arXiv subjects

Fabrizio Lillo

Publications and source records attributed to Fabrizio Lillo.

At least 19 recordsLinked to original sources

arXiv2026

Deep Learning of Robust Market Making under Regime-Switching Order Flow

Classical market-making strategies based on stochastic control, such as the Avellaneda-Stoikov and the Gu\'{e}ant-Lehalle-Fernandez-Tapia (GLFT) extension, provide closed-form quoting rules, but rest on assumptions that break down at realistic microstructure timescales. One of them is that order flow is stationary, while empirical evidence points to the existence of regimes, possibly associated with algorithmic execution of metaorders. In this case, existing methods provide negative PnL. In this paper, we develop a deep reinforcement-learning market maker (RLMM) - a Rainbow-style distributional DQN (C51) which is calibrated and tested in a zero-intelligence limit order book. We find that, in the stationary setting, RLMM outperforms GLFT across the entire observed risk-return frontier. The RLMM is more robust to flow asymmetry than GLFT, but, like any stationarily trained strategy, it still suffers large drawdowns from inventory saturation under persistent directional imbalance. Augmenting the state of RLMM with two auxiliary signals - a Bayesian online change-point filter over the directional flow bias and a queue-adjusted quote-exposure imbalance -restores profitability. A final scenario-bandit step that reweights low-return regime scenarios further improves performance under random-persistence and correlated-direction stress.

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arXiv2026

Explainable Deep Learning for Price-Trade Dynamics: From Black-Box Forecasts to Effective Parametric Models

Understanding the joint dynamics of prices and trades is central to market microstructure, where returns and order flow interact through nonlinear and state-dependent mechanisms. Linear models are interpretable but may miss these effects, while deep neural networks improve forecasting at the cost of transparency. We use neural networks as tools for structural discovery rather than only for prediction. A deep feed-forward network is trained on high-frequency returns and signed volumes for large- and small-tick stocks and compared with a linear VAR benchmark. The neural network improves predictive performance, especially for returns, revealing nonlinear dependencies beyond the linear specification. Using Shapley-based explainability, we show that the dominant contributions are concentrated at the most recent lags. Model-implied responses are consistent with conditional averages reconstructed from the data. Unlike empirical averages, however, the neural-network decomposition isolates individual regressor contributions to the aggregate dependence. Lagged signed volume generates sign-preserving and saturating effects, consistent with nonlinear price impact and order-flow persistence. Lagged returns act as state variables: when the previous trade does not move the price, the model predicts continuation in the direction of past order flow, whereas non-zero returns generate attenuation or reversal. Building on these findings, we introduce a parsimonious SHAP-inspired nonlinear parametric model. It reproduces the main return-volume dependencies, outperforms the linear VAR benchmark, and achieves performance comparable to the neural network. A multi-lag extension captures residual longer-memory effects while preserving interpretability. Overall, explainability offers a route from black-box prediction to economically meaningful parametric models of price and trade dynamics.

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arXiv2026

Online Learning of Scale Parameters in Score-Driven Filters

A score-driven filter multiplies its scaled log-likelihood score by a scale parameter. We call this coefficient the gain and learn it online. Given the current state and realised scaled score, each admissible gain selects a reachable next state and predictive density. A scalar gain moves along a line; diagonal gains control coordinatewise transmission and may change direction. We evaluate gain selection using a one-step predictive Kullback--Leibler objective. In the scalar unscaled case, the negative consecutive-score product is a stochastic gradient; the positive product used in accelerated recursions is a descent direction. Positive scalar score scaling changes only the effective learning rate. Monotone differentiable gain links induce mirror-descent geometry, while persistence adds a Bregman pull towards a reference gain. Under convexity, compactness, integrability, and schedule conditions, projected and discounted mirror updates satisfy dynamic-regret bounds relative to time-varying, current-information comparators. Simulations isolate score scaling, link geometry, persistence, and coordinatewise gains. Across twelve equity indices, the bounded discounted-logistic gain records a lower out-of-sample mean negative log score than the constant gain in eleven markets, although market-level evidence is mixed. It also avoids the extreme transients of the numerically capped exponential-link benchmark. Improvements are largest in markets spanning multiple crises.

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arXiv2026

Can Reinforcement Learning Efficiently Discover Price Manipulation?

