Michael Ludkovski

Erhan Bayraktar, Michael Ludkovski

We consider a framework for solving optimal liquidation problems in limit order books. In particular, order arrivals are modeled as a point process whose intensity depends on the liquidation price. We set up a stochastic control problem in which the goal is to maximize the expected revenue from liquidating the entire position held. We solve this optimal liquidation problem for power-law and exponential-decay order book models and discuss several extensions. We also consider the continuous selling (or fluid) limit when the trading units are ever smaller and the intensity is ever larger. This limit provides an analytical approximation to the value function and the optimal solution. Using techniques from viscosity solutions we show that the discrete state problem and its optimal solution converge to the corresponding quantities in the continuous selling limit uniformly on compacts.

Tao Chen, Mike Ludkovski, Moritz Voß

We investigate optimal order execution problems in discrete time with instantaneous price impact and stochastic resilience. First, in the setting of linear transient price impact we derive a closed-form recursion for the optimal strategy, extending the deterministic results from Obizhaeva and Wang (J Financial Markets, 2013). Second, we develop a numerical algorithm based on dynamic programming and deep learning for the case of nonlinear transient price impact as proposed by Bouchaud et al. (Quant. Finance, 2004). Specifically, we utilize an actor-critic framework that constructs two neural-network (NN) surrogates for the value function and the feedback control. The flexible scalability of NN functional approximators enables parametric learning, i.e., incorporating several model or market parameters as part of the input space. Precise calibration of price impact, resilience, etc., is known to be extremely challenging and hence it is critical to understand sensitivity of the execution policy to these parameters. Our NN learner organically scales across multiple input dimensions and is shown to accurately approximate optimal strategies across a wide range of parameter configurations. We provide a fully reproducible Jupyter Notebook with our NN implementation, which is of independent pedagogical interest, demonstrating the ease of use of NN surrogates in (parametric) stochastic control problems.

Mike Ludkovski

We introduce mlOSP, a computational template for Machine Learning for Optimal Stopping Problems. The template is implemented in the R statistical environment and publicly available via a GitHub repository. mlOSP presents a unified numerical implementation of Regression Monte Carlo (RMC) approaches to optimal stopping, providing a state-of-the-art, open-source, reproducible and transparent platform. Highlighting its modular nature, we present multiple novel variants of RMC algorithms, especially in terms of constructing simulation designs for training the regressors, as well as in terms of machine learning regression modules. Furthermore, mlOSP nests most of the existing RMC schemes, allowing for a consistent and verifiable benchmarking of extant algorithms. The article contains extensive R code snippets and figures, and serves as a vignette to the underlying software package.

Qiqi Li, Mike Ludkovski

We design a Gaussian Process (GP) spatiotemporal model to capture features of day-ahead wind power forecasts. We work with hourly-scale day-ahead forecasts across hundreds of wind farm locations, with the main aim of constructing a fully probabilistic joint model across space and hours of the day. To this end, we design a separable space-time kernel, implementing both temporal and spatial input warping to capture the non-stationarity in the covariance of wind power. We conduct synthetic experiments to validate our choice of the spatial kernel and to demonstrate the effectiveness of warping in addressing nonstationarity. The second half of the paper is devoted to a detailed case study using a realistic, fully calibrated dataset representing wind farms in the ERCOT region of Texas.

Guillermo Terrén-Serrano, Michael Ludkovski

We propose and analyze the application of statistical functional depth metrics for the selection of extreme scenarios in day-ahead grid planning. Our primary motivation is screening of probabilistic scenarios for realized load and renewable generation, in order to identify scenarios most relevant for operational risk mitigation. To handle the high-dimensionality of the scenarios across asset classes and intra-day periods, we employ functional measures of depth to sub-select outlying scenarios that are most likely to be the riskiest for the grid operation. We investigate a range of functional depth measures, as well as a range of operational risks, including load shedding, operational costs, reserves shortfall and variable renewable energy curtailment. The effectiveness of the proposed screening approach is demonstrated through a case study on the realistic Texas-7k grid.

