arXiv subjects

Fabrizio Lillo

Publications and source records attributed to Fabrizio Lillo.

At least 19 recordsLinked to original sources

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. Strictly increasing, continuously differentiable gain links with positive derivative 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.

cs.LG↗

arXiv2026

Filtering and Smoothing with Score-Driven Models

Score-driven models are, by construction, purely predictive filters. When such models are read as filters for an underlying latent process, rather than as data generating processes, this is a restriction rather than a feature, because the contemporaneous and the future observations also carry information on the current state. Starting from the observation that the Kalman filter and smoother recursions for linear Gaussian models can be written in terms of the score of the conditional log-likelihood, and that the predictive step then takes the form of a score-driven recursion, we generalize score-driven models along this direction by deriving an update filter and a smoother, which exploit the contemporaneous observation and the whole sample respectively, together with the corresponding conditional variances. In extensive Monte Carlo analyses the update filter lowers the mean square error of the predictive filter by 3.5% to 8.8%, and the smoother by 32% to 44%, with the ordering holding in every replication; confidence bands built on the associated conditional variances attain their nominal coverage, whereas bands that ignore filtering uncertainty capture less than a third of it. Empirically, we demonstrate the benefits of employing score-driven models as filters rather than as purely predictive processes, showing that the resulting smoothed estimates align more closely with realized quantities than their predictive counterparts.

stat.ME↗

arXiv2026

Robust filtering and smoothing via perturbation methods

Using a perturbation technique, we derive a new approximate filtering and smoothing methodology generalizing along different directions several existing approaches to robust filtering based on the score and the Hessian matrix of the observation density. The main advantages of the methodology can be summarized as follows: (i) it relaxes the critical assumption of a Gaussian conditional distribution for the latent states underlying such approaches; (ii) can be applied to a general class of state-space models including location, scale and count data models; (iii) rationalizes the approximation to the likelihood function within the same perturbation approach, thus allowing for straightforward inference of the model parameters; (iv) enables the computation of confidence bands around the state estimates reflecting the combination of parameter and filtering uncertainty. We show through an extensive Monte Carlo study that the mean square loss with respect to exact simulation-based methods is small in a wide range of scenarios. We finally illustrate empirically the application of the methodology to the estimation of stochastic volatility and correlations in financial time-series.

stat.ME↗

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 Guéant-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.

q-fin.TR↗

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.

q-fin.TR↗

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.

q-fin.TR↗

arXiv2026

CAESar: Conditional Autoregressive Expected Shortfall

In financial risk management, Value at Risk (VaR) estimates potential portfolio losses but fails to account for losses beyond a certain threshold. Expected Shortfall (ES) addresses this limitation by providing the conditional expectation of such exceedances, providing a better measure of tail risk. However, ES is not elicitable on its own, meaning that it cannot be estimated by minimizing some scoring function, although its joint elicitability with VaR allows for combined estimation. Building on this property, we propose the Conditional Autoregressive Expected Shortfall (CAESar) model, which flexibly handles dynamic patterns and heteroskedasticity, without making distributional assumptions on price returns. The optimization of CAESar coefficients involves three steps: fitting the VaR component via CAViaR regression, formulating ES as an autoregressive process, and jointly estimating VaR and ES coefficients while ensuring a monotonicity constraint to avoid crossing quantiles. Through extensive backtesting, CAESar outperforms existing methods, proving highly effective for risk forecasting.

q-fin.RM↗

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.

q-fin.TR↗

arXiv2026

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.

math.ST↗

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.

cs.LG↗

arXiv2026

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.

math.OC↗

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.

q-fin.ST↗

arXiv2026

Deviations from the Nash equilibrium in a two-player optimal execution game with reinforcement learning

The use of reinforcement learning algorithms in financial trading is becoming increasingly prevalent. However, the autonomous nature of these algorithms can lead to unexpected outcomes that deviate from traditional game-theoretical predictions and may even destabilize markets. In this study, we examine a scenario in which two autonomous agents, modelled with Double Deep Q-Learning, learn to liquidate the same asset optimally in the presence of market impact, under the Almgren-Chriss (2000) framework. We show that the strategies learned by the agents deviate significantly from the Nash equilibrium of the corresponding market impact game. Notably, the learned strategies exhibit supra-competitive solution, {which might be compatible with a tacit collusive behaviour}, closely aligning with the Pareto-optimal solution. We further explore how different levels of market volatility influence the agents' performance and the equilibria they discover, including scenarios where volatility differs between the training and testing phases.

q-fin.TR↗

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.

q-fin.TR↗

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.

q-fin.CP↗

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.

q-fin.TR↗

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.

q-fin.MF↗