Solving high-dimensional optimal stopping problems using deep learning

Nowadays many financial derivatives which are traded on stock and futures exchanges, such as American or Bermudan options, are of early exercise type. Often the pricing of early exercise options gives rise to high-dimensional optimal stopping problems, since the dimension corresponds to the number of underlyings in the associated hedging portfolio. High-dimensional optimal stopping problems are, however, notoriously difficult to solve due to the well-known curse of dimensionality. In this work we propose an algorithm for solving such problems, which is based on deep learning and computes, in the context of early exercise option pricing, both approximations for an optimal exercise strategy and the price of the considered option. The proposed algorithm can also be applied to optimal stopping problems that arise in other areas where the underlying stochastic process can be efficiently simulated. We present numerical results for a large number of example problems, which include the pricing of many high-dimensional American and Bermudan options such as, for example, Bermudan max-call options in up to 5000 dimensions. Most of the obtained results are compared to reference values computed by exploiting the specific problem design or, where available, to reference values from the literature. These numerical results suggest that the proposed algorithm is highly effective in the case of many underlyings, in terms of both accuracy and speed.Comment: 42 pages, 1 figur

High-dimensional dynamics of generalization error in neural networks

We perform an average case analysis of the generalization dynamics of large neural networks trained using gradient descent. We study the practically-relevant "high-dimensional" regime where the number of free parameters in the network is on the order of or even larger than the number of examples in the dataset. Using random matrix theory and exact solutions in linear models, we derive the generalization error and training error dynamics of learning and analyze how they depend on the dimensionality of data and signal to noise ratio of the learning problem. We find that the dynamics of gradient descent learning naturally protect against overtraining and overfitting in large networks. Overtraining is worst at intermediate network sizes, when the effective number of free parameters equals the number of samples, and thus can be reduced by making a network smaller or larger. Additionally, in the high-dimensional regime, low generalization error requires starting with small initial weights. We then turn to non-linear neural networks, and show that making networks very large does not harm their generalization performance. On the contrary, it can in fact reduce overtraining, even without early stopping or regularization of any sort. We identify two novel phenomena underlying this behavior in overcomplete models: first, there is a frozen subspace of the weights in which no learning occurs under gradient descent; and second, the statistical properties of the high-dimensional regime yield better-conditioned input correlations which protect against overtraining. We demonstrate that naive application of worst-case theories such as Rademacher complexity are inaccurate in predicting the generalization performance of deep neural networks, and derive an alternative bound which incorporates the frozen subspace and conditioning effects and qualitatively matches the behavior observed in simulation

Solving Nonlinear and High-Dimensional Partial Differential Equations via Deep Learning

In this work we apply the Deep Galerkin Method (DGM) described in Sirignano and Spiliopoulos (2018) to solve a number of partial differential equations that arise in quantitative finance applications including option pricing, optimal execution, mean field games, etc. The main idea behind DGM is to represent the unknown function of interest using a deep neural network. A key feature of this approach is the fact that, unlike other commonly used numerical approaches such as finite difference methods, it is mesh-free. As such, it does not suffer (as much as other numerical methods) from the curse of dimensionality associated with highdimensional PDEs and PDE systems. The main goals of this paper are to elucidate the features, capabilities and limitations of DGM by analyzing aspects of its implementation for a number of different PDEs and PDE systems. Additionally, we present: (1) a brief overview of PDEs in quantitative finance along with numerical methods for solving them; (2) a brief overview of deep learning and, in particular, the notion of neural networks; (3) a discussion of the theoretical foundations of DGM with a focus on the justification of why this method is expected to perform well

RLOC: Neurobiologically Inspired Hierarchical Reinforcement Learning Algorithm for Continuous Control of Nonlinear Dynamical Systems