In this paper, we investigate whether a model-free RL agent can identify and exploit price manipulation opportunities more effectively than a traditional model-based approach that assumes correct specification of the data-generating process but relies on noisy parameter estimates. We consider a single-asset market in which prices evolve according to an Almgren-Chriss framework with non-linear permanent impact and linear temporary impact. We first establish the existence of price-manipulative strategies in discrete time and compute the optimal benchmark strategy using Sequential Least Squares Quadratic Programming under full information. We then compare two finite-sample learning approaches: a model-based procedure that estimates impact parameters from simulated execution data and an agnostic RL approach based on Deep Deterministic Policy Gradient, trained directly on the same amount of data. For intermediate volatility, the RL agent successfully discovers profitable manipulative strategies without explicit knowledge of the underlying model, even when training data are quite limited. More importantly, RL consistently outperforms the model-based approach when parameter estimates are affected by sampling error, despite the latter benefiting from the correct model specification. For large volatility, all methods are unable to identify manipulation opportunities, while for small volatility, the model based approach outperforms RL. These findings highlight both the effectiveness of RL in complex control problems and the risks associated with deploying learning algorithms in financial markets without appropriate safeguards.

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arXiv2026

Mitigating Adverse Selection in Concentrated Liquidity AMMs with Dynamic Fees: An Agent-Based Model Approach

Automated Market Makers based on concentrated liquidity, such as Uniswap v3, significantly improve capital efficiency but expose Liquidity Providers (LPs) to adverse selection costs, formalized as Loss-Versus-Rebalancing (LVR). While theoretical literature quantifies these costs, the interplay between realistic blockchain microstructure and endogenous pricing mechanisms remains under-explored. This paper develops a granular Agent-Based Model of a Uniswap v3 pool interacting with a stochastic reference market governed by Heston volatility dynamics. The framework incorporates discrete block propagation, mempool latency, and a heterogeneous population of agents, including latency-sensitive arbitrageurs, smart routers, Maximal Extractable Value searchers, and active LPs benchmarked against a frictionless rebalancing strategy. We propose and evaluate dynamic fee schedules driven by volatility and order-flow toxicity proxies intended to compensate LPs for adverse-selection losses. Our simulations investigate the conditions under which LPs can achieve positive hedged Profit and Loss (fees minus LVR). The analysis suggests that dynamic fee adjustments can improve hedged LP profitability mainly by increasing fee income in states associated with stale-price risk. Depending on the configuration, these rules may also affect realized LVR, but the current aggregate results support compensation for LVR more directly than a reduction of LVR itself.

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arXiv2026

Trading in the Sunshine or in the Shade: Market Impact and Adverse Selection on Hyperliquid

Sunshine trading theory predicts that publicly disclosing trading intentions can reduce adverse selection and attract liquidity provision, lowering execution costs. Evidence is scarce, because explicit preannouncement of large orders is rare in traditional markets. We study Hyperliquid, a fully on-chain limit order book for cryptocurrency perpetual futures, where protocol-native TWAP orders disclose their terms from inception and remain visible while active, a natural form of sunshine trading. Using address-level data, we reconstruct 4.3 million hidden metaorders and compare them with 465,000 visible TWAP executions. The two execution styles differ sharply: hidden metaorders follow front-loaded, U-shaped schedules consistent with transient-impact optimal execution, whereas TWAPs trade nearly uniformly. We test the preannouncement predictions of Admati and Pfleiderer (1991). Visible TWAPs face lower execution costs than comparable hidden metaorders and leave a smaller permanent price impact. Hidden metaorders executed alongside already-visible same-direction TWAP flow incur higher permanent costs: adverse-selection costs shift toward non-announcers. Finally, visible TWAP programs elicit liquidity provision: while active, displayed depth rises and the book tilts toward the absorbing side, the more so the larger the announced order.

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arXiv2026

GravityGraphSAGE: Link Prediction in Directed Attributed Graphs

Link prediction (inferring missing or future connections between nodes in a graph) is a fundamental problem in network science with widespread applications in, e.g., biological systems, recommender systems, finance and cybersecurity. The ability to accurately predict links has significant real-world applications, such as detecting fraudulent financial transactions or identifying drug-target interactions in biomedicine. Despite a rich literature, link prediction is still challenging, especially for graphs enriched with information on edges (direction) and nodes (attributes). In fact, research on link prediction, especially the one based on Graph Deep Learning (GDL), has mostly focused on undirected graphs, without fully leveraging node attributes. Here, we fill this gap by proposing Gravity-GraphSAGE (GG-SAGE), a modified version of GraphSAGE, a GDL model for node embeddings, composed of a gravity-inspired decoder. This implementation is the first example in the literature of a GraphSAGE backbone adopted for directed link prediction. Using the benchmark datasets Cora, Citeseer, PubMed and 16 real-world graphs from the online Netzschleuder repository, we show that our proposed model outperforms state-of-the-art GDL link prediction techniques. Using further experimental evidence, we relate the quality of the output of our model with various characteristics of the graph, suggesting that our framework scales well when applied to data of increasing complexity.

