32 research outputs found
Optimal Execution with Dynamic Order Flow Imbalance
We examine optimal execution models that take into account both market
microstructure impact and informational costs. Informational footprint is
related to order flow and is represented by the trader's influence on the flow
imbalance process, while microstructure influence is captured by instantaneous
price impact. We propose a continuous-time stochastic control problem that
balances between these two costs. Incorporating order flow imbalance leads to
the consideration of the current market state and specifically whether one's
orders lean with or against the prevailing order flow, key components often
ignored by execution models in the literature. In particular, to react to
changing order flow, we endogenize the trading horizon . After developing
the general indefinite-horizon formulation, we investigate several tractable
approximations that sequentially optimize over price impact and over . These
approximations, especially a dynamic version based on receding horizon control,
are shown to be very accurate and connect to the prevailing Almgren-Chriss
framework. We also discuss features of empirical order flow and links between
our model and "Optimal Execution Horizon" by Easley et al (Mathematical
Finance, 2013).Comment: 31 pages, 8 figure
Sequential Design for Ranking Response Surfaces
We propose and analyze sequential design methods for the problem of ranking
several response surfaces. Namely, given response surfaces over a
continuous input space , the aim is to efficiently find the index of
the minimal response across the entire . 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.Comment: 26 pages, 7 figures (updated several sections and figures
Sequential Design for Optimal Stopping Problems
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.Comment: 24 page
mlOSP: Towards a Unified Implementation of Regression Monte Carlo Algorithms
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. At the same time, 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 the dual role of presenting new RMC features and as a vignette to the
underlying software package.Comment: Package repository is at http://github.com/mludkov/mlOS
Sequential Tracking of a Hidden Markov Chain Using Point Process Observations
We study finite horizon optimal switching problems for hidden Markov chain
models under partially observable Poisson processes. The controller possesses a
finite range of strategies and attempts to track the state of the unobserved
state variable using Bayesian updates over the discrete observations. Such a
model has applications in economic policy making, staffing under variable
demand levels and generalized Poisson disorder problems. We show regularity of
the value function and explicitly characterize an optimal strategy. We also
provide an efficient numerical scheme and illustrate our results with several
computational examples.Comment: Key words and phrases. Markov Modulated Poisson processes, optimal
switchin
Adaptive Batching for Gaussian Process Surrogates with Application in Noisy Level Set Estimation
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.Comment: 36 pages, 6 figure