Check In & Breakfast
Opening Remarks
Mihai Cucuringu (UCLA)
Graph Clustering and Ranking for Multivariate Time Series with Applications to Statistical Arbitrage and Lead-Lag Detection in Equity Markets
We develop spectral methods for clustering heterogeneous networks, in the setting of signed and directed networks, and demonstrate their benefits on networks arising from stochastic block models and financial multivariate time series data, where one is often interested in clustering assets that exhibit similar contemporaneous behavior. We demonstrate the economic benefits of the proposed graph clustering algorithms in statistical arbitrage and portfolio construction applications. Both signed and directed graph clustering problems share an important common feature: they can be solved by exploiting the spectrum of certain graph Laplacian matrices or derivations thereof, allowing for performance guarantees under suitably defined stochastic block models.
A task of major interest in financial applications is that of uncovering lead-lag relationships in high-dimensional multivariate time series. In such settings, certain groups of variables partially lead the evolution of the system, while other variables follow this evolution with a time delay, resulting in a lead-lag structure, which, at the pairwise level, can be encoded as edges of a directed network. Detecting clusters which exhibit a certain notion of pairwise flow imbalance amounts to identifying baskets of assets which lead-lag each other. We leverage graph clustering and ranking algorithms for the task of lead-lag detection, and demonstrate that our proposed methodology is able to detect statistically significant lead-lag clusters in the US equity market. We study the composition of the uncovered lead-lag equity clusters, compare performance at different time frequencies and against established approaches from the lead-lag literature for portfolio construction.
Coffee Break
Junting Duan (Stanford University)
Learning the Values of Illiquid Bonds with Statistical Guarantees
Mortgage, corporate, municipal and other bonds are notoriously illiquid. We develop a method for learning the values of non-traded bonds that harnesses the information contained in contemporaneous trades of other bonds. We propose a conditional factor model for the cross-section of price quantiles, which is characterized by latent factors and factor exposures depending on observable bond and other characteristics through flexible, potentially nonlinear functions. Our training procedure leverages deep neural networks for factor exposures combined with an alternating optimization procedure. Simultaneous training for several quantile levels delivers a point estimate for the bond price in the form of the median along with a confidence interval. Conformal inference techniques yield valid finite-sample coverage guarantees. We test the performance of our estimators for pay-ups of Agency MBS pools using over 4.4 million TRACE trade records during 2011--22 and data on pool characteristics from eMBS/ICE. Our method significantly outperforms benchmark methods in terms of R-squared for point estimation while providing valid confidence intervals.
Haosheng Zhao (UCSB)
Stochastic Differential Games on Graphs
In this talk, we present a new model for stochastic differential games on graphs, aiming to bridge game theory with network structures to capture the influence of graph structures on strategic interactions. Our framework supports heterogeneous player interactions across general graph structures, extending current models to encompass more complex, network-driven dynamics.
We establish two main results: firstly, we demonstrate the convergence of fictitious play, along with numerical estimates of convergence rates that reflect key aspects of the graph structure. Secondly, we provide a semi-explicit construction of the Nash equilibrium, validated through numerical simulations and offering a reliable computational baseline for future applications in deep learning.
Building on our theoretical findings and leveraging recent developments in graph neural networks, we propose a graph-dependent, non-trainable modification of the neural network architecture. By integrating this interpretable novel architecture with state-of-the-art game-solving algorithms, we demonstrate through numerical experiments that this architecture achieves comparable performance to standard architectures while using fewer parameters. This is joint work with Ruimeng Hu and Jihao Long.
Zhaoyu Zhang (UCLA)
Mean Field Game on Graphs
In this work, we study the mean field game on a graph. Specifically, we derive the master equation on a graph by investigating the Hamilton-Jacobi equation on a Wasserstein space from a potential mean field game perspective. Additionally, we derive the master equation on the graph from a probabilistic viewpoint using the transition matrix of a Markov chain. Lastly, under appropriate convexity assumptions, we provide some regularity results.
Lunch Break
Bruno Dupire (Bloomberg)
Some Financial Applications of the Functional Itô Calculus
Path dependence is ubiquitous in finance, sometimes explicitly as the payoff of an exotic option may depend on the whole path of the asset price, not only at maturity, other times through the dynamics of the underlying (volatility, dividends…).
The framework to model path dependence is the Functional Itô Calculus and we review its basic concepts before offering a partial panorama of its applications: computation of the Greeks of path dependent options, perturbation analysis, volatility risk decomposition, Taylor expansion with signatures for fast computation of VaR and characterization of attainable claims, amongst other ones.