Michael Ludkovski, Changgen Xie, Zimu Zhu

We consider numerical resolution of principal-agent (PA) problems in continuous time. We formulate a generic PA model with continuous and lump payments and a multi-dimensional strategy of the agent. To tackle the resulting Hamilton-Jacobi-Bellman equation with an implicit Hamiltonian we develop a novel deep learning method: the Deep Principal-Agent Actor Critic (DeepPAAC) Actor-Critic algorithm. DeepPAAC is able to handle multi-dimensional states and controls, as well as constraints. We investigate the role of the neural network architecture, training designs, loss functions, etc. on the convergence of the solver, presenting five different case studies.

Olivier Mulkin, Miguel Heleno, Mike Ludkovski

We develop a new methodology to select scenarios of DER adoption most critical for distribution grids. Anticipating risks of future voltage and line flow violations due to additional PV adopters is central for utility investment planning but continues to rely on deterministic or ad hoc scenario selection. We propose a highly efficient search framework based on multi-objective Bayesian Optimization. We treat underlying grid stress metrics as computationally expensive black-box functions, approximated via Gaussian Process surrogates and design an acquisition function based on probability of scenarios being Pareto-critical across a collection of line- and bus-based violation objectives. Our approach provides a statistical guarantee and offers an order of magnitude speed-up relative to a conservative exhaustive search. Case studies on realistic feeders with 200-400 buses demonstrate the effectiveness and accuracy of our approach.

Mike Ludkovski, Jimmy Risk

We develop a flexible Gaussian Process (GP) framework for learning the covariance structure of Age- and Year-specific mortality surfaces. Utilizing the additive and multiplicative structure of GP kernels, we design a genetic programming algorithm to search for the most expressive kernel for a given population. Our compositional search builds off the Age-Period-Cohort (APC) paradigm to construct a covariance prior best matching the spatio-temporal dynamics of a mortality dataset. We apply the resulting genetic algorithm (GA) on synthetic case studies to validate the ability of the GA to recover APC structure, and on real-life national-level datasets from the Human Mortality Database. Our machine-learning based analysis provides novel insight into the presence/absence of Cohort effects in different populations, and into the relative smoothness of mortality surfaces along the Age and Year dimensions. Our modelling work is done with the PyTorch libraries in Python and provides an in-depth investigation of employing GA to aid in compositional kernel search for GP surrogates.

Mike Ludkovski, Yuri Saporito

We investigate a machine learning approach to option Greeks approximation based on Gaussian process (GP) surrogates. The method takes in noisily observed option prices, fits a nonparametric input-output map and then analytically differentiates the latter to obtain the various price sensitivities. Our motivation is to compute Greeks in cases where direct computation is expensive, such as in local volatility models, or can only ever be done approximately. We provide a detailed analysis of numerous aspects of GP surrogates, including choice of kernel family, simulation design, choice of trend function and impact of noise. We further discuss the application to Delta hedging, including a new Lemma that relates quality of the Delta approximation to discrete-time hedging loss. Results are illustrated with two extensive case studies that consider estimation of Delta, Theta and Gamma and benchmark approximation quality and uncertainty quantification using a variety of statistical metrics. Among our key take-aways are the recommendation to use Matern kernels, the benefit of including virtual training points to capture boundary conditions, and the significant loss of fidelity when training on stock-path-based datasets.

Xiong Lyu, Mike Ludkovski

We develop adaptive replicated designs for Gaussian process metamodels of stochastic experiments. Adaptive batching is a natural extension of sequential design heuristics with the benefit of replication growing as response features are learned, inputs concentrate, and the metamodeling overhead rises. Motivated by the problem of learning the level set of the mean simulator response we develop four novel schemes: Multi-Level Batching (MLB), Ratchet Batching (RB), Adaptive Batched Stepwise Uncertainty Reduction (ABSUR), Adaptive Design with Stepwise Allocation (ADSA) and Deterministic Design with Stepwise Allocation (DDSA). Our algorithms simultaneously (MLB, RB and ABSUR) or sequentially (ADSA and DDSA) determine the sequential design inputs and the respective number of replicates. Illustrations using synthetic examples and an application in quantitative finance (Bermudan option pricing via Regression Monte Carlo) show that adaptive batching brings significant computational speed-ups with minimal loss of modeling fidelity.