Nonlinear optimal control problems are often solved with numerical methods that require knowledge of system's dynamics which may be difficult to infer, and that carry a large computational cost associated with iterative calculations. We present a novel neurobiologically inspired hierarchical learning framework, Reinforcement Learning Optimal Control, which operates on two levels of abstraction and utilises a reduced number of controllers to solve nonlinear systems with unknown dynamics in continuous state and action spaces. Our approach is inspired by research at two levels of abstraction: first, at the level of limb coordination human behaviour is explained by linear optimal feedback control theory. Second, in cognitive tasks involving learning symbolic level action selection, humans learn such problems using model-free and model-based reinforcement learning algorithms. We propose that combining these two levels of abstraction leads to a fast global solution of nonlinear control problems using reduced number of controllers. Our framework learns the local task dynamics from naive experience and forms locally optimal infinite horizon Linear Quadratic Regulators which produce continuous low-level control. A top-level reinforcement learner uses the controllers as actions and learns how to best combine them in state space while maximising a long-term reward. A single optimal control objective function drives high-level symbolic learning by providing training signals on desirability of each selected controller. We show that a small number of locally optimal linear controllers are able to solve global nonlinear control problems with unknown dynamics when combined with a reinforcement learner in this hierarchical framework. Our algorithm competes in terms of computational cost and solution quality with sophisticated control algorithms and we illustrate this with solutions to benchmark problems.Comment: 33 pages, 8 figure

Totally Corrective Boosting with Cardinality Penalization

We propose a totally corrective boosting algorithm with explicit cardinality regularization. The resulting combinatorial optimization problems are not known to be efficiently solvable with existing classical methods, but emerging quantum optimization technology gives hope for achieving sparser models in practice. In order to demonstrate the utility of our algorithm, we use a distributed classical heuristic optimizer as a stand-in for quantum hardware. Even though this evaluation methodology incurs large time and resource costs on classical computing machinery, it allows us to gauge the potential gains in generalization performance and sparsity of the resulting boosted ensembles. Our experimental results on public data sets commonly used for benchmarking of boosting algorithms decidedly demonstrate the existence of such advantages. If actual quantum optimization were to be used with this algorithm in the future, we would expect equivalent or superior results at much smaller time and energy costs during training. Moreover, studying cardinality-penalized boosting also sheds light on why unregularized boosting algorithms with early stopping often yield better results than their counterparts with explicit convex regularization: Early stopping performs suboptimal cardinality regularization. The results that we present here indicate it is beneficial to explicitly solve the combinatorial problem still left open at early termination

Stable Distribution Alignment Using the Dual of the Adversarial Distance

Methods that align distributions by minimizing an adversarial distance between them have recently achieved impressive results. However, these approaches are difficult to optimize with gradient descent and they often do not converge well without careful hyperparameter tuning and proper initialization. We investigate whether turning the adversarial min-max problem into an optimization problem by replacing the maximization part with its dual improves the quality of the resulting alignment and explore its connections to Maximum Mean Discrepancy. Our empirical results suggest that using the dual formulation for the restricted family of linear discriminators results in a more stable convergence to a desirable solution when compared with the performance of a primal min-max GAN-like objective and an MMD objective under the same restrictions. We test our hypothesis on the problem of aligning two synthetic point clouds on a plane and on a real-image domain adaptation problem on digits. In both cases, the dual formulation yields an iterative procedure that gives more stable and monotonic improvement over time.Comment: ICLR 2018 Conference Invite to Worksho

Deep Decoder: Concise Image Representations from Untrained Non-convolutional Networks

Deep neural networks, in particular convolutional neural networks, have become highly effective tools for compressing images and solving inverse problems including denoising, inpainting, and reconstruction from few and noisy measurements. This success can be attributed in part to their ability to represent and generate natural images well. Contrary to classical tools such as wavelets, image-generating deep neural networks have a large number of parameters---typically a multiple of their output dimension---and need to be trained on large datasets. In this paper, we propose an untrained simple image model, called the deep decoder, which is a deep neural network that can generate natural images from very few weight parameters. The deep decoder has a simple architecture with no convolutions and fewer weight parameters than the output dimensionality. This underparameterization enables the deep decoder to compress images into a concise set of network weights, which we show is on par with wavelet-based thresholding. Further, underparameterization provides a barrier to overfitting, allowing the deep decoder to have state-of-the-art performance for denoising. The deep decoder is simple in the sense that each layer has an identical structure that consists of only one upsampling unit, pixel-wise linear combination of channels, ReLU activation, and channelwise normalization. This simplicity makes the network amenable to theoretical analysis, and it sheds light on the aspects of neural networks that enable them to form effective signal representations.Comment: International Conference on Learning Representations 201