Coffee Break
Thiha Aung (UCSB)
Optimal Dispatch of hybrid wind-storage assets: From stochastic control to experiments on a synthetic grid
We develop a mathematical model for intraday dispatch of co-located wind-battery energy assets. Focusing on the primary objective of firming grid-side actual production vis-a-vis the preset day-ahead hourly generation targets, we conduct a comprehensive study of the resulting stochastic control problem across different firming formulations and wind generation dynamics. Among others, we provide a closed-form solution in the special case of a quadratic objective and linear dynamics, as well as design a novel adaptation of a Gaussian Process-based Regression Monte Carlo algorithm for our setting. Extensions studied include asymmetric loss function for peak shaving, capturing the cost of battery cycling and the role of battery duration. We calibrate our model to a collection of 100+ wind-battery assets in Texas, benchmarking the economic benefits of firming based on outputs of a realistic unit commitment and economic dispatch solver for ERCOT.
Alberto Gennaro (UC Berkeley)
Principal agent under random time
We are considering the problem of optimal portfolio delegation between an investor and a portfolio manager under a random default time. We focus on a novel variation of the Principal-Agent problem adapted to this framework. We address the challenge of an uncertain investment horizon caused by an exogenous random default time, after which neither the agent nor the principal can access the market. We apply traditional results from Backward Stochastic Differential Equations (BSDEs) and control theory to address the agent problem and we demonstrate that the contracting problem can be resolved by examining the existence of solutions to integro-partial Hamilton-Jacobi-Bellman (HJB) equation. We develop a deep-learning algorithm to solve the problem with no access to the optimizer of the Hamiltonian function.
Camilo Hernández (USC)
A unifying framework for a class of principal-agent problems
In this talk, we provide a framework to address extensions of Principal–(multi-)agent problems in contract theory in the context of moral hazard. These include: (i) incorporating constraints on the terminal payment $\xi$, such as $\xi = g(X_T)$, $\xi \leq R$; (ii) equal terminal payments in a Principal–multi-agent problem; (iii) settings in which both the principal and the agent act on the system in a competitive, i.e., Nash, fashion.
Building upon our results in [2] for Stackelberg games, one is able to reformulate the Principal’s unconventional problem as a standard optimal stochastic control problem, yet with stochastic target constraints. Crucially, in the previous scenarii the target sets are described by level set equations of suitable—application dependent—functions $\Psi$. Leveraging the stochastic representation of level set equations in [3], we extend the ideas in [1] and show that the optimal strategies and the value of the Principal’s reformulated problem can then be obtained by solving a well-specified system of Hamilton-Jacobi-Bellman equations. We will illustrate our results in the above scenarii.
Joint work with Nicolás Hernández-Santibáñez, Emma Hubert and Dylan Possamaï.
References:
[1] B. Bouchard, R. Élie, and C. Imbert. Optimal control under stochastic target constraints. SIAM Journal on Control and Optimization, 48(5):3501–3531, 2010.
[2] C. Hernández, N. Hernández-Santibáñez, E. Hubert, and D. Possamaï. Closed-loop equilibria for Stackelberg games: it’s all about stochastic targets. ArXiv preprint arXiv:2406.19607, 2024.
[3] H. M. Soner and N. Touzi. A stochastic representation for the level set equations. Communications in Partial Differential Equations, 27(9 & 10):2031–2053, 2002.
Check In & Breakfast
Agostino Capponi (Columbia University)
Data-Driven Dynamic Factor Modeling via Manifold Learning
We introduce a novel data-driven dynamic factor framework where a response variable $y(t) \in \mathbb{R}^m$ depends on a high-dimensional set of covariates $x(t) \in \mathbb{R}^d$ without imposing any parametric model on the joint covariate dynamics. Leveraging diffusion maps – a nonlinear manifold learning technique introduced in Coifman and Lafon [2006] – our framework uncovers the joint dynamics of the covariates in a purely data-driven way. It achieves this by constructing lower-dimensional covariate embeddings that retain most of the explanatory power for the time series of responses $y(t)$, while exhibiting simple linear dynamics. We combine diffusion maps with Kalman filtering techniques to infer the latent dynamic covariate embeddings, and predict the response variable directly from the diffusion map embedding space.