Xiong Lyu, Mickael Binois, Michael Ludkovski

We consider the problem of learning the level set for which a noisy black-box function exceeds a given threshold. To efficiently reconstruct the level set, we investigate Gaussian process (GP) metamodels. Our focus is on strongly stochastic samplers, in particular with heavy-tailed simulation noise and low signal-to-noise ratio. To guard against noise misspecification, we assess the performance of three variants: (i) GPs with Student-$t$ observations; (ii) Student-$t$ processes (TPs); and (iii) classification GPs modeling the sign of the response. In conjunction with these metamodels, we analyze several acquisition functions for guiding the sequential experimental designs, extending existing stepwise uncertainty reduction criteria to the stochastic contour-finding context. This also motivates our development of (approximate) updating formulas to efficiently compute such acquisition functions. Our schemes are benchmarked by using a variety of synthetic experiments in 1--6 dimensions. We also consider an application of level set estimation for determining the optimal exercise policy of Bermudan options in finance.

Sergio Rodriguez, Michael Ludkovski

We consider numerical schemes for root finding of noisy responses through generalizing the Probabilistic Bisection Algorithm (PBA) to the more practical context where the sampling distribution is unknown and location-dependent. As in standard PBA, we rely on a knowledge state for the approximate posterior of the root location. To implement the corresponding Bayesian updating, we also carry out inference of oracle accuracy, namely learning the probability of correct response. To this end we utilize batched querying in combination with a variety of frequentist and Bayesian estimators based on majority vote, as well as the underlying functional responses, if available. For guiding sampling selection we investigate both Information Directed sampling, as well as Quantile sampling. Our numerical experiments show that these strategies perform quite differently; in particular we demonstrate the efficiency of randomized quantile sampling which is reminiscent of Thompson sampling. Our work is motivated by the root-finding sub-routine in pricing of Bermudan financial derivatives, illustrated in the last section of the paper.

Ruimeng Hu, Mike Ludkovski

We propose and analyze sequential design methods for the problem of ranking several response surfaces. Namely, given $L \ge 2$ response surfaces over a continuous input space $\cal X$, the aim is to efficiently find the index of the minimal response across the entire $\cal X$. The response surfaces are not known and have to be noisily sampled one-at-a-time. This setting is motivated by stochastic control applications and requires joint experimental design both in space and response-index dimensions. To generate sequential design heuristics we investigate stepwise uncertainty reduction approaches, as well as sampling based on posterior classification complexity. We also make connections between our continuous-input formulation and the discrete framework of pure regret in multi-armed bandits. To model the response surfaces we utilize kriging surrogates. Several numerical examples using both synthetic data and an epidemics control problem are provided to illustrate our approach and the efficacy of respective adaptive designs.

Robert B. Gramacy, Mike Ludkovski

We propose a new approach to solve optimal stopping problems via simulation. Working within the backward dynamic programming/Snell envelope framework, we augment the methodology of Longstaff-Schwartz that focuses on approximating the stopping strategy. Namely, we introduce adaptive generation of the stochastic grids anchoring the simulated sample paths of the underlying state process. This allows for active learning of the classifiers partitioning the state space into the continuation and stopping regions. To this end, we examine sequential design schemes that adaptively place new design points close to the stopping boundaries. We then discuss dynamic regression algorithms that can implement such recursive estimation and local refinement of the classifiers. The new algorithm is illustrated with a variety of numerical experiments, showing that an order of magnitude savings in terms of design size can be achieved. We also compare with existing benchmarks in the context of pricing multi-dimensional Bermudan options.