bayesian and consistency-based adaptivity: Cross-market LMSR–BMM hybrids, where variance adjustment (consistency indices) is driven by the joint distribution of order flows across markets, facilitate rapid adaptation to shocks (Brahma et al., 2010).
Multi-objective and bandit learning: Cost-function AMMs combined with bandit algorithms enable profit optimization across markets, distribution-free regret, and dynamic overround adjustment for regime-specific conditions (Penna et al., 2011).
Reinforcement learning (RL) and policy weighting: RL agents designed for ultra-high-frequency market making (deep Q-learning, multi-agent) optimize discrete action spaces representing spread adjustments and hedging (Vicente, 24 Jul 2025). Non-stationarity is managed by POW-dTS (policy weighting via discounted Thompson sampling), which dynamically selects from a library of pre-trained policies according to market conditions. Multi-objective RL further enables explicit trade-off optimization between profitability and inventory/risk control via Pareto front analysis, facilitating robust transfer of strategies in cross-market environments.

Performance metrics in cross-market RL settings encompass hypervolume, sparsity, and undominated solution counts in objective space, evaluating the capacity of strategies to efficiently and safely supply liquidity under diverse, rapidly changing conditions.

7. Applications: Decentralized, Energy, and Exotic Markets

Automated market makers (AMMs) for fiat currencies and digital assets extend cross-market concepts into decentralized settings (Lipton et al., 2021, Port et al., 2022, Tolstikov, 17 Oct 2024). Cross-settlement mechanisms (CFMMs) use invariant functions (constant sum, constant product, or hybrids) to enable stable exchange rates and liquidity between CBDCs, stablecoins, or assets with negative prices (e.g., in energy or derivatives markets). Design features include:

Parametric invariants: Homotopy and geometric mixing between CSMM and CPMM, dynamically adjustable by amplification coefficients or parameterizations, to control slippage and liquidity concentration (Port et al., 2022).
Negative prices: The concentrated superelliptical market maker (CSEMM) facilitates swaps between assets of either sign, essential in electricity markets where negative prices are observed (Tolstikov, 17 Oct 2024).
Liquidity fingerprint: The CSEMM offers adjustable tail characteristics (interpolating between covered-call and LMSR profiles), providing practitioners with flexible liquidity allocation matching empirical patterns.

Energy and spot-futures markets employ nested mean-reverting models (e.g., multi-scale Ornstein-Uhlenbeck processes for EFP spreads in gold) combined with HJB-based strategy optimization (Barzykin et al., 23 Apr 2024). These frameworks enable nearly real-time updates of quoting and hedging policies, fully exploiting cross-market co-integration and spot-futures liquidity relationships.

In summary, cross-market making is characterized by the interaction of inventory dynamics, optimal design (cost-function or Bayesian), learning and adaptivity (bandit, RL, policy weighting), and detailed microstructural modeling (non-Markovian, limit order book, regime switching). State-of-the-art research produces unified models and practical algorithms supporting efficient and robust liquidity provision across multiple correlated markets, asset classes, and venues, with extensive applications in prediction, OTC, energy, and decentralized finance contexts.