We showcase our framework on the problem of stress testing equity portfolios, using a combination of financial and macroeconomic factors from the FED's supervisory scenarios. Unlike standard scenario analysis (SSA), where one assumes that the conditional expectation of the unstressed factors given the scenario is zero, we account for dynamic correlation between stressed and unstressed risk factors through a novel conditional sampling procedure. We demonstrate that our data-driven stress testing procedure outperforms SSA- and PCA-based benchmarks through historical backtests spanning three major financial crises, achieving reductions in mean absolute error (MAE) of up to 52% and 57% for scenario-based portfolio return prediction, respectively. (joint work with Graeme Baker and Jose Sidaoui Gali).
Coffee Break
Anna De Crescenzo (UC Berkeley & Université Paris Cité)
Mean-field control of non-exchangeable systems
We study the optimal control of mean-field systems with heterogeneous interactions. By employing graphon theory, we analyze the large-population limit, proving a propagation of chaos result that yields a collection of mean-field stochastic differential equations. We further address the control of these non-exchangeable McKean-Vlasov systems from the perspective of a central planner. Leveraging tools tailored for this framework, such as derivatives along flows of measures and the corresponding Itô calculus, we establish that the value function of this control problem satisfies a Bellman dynamic programming equation in a function space over the Wasserstein space. To illustrate the applicability of our approach, we present a linear-quadratic graphon model with analytical solutions and apply it to a systemic risk example involving heterogeneous banks. Based on joint works with F. Coppini, F. de Feo, M. Fuhrman, I. Kharroubi, H. Pham.
Xinyu Li (UC Berkeley)
An α-potential game framework for N-player dynamic games
This paper proposes and studies a general form of dynamic N-player non-cooperative games called α-potential games, where the change of a player's value function upon her unilateral deviation from her strategy is equal to the change of an α-potential function up to an error α. Analogous to the static potential game (which corresponds to α=0), the α-potential game framework is shown to reduce the challenging task of finding α-Nash equilibria for a dynamic game to minimizing the α-potential function. Moreover, an analytical characterization of α-potential functions is established, with α represented in terms of the magnitude of the asymmetry of value functions' second-order derivatives. For stochastic differential games in which the state dynamic is a controlled diffusion, α is characterized in terms of the number of players, the choice of admissible strategies, and the intensity of interactions and the level of heterogeneity among players. Two classes of stochastic differential games, namely distributed games and games with mean field interactions, are analyzed to highlight the dependence of αon general game characteristics that are beyond the mean-field paradigm, which focuses on the limit of N with homogeneous players. To analyze the α-NE, the associated optimization problem is embedded into a conditional McKean-Vlasov control problem. A verification theorem is established to construct α-NE based on solutions to an infinite-dimensional Hamilton-Jacobi-Bellman equation, which is reduced to a system of ordinary differential equations for linear-quadratic games.
Ying Tan (UCSB)
Mean-field Schrödinger bridge for generative AI: Relaxed formulation and convergence
Flow matching has emerged as a popular framework in generative AI, aiming to transport a flow from a noise distribution to a target data distribution. However, its current success largely relies on empirical heuristics rather than on a solid mathematical foundation. Motivated by this, we propose a relaxed version of the Schrödinger bridge problem, replacing the terminal distribution constraint with an additional penalty function, which facilitates computationally tractable implementations compared to the classic Schrödinger bridge problem. This relaxation leads to a McKean–Vlasov type stochastic control problem with a special structure, whose well-posedness is established through the analysis of the corresponding forward-backward stochastic differential equations (FBSDEs). Furthermore, we prove a linear convergence rate for both the optimal strategy and the value function as the penalty term tends to infinity, leveraging a novel framework based on Doob’s h-transform and a static optimization problem in measure space. Finally, to support numerical experiments, we examine the convergence of data-driven approximations and the finite-particle system.
Lunch Break
Bixing Qiao (USC)
A New Approach For The Kyle-Back Strategic Insider Equilibrium Problem
Kyle-Back (KB) model has been widely studied by the math finance community. Most of the works assume Markovian structure for pricing rules in order to use the PDE approach. We provide a new formulation for the KB model that does not assume Markovian structure for pricing function of the market maker. When requiring integrability for the insider's strategy, the equilibrium of the KB model can be characterized by the solution of the Forward-Backward Stochastic Differential Equations (FBSDE). By analyzing the FBSDE, we find an example such that a unique equilibrium exists with non-Markovian pricing function. We further study the set value of the insider payoff. By applying the so-called duality property, we can characterize the set value of the insider payoff through an optimal control problem. The well known bridge strategy with linear pricing function is an element of our set value even though the insider strategy is not integrable.