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arXiv2026

Predicting the success of new crypto-tokens: the Pump.fun case

We study the dynamics of token launched on Pump.fun, a Solana-based launchpad platform, to identify the determinants of the token success. Pump.fun employs a bonding curve mechanism to bootstrap initial liquidity possibly leading to graduation to the on-chain market, which can be seen as a token success. We build predictive models of the probability of graduation conditional on the current amount of Solana locked in the bonding curve and a set of explanatory variables that capture structural and behavioral aspects of the launch process. Conditioning the graduation probability on these variables significantly improves its predictive power, providing insights into early-stage market behavior, speculative and manipulative dynamics, and the informational efficiency of bonding-curve-based token launches.

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arXiv2025

Reinforcement Learning in Queue-Reactive Models: Application to Optimal Execution

We investigate the use of Reinforcement Learning for the optimal execution of meta-orders, where the objective is to execute incrementally large orders while minimizing implementation shortfall and market impact over an extended period of time. Departing from traditional parametric approaches to price dynamics and impact modeling, we adopt a model-free, data-driven framework. Since policy optimization requires counterfactual feedback that historical data cannot provide, we employ the Queue-Reactive Model to generate realistic and tractable limit order book simulations that encompass transient price impact, and nonlinear and dynamic order flow responses. Methodologically, we train a Double Deep Q-Network agent on a state space comprising time, inventory, price, and depth variables, and evaluate its performance against established benchmarks. Numerical simulation results show that the agent learns a policy that is both strategic and tactical, adapting effectively to order book conditions and outperforming standard approaches across multiple training configurations. These findings provide strong evidence that model-free Reinforcement Learning can yield adaptive and robust solutions to the optimal execution problem.

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arXiv2025

Deep reinforcement learning for optimal trading with partial information

Reinforcement Learning (RL) applied to financial problems has been the subject of a lively area of research. The use of RL for optimal trading strategies that exploit latent information in the market is, to the best of our knowledge, not widely tackled. In this paper we study an optimal trading problem, where a trading signal follows an Ornstein-Uhlenbeck process with regime-switching dynamics. We employ a blend of RL and Recurrent Neural Networks (RNN) in order to make the most at extracting underlying information from the trading signal with latent parameters. The latent parameters driving mean reversion, speed, and volatility are filtered from observations of the signal, and trading strategies are derived via RL. To address this problem, we propose three Deep Deterministic Policy Gradient (DDPG)-based algorithms that integrate Gated Recurrent Unit (GRU) networks to capture temporal dependencies in the signal. The first, a one -step approach (hid-DDPG), directly encodes hidden states from the GRU into the RL trader. The second and third are two-step methods: one (prob-DDPG) makes use of posterior regime probability estimates, while the other (reg-DDPG) relies on forecasts of the next signal value. Through extensive simulations with increasingly complex Markovian regime dynamics for the trading signal's parameters, as well as an empirical application to equity pair trading, we find that prob-DDPG achieves superior cumulative rewards and exhibits more interpretable strategies. By contrast, reg-DDPG provides limited benefits, while hid-DDPG offers intermediate performance with less interpretable strategies. Our results show that the quality and structure of the information supplied to the agent are crucial: embedding probabilistic insights into latent regimes substantially improves both profitability and robustness of reinforcement learning-based trading strategies.

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arXiv2025

Deviations from Tradition: Stylized Facts in the Era of DeFi

Decentralized Exchanges (DEXs) are now a significant component of the financial world where billions of dollars are traded daily. Differently from traditional markets, which are typically based on Limit Order Books, DEXs typically work as Automated Market Makers, and, since the implementation of Uniswap v3, feature concentrated liquidity. By investigating the twenty-four most active pools in Uniswap v3 during 2023 and 2024, we empirically study how this structural change in the organization of the markets modifies the well-studied stylized facts of prices, liquidity, and order flow observed in traditional markets. We find a series of new statistical regularities in the distributions and cross-autocorrelation functions of these variables that we are able to associate either with the market structure (e.g., the execution of orders in blocks) or with the intense activity of Maximal Extractable Value searchers, such as Just-in-Time liquidity providers and sandwich attackers.

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arXiv2025

A high-frequency approach to Realized Risk Measures

We propose a new approach, termed Realized Risk Measures (RRM), to estimate Value-at-Risk (VaR) and Expected Shortfall (ES) using high-frequency financial data. It extends the Realized Quantile (RQ) approach proposed by Dimitriadis and Halbleib by lifting the assumption of return self-similarity, which displays some limitations in describing empirical data. More specifically, as the RQ, the RRM method transforms intra-day returns in intrinsic time using a subordinator process, in order to capture the inhomogeneity of trading activity and/or volatility clustering. Then, microstructural effects resulting in non-zero autocorrelation are filtered out using a suitable moving average process. Finally, a fat-tailed distribution is fitted on the cleaned intra-day returns. The return distribution at low frequency (daily) is then extrapolated via either a characteristic function approach or Monte Carlo simulations. VaR and ES are estimated as the quantile and the tail mean of the distribution, respectively. The proposed approach is benchmarked against the RQ through several experiments. Extensive numerical simulations and an empirical study on 18 US stocks show the outperformance of our method, both in terms of the in-sample estimated risk measures and in the out-of-sample risk forecasting

