Performative Market Maker

• Performative Market Maker is a system where agents’ quotes actively influence market outcomes through intrinsic feedback loops.
• It integrates closed-loop stochastic models and algorithmic learning to adjust quotes based on endogenous market dynamics.
• Research shows feedback-aware strategies can enhance profit stability and reduce adverse selection in high-frequency trading.

A performative market maker is a market participant, algorithm, or institutional agent whose quoting and trading actions not only respond to but actively shape market outcomes through intrinsic feedback mechanisms, leading to a coupling between the market’s evolution and the strategies used within it. This explicit performativity breaks with purely descriptive models that assume market prices evolve independently of agent actions, instead positing that financial models and the strategies they inform become entangled with the realized market processes. The literature formalizes this through closed-loop stochastic systems, equilibrium game-theoretic structures, and algorithmic learning models that internalize the influence of strategic action on realized prices, liquidity, and welfare.

1. Mathematical Formulation of Performative Feedback

A central mathematical framework for performative market making is based on embedding the dominant quoting model used by market participants back into the market’s price formation mechanism. In classical settings, mid-price $s(t)$ evolution is modeled by exogenous diffusions,

$ds(t) = \mu\,dt + \sigma\,dW(t)$

with strategy-independent drift and volatility.

The performative formulation (Kleitsikas et al., 6 Aug 2025) introduces a feedback process:

$$ds(t) = [r(t) - \varepsilon s(t)]\,dt + \sigma\,dW(t)$$

where $r(t)$ is a prevailing reference or reservation price (often itself a function of $s(t)$ and an endogenous model adjustment $g(t)$), and $\varepsilon > 0$ is the performative sensitivity parameter. Setting $r(t) = s(t) + g(t)$ yields

$$ds(t) = [g(t) - \xi s(t)]\,dt + \sigma\,dW(t) \qquad \text{with}\ \xi := \varepsilon - 1 > 0,$$

which is a Hull–White-type Ornstein–Uhlenbeck process. Here, $g(t)$ typically captures the model-driven adjustment such as an inventory risk correction (e.g., $g(t) = -\gamma \sigma^2 q (T-t)$ in Avellaneda–Stoikov). The parameter $\xi$ governs the rate at which price dynamics are "forced" to conform to the dominant financial model, creating a closed feedback loop between agents’ strategies and the market trajectory.

A closed-form solution for the terminal price is

$$s(T) = e^{-\xi (T-t)} s(t) + \int_t^T e^{-\xi (T-u)} g(u)\,du + \sigma \int_t^T e^{-\xi (T-u)} dW(u)$$

demonstrating that as $\xi$ increases, prices converge more tightly to the strategy-induced path encoded by $g(t)$. The mean and variance explicitly show the degree of performative conformity.

2. Closed-Form Solutions and Analytical Properties

Within the performative feedback model, explicit characterizations of optimal quoting policies can be derived. For a market maker operating under risk aversion $\gamma$ and with inventory $q$, the solution for the performative optimal reservation price and spread (Kleitsikas et al., 6 Aug 2025) is

$$r^*_\text{perf} = e^{-\xi (T-t)} s - \gamma \sigma^2 \left[q \cdot \Delta_\xi + q_\text{perf}\cdot \frac{1 - e^{-2\xi (T-t)}}{2\xi}\right]$$

$$\text{spread}^*_\text{perf} = \frac{2}{\gamma} \ln(1+\gamma/k) + \frac{\gamma \sigma^2}{2\xi} [1 - e^{-2\xi (T-t)}]$$

where $k$ is an order-flow sensitivity parameter, and $\Delta_\xi$ as well as $q_\text{perf}$ are analytical corrections arising from the feedback structure. These formulas reveal that both the reservation price and the spread incorporate not only classical inventory risk terms but also deterministic corrections that encode the endogenous feedback from aggregated quoting strategies.

As $\xi \to 0$, the model reduces to the classical Avellaneda–Stoikov solution with the standard mean-reverting price process; as $\xi \to \infty$, price dynamics are dominated by model feedback. This suggests that, depending on market structure and strategy prevalence, performative effects can induce systematic, predictable drifts breaking martingale conditions.

3. Machine Learning and Reverse Engineering Dominant Strategies

Analytical approximations may rely on linearizations or misspecification in arrival rate models, so the literature advances machine learning-enhanced performative strategies. The "theta-enhanced performative strategy" (Kleitsikas et al., 6 Aug 2025) introduces learnable multiplicative and additive parameters $\theta = (\theta_0, \theta_1, \theta_2)$: $$r^*_{\text{perf},\theta} = \theta_0 e^{-\xi (T-t)} s - \gamma\sigma^2 [\theta_1 q_{\text{market}} \Delta_\xi + \theta_2 q_{\text{perf}} E_\xi]$$ Here, $E_\xi$ is a feedback-dependent correction, and the $\theta$ parameters are tuned via meta-heuristic optimization to maximize expected utility or P&L. This approach allows the market maker to "reverse engineer" prevailing feedback structures and adaptively respond to the embedded systematic bias, outperforming agents deploying only classical models, particularly when $\xi$ is small and feedback signals are subtle.