Weixuan Xia (USC)
Wealth or Stealth? The Camouflage Effect in Insider Trading
We consider a Kyle-type model where insider trading occurs within a potentially large pool of liquidity traders and is subject to legal penalties. Insiders exploit the liquidity provided by the trading masses to "camouflage" their actions and balance wealth maximization with stealth to evade detection. We establish equilibrium existence for any population size and a unique limiting equilibrium under mild conditions. A stealth index governs the scale of insider trading, revealing that as traders prioritize concealment, price informativeness declines, leading to an equilibrium closely approximated by a simple limit with minimal price impact. Empirical calibration (1980–2018) highlights the critical role of large populations in insider trading models under legal risk, with key insights on stealth trading and legal enforcement. This talk is based on joint work with Jin Ma and Jianfeng Zhang.
Hubeyb Gurdogan (UCLA)
The Quadratic Optimization Bias Of Large Covariance Matrices
We describe a puzzle involving the interactions between an optimization of a multivariate quadratic function and a "plug-in" estimator of a spiked covariance matrix. When the largest eigenvalues (i.e., the spikes) diverge with the dimension, the gap between the true and the out-of-sample optima typically also diverges. We show how to "fine-tune" the plug-in estimator in a precise way to avoid this outcome. Central to our description is a "quadratic optimization bias" function, the roots of which determine this fine-tuning property. We derive an estimator of this root from a finite number of observations of a high dimensional vector. This leads to a new covariance estimator designed specifically for applications involving quadratic optimization. Our theoretical results have further implications for improving low dimensional representations of data, and principal component analysis in particular.
Coffee Break
Lukas Fiechtner (Stanford University)
Wasserstein Distributionally Robust Regret Optimisation
Distributionally Robust Optimization (DRO) provides an approach for dealing with optimal decision making in the context of distributional uncertainty. While DRO has been successfully applied in a wide range of areas, its adversarial nature can also lead to decisions that tend to be too conservative. In order to mitigate overconservative decisions, we investigate ex-ante Distributionally Robust Regret Optimization (DRRO). We focus on Wasserstein-based distributional uncertainty sets, which are popular in DRO due to connections to traditional machine learning techniques including norm regularization, among others. We provide a systematic study of fundamental properties of Wasserstein DRRO in the spirit of what is known for the Wasserstein DRO. We develop sensitivity analysis results for Wasserstein DRRO, showing that under certain smoothness and regularity assumptions, it is equivalent up to first-order terms to ERM. This equivalence is exact in convex quadratic optimization settings. Next, we revisit the classical Wasserstein DRRO newsvendor problem and recover a result stated in [14] showing that, in this setting, the optimal DRRO policy can be obtained by solving two convex problems. However, when the newsvendor problem involves multiple supply and demand sources, the problem becomes NP-hard. Thus, we propose a convex relaxation for general Wasserstein DRRO problems and verify that it has excellent empirical performance in various settings of interest. Moreover, we obtain a bound on the optimality gap of our convex relaxation and are able to show that it is superior to recently developed Wasserstein DRRO relaxations.
Moritz Voss (UCLA)
In-Context Operator Learning for Linear Propagator Models
We study operator learning in the context of linear propagator models for optimal order execution problems with transient price impact à la Bouchaud et al. (2004) and Gatheral (2010). Transient price impact persists and decays over time according to some propagator kernel. Specifically, we propose to use In-Context Operator Networks (ICON), a novel transformer-based neural network architecture introduced by Yang et al. (2023), which facilitates data-driven learning of operators by merging offline pre-training with an online few-shot prompting inference. First, we train ICON to learn the operator from various propagator models that maps the trading rate to the induced transient price impact. The inference step is then based on in-context prediction, where ICON is presented only with a few examples. We illustrate that ICON is capable of accurately inferring the underlying price impact model from the data prompts, even with propagator kernels not seen in the training data. In a second step, we employ the pre-trained ICON model provided with context as a surrogate operator in solving an optimal order execution problem via a neural network control policy, and demonstrate that the exact optimal execution strategies from Abi Jaber and Neuman (2022) for the models generating the context are correctly retrieved. Our introduced methodology is very general, offering a new approach to solving optimal stochastic control problems with unknown state dynamics, inferred data-efficiently from a limited number of examples by leveraging the few-shot and transfer learning capabilities of transformer networks.
This is joint work with Tingwei Meng, Nils Detering, Giulio Farolfi, Stanley Osher, and Georg Menz.
Closing Remarks