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arXiv2025

Tackling estimation risk in Kelly investing using options

The Kelly criterion provides a general framework for optimizing the growth rate of an investment portfolio over time by maximizing the expected logarithmic utility of wealth. However, the optimality condition of the Kelly criterion is highly sensitive to accurate estimates of the probabilities and investment payoffs. Estimation risk can lead to greatly suboptimal portfolios. In a simple binomial model, we show that the introduction of a European option in the Kelly framework can be used to construct a class of growth optimal portfolios that are robust to estimation risk.

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arXiv2025

Modeling metaorder impact with a Non-Markovian Zero Intelligence model

Devising models of the limit order book that realistically reproduce the market response to exogenous trades is extremely challenging and fundamental in order to test trading strategies. We propose a novel explainable model for small tick assets, the Non-Markovian Zero Intelligence, which is a variant of the well-known Zero Intelligence model. The main modification is that the probability of limit orders' signs (buy/sell) is not constant but is a function of the exponentially weighted mid-price return, representing the past price dynamics, and can be interpreted as the reaction of traders with reservation prices to the price trend. With numerical simulations and analytical arguments, we show that the model predicts a concave price path during a metaorder execution and to a price reversion after the execution ends, as empirically observed. We analyze in-depth the mechanism at the root of the arising concavity, the components which constitute the price impact in our model, and the dependence of the results on the two main parameters, namely the time scale and the strength of the reaction of traders to the price trend.

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arXiv2025

Price manipulation schemes of new crypto-tokens in decentralized exchanges

Blockchain technology has revolutionized financial markets by enabling decentralized exchanges (DEXs) that operate without intermediaries. Uniswap V2, a leading DEX, facilitates the rapid creation and trading of new tokens, which offer high return potential but exposing investors to significant risks. In this work, we analyze the financial impact of newly created tokens, assessing their market dynamics, profitability and liquidity manipulations. Our findings reveal that a significant portion of market liquidity is trapped in honeypots, reducing market efficiency and misleading investors. Applying a simple buy-and-hold strategy, we are able to uncover some major risks associated with investing in newly created tokens, including the widespread presence of rug pulls and sandwich attacks. We extract the optimal sandwich amount, revealing that their proliferation in new tokens stems from higher profitability in low-liquidity pools. Furthermore, we analyze the fundamental differences between token price evolution in swap time and physical time. Using clustering techniques, we highlight these differences and identify typical patterns of honeypot and sellable tokens. Our study provides insights into the risks and financial dynamics of decentralized markets and their challenges for investors.

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arXiv2025

Why is the estimation of metaorder impact with public market data so challenging?

Estimating market impact and transaction costs of large trades (metaorders) is a very important topic in finance. However, using models of price and trade based on public market data provide average price trajectories which are qualitatively different from what is observed during real metaorder executions: the price increases linearly, rather than in a concave way, during the execution and the amount of reversion after its end is very limited. We claim that this is a generic phenomenon due to the fact that even sophisticated statistical models are unable to correctly describe the origin of the autocorrelation of the order flow. We propose a modified Transient Impact Model which provides more realistic trajectories by assuming that only a fraction of the metaorder trading triggers market order flow. Interestingly, in our model there is a critical condition on the kernels of the price and order flow equations in which market impact becomes permanent.

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arXiv2024

Filtering and Statistical Properties of Unimodal Maps Perturbed by Heteroscedastic Noises

We propose a theory of unimodal maps perturbed by an heteroscedastic Markov chain noise and experiencing another heteroscedastic noise due to uncertain observation. We address and treat the filtering problem showing that by collecting more and more observations, one would predict the same distribution for the state of the underlying Markov chain no matter one's initial guess. Moreover we give other limit theorems, emphasizing in particular concentration inequalities and extreme value and Poisson distributions. Our results apply to a family of maps arising from a model of systemic risk in finance.

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arXiv2024

Optimal execution with deterministically time varying liquidity: well posedness and price manipulation

We investigate the well-posedness in the Hadamard sense and the absence of price manipulation in the optimal execution problem within the Almgren-Chriss framework, where the temporary and permanent impact parameters vary deterministically over time. We present sufficient conditions for the existence of a unique solution and provide second-order conditions for the problem, with a particular focus on scenarios where impact parameters change monotonically over time. Additionally, we establish conditions to prevent transaction-triggered price manipulation in the optimal solution, i.e. the occurence of buying and selling in the same trading program. Our findings are supported by numerical analyses that explore various regimes in simple parametric settings for the dynamics of impact parameters.

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