A plausible implication is that in high-frequency or AI-dominated markets, incorporating adaptive model detection and calibration into market making can improve both stability and profitability under performativity.

4. Strategic Behavior, Arbitrage, and Profitability

The theoretical model leads to qualitative shifts in quoting and arbitrage opportunities for a performative market maker. Unlike classical inventory-driven strategies that passively balance risk, a performative agent recognizes and exploits deterministic drifts in price dynamics induced by the prevalence of specific strategies across the market (Kleitsikas et al., 6 Aug 2025). The resulting quotes are not only adjusted for risk, but also "tilted" to preemptively benefit from self-fulfilling feedback, anticipating the collective action of other market makers or liquidity providers.

Empirical evidence from simulation experiments (Kleitsikas et al., 6 Aug 2025) shows that performative market making with feedback-aware strategies achieves superior terminal P&L and Sharpe ratios compared to agents implementing naive or non-performative models. The observed P&L stability results from capturing predictable drifts, lowering exposure to adverse selection, and reducing variance by exploiting the endogenous structure imposed by the market’s own dominant strategies.

5. Broader Implications: Performativity in Quantitative Markets

The formalization of performativity connects financial market microstructure with sociological theories of self-fulfilling prophecy and reflexivity. The operational consequence is that trading algorithms—and, specifically, market-making models—cease to be exogenous inputs, instead becoming drivers of market evolution. As more agents adopt similar models and quoting rules, the endogenous feedback loops tighten, increasing the conformity of observable prices to the path predicted or prescribed by these models.

This observation has direct ramifications for market design and regulation. For instance, excessive conformity of prices to specific quoting strategies may indicate vulnerability to endogenous bubbles, lack of price discovery, or unintentional tacit collusion. Conversely, adaptive or adversarial agents that "reverse engineer" and exploit prevailing performative strategies may introduce stabilizing or destabilizing forces, depending on their prevalence and capital.

A plausible implication is that as reinforcement learning and agent-based strategies become widespread and increasingly performative, the collective feedback may magnify model risk and call for explicit performative risk management in market surveillance.

6. Connections with Game-Theoretic and Networked Settings

In broader settings, performative market making is intimately concerned with strategic feedback in game-theoretic or network-constrained markets. For networked Cournot competition (Como et al., 31 Dec 2024), a centralized market maker dispatches flows to maximize a welfare function subject to capacity constraints. The optimal action of the market maker is determined through Euler–Lagrange-type first-order conditions linking local welfare derivatives and network flows. When bottlenecks (saturated links) appear in the network, persistent price differences arise between markets, and the market maker’s actions segment the network into price groups. The Nash equilibrium is characterized via the maximization of a potential function $P(x, y)$, with unique equilibria under standard assumptions.

The performativity here is manifest: the market maker's allocations set local prices, affect producer best-responses, and shape market stratification. Real-world validation, such as in the Italian day-ahead electricity market (Como et al., 31 Dec 2024), confirms that capacity saturations and congestion pricing governed by performative market maker policies explain observed price divergences across zones.

In both the single-asset and networked contexts, performative market making underscores the causal influence of strategic design, equilibrium selection, and model feedback on market outcomes.

7. Future Research Directions and Open Problems

Several lines of inquiry are suggested by the mathematical and empirical results on performative market making:

• Quantitative characterization of phase transitions in equilibrium outcomes as a function of performative feedback parameters, prevalence of dominant models, or policy harmonization rates.
• Analysis of stability vs. fragility trade-offs: under what conditions does performativity induce instabilities, endogenous volatility, or excessive market consensus?
• Systematic calibration and detection of performative sensitivity ($\xi$) and strategic weights ($\theta$) in empirical data, especially in high-frequency and multi-agent markets (A plausible implication is that performance monitoring tools need to be upgraded to detect endogenous risk from feedback loops).
• Regulatory implications for transparency, disclosure, and control of algorithmic “herding” or model-induced price formation.

A plausible inference is that in the era of AI-driven trading, the awareness and explicit internalization of performative feedback in market making may be essential for both robustness and profitability.

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In summary, performative market making formalizes the feedback between agent strategies and market processes, encapsulates pricing and quoting models that recursively shape observed prices, and provides both novel analytic tools and practical design implications for adaptive, feedback-aware trading in modern financial markets (Kleitsikas et al., 6 Aug 2025, Como et al., 31 Dec 2024).