Deep-learning based numerical BSDE method for barrier options

As is known, an option price is a solution to a certain partial differential equation (PDE) with terminal conditions (payoff functions). There is a close association between the solution of PDE and the solution of a backward stochastic differential equation (BSDE). We can either solve the PDE to obtain option prices or solve its associated BSDE. Recently a deep learning technique has been applied to solve option prices using the BSDE approach. In this approach, deep learning is used to learn some deterministic functions, which are used in solving the BSDE with terminal conditions. In this paper, we extend the deep-learning technique to solve a PDE with both terminal and boundary conditions. In particular, we will employ the technique to solve barrier options using Brownian motion bridges

Deep Fictitious Play for Stochastic Differential Games

In this paper, we apply the idea of fictitious play to design deep neural networks (DNNs), and develop deep learning theory and algorithms for computing the Nash equilibrium of asymmetric $N$-player non-zero-sum stochastic differential games, for which we refer as \emph{deep fictitious play}, a multi-stage learning process. Specifically at each stage, we propose the strategy of letting individual player optimize her own payoff subject to the other players' previous actions, equivalent to solve $N$ decoupled stochastic control optimization problems, which are approximated by DNNs. Therefore, the fictitious play strategy leads to a structure consisting of $N$ DNNs, which only communicate at the end of each stage. The resulted deep learning algorithm based on fictitious play is scalable, parallel and model-free, {\it i.e.}, using GPU parallelization, it can be applied to any $N$-player stochastic differential game with different symmetries and heterogeneities ({\it e.g.}, existence of major players). We illustrate the performance of the deep learning algorithm by comparing to the closed-form solution of the linear quadratic game. Moreover, we prove the convergence of fictitious play under appropriate assumptions, and verify that the convergent limit forms an open-loop Nash equilibrium. We also discuss the extensions to other strategies designed upon fictitious play and closed-loop Nash equilibrium in the end

A Fast Deep Learning Approach for Beam Orientation Optimization for Prostate Cancer IMRT Treatments

We propose a fast beam orientation selection method, based on deep neural networks (DNN), capable of developing a plan comparable to those by the state-of-the-art column generation method. The novelty of Our model lies in its supervised learning structure, the DNN architecture, and ability to learn from anatomical features to predict dosimetrically suitable beam orientations without using the dosimetric information from the candidate beams, a time consuming and computationally expensive process. This may save hours of computation. A supervised DNN is trained to mimic the column generation algorithm, which iteratively chooses beam orientations by calculating beam fitness values based on the KKT optimality conditions. The dataset contains 70 prostate cancer patients. The DNN trained over 400 epochs, each with 2500 steps, using the Adam optimizer and a 6-fold cross-validation technique. The average and standard deviation of training, validation, and testing loss functions among the 6-folds were at most 1.44%. The differences in the dose coverage of PTV between plans generated by column generation and by DNN were 0.2%. The average dose differences received by organs at risk were between 1 and 6 percent: bladder had the smallest average difference, then rectum, left and right femoral heads. The dose received by body had an average difference of 0.1%. In the training phase of the proposed method, the model learns the suitable beam orientations based on the anatomical features of patients and omits time intensive calculations of dose influence matrices for all possible candidate beams. Solving the Fluence Map Optimization to get the final treatment plan requires calculating dose influence matrices only for the selected beams. The proposed DNN is a fast beam orientation selection method based that selects beam orientations in seconds and is therefore suitable for clinical routines.Comment: 28 pages, 9 figure