### Quant GANs: Deep Generation of Financial Time Series

Modeling financial time series by stochastic processes is a challenging task and a central area of research in financial mathematics. In this paper, we break through this barrier and present Quant GANs, a data-driven model which is inspired by the recent success of generative adversarial networks (GANs). Quant GANs consist of a generator and discriminator function which utilize temporal convolutional networks (TCNs) and thereby achieve to capture longer-ranging dependencies such as the presence of volatility clusters. Furthermore, the generator function is explicitly constructed such that the induced stochastic process allows a transition to its risk-neutral distribution. Our numerical results highlight that distributional properties for small and large lags are in an excellent agreement and dependence properties such as volatility clusters, leverage effects, and serial autocorrelations can be generated by the generator function of Quant GANs, demonstrably in high fidelity.

### Comparing Generative Adversarial Network Techniques for Image Creation and Modification

Generative adversarial networks (GANs) have demonstrated to be successful at generating realistic real-world images. In this paper we compare various GAN techniques, both supervised and unsupervised. The effects on training stability of different objective functions are compared. We add an encoder to the network, making it possible to encode images to the latent space of the GAN. The generator, discriminator and encoder are parameterized by deep convolutional neural networks. For the discriminator network we experimented with using the novel Capsule Network, a state-of-the-art technique for detecting global features in images. Experiments are performed using a digit and face dataset, with various visualizations illustrating the results. The results show that using the encoder network it is possible to reconstruct images. With the conditional GAN we can alter visual attributes of generated or encoded images. The experiments with the Capsule Network as discriminator result in generated images of a lower quality, compared to a standard convolutional neural network.

### Regression with Conditional GAN

In recent years, impressive progress has been made in the design of implicit probabilistic models via Generative Adversarial Networks (GAN) and its extension, the Conditional GAN (CGAN). Excellent solutions have been demonstrated mostly in image processing applications which involve large, continuous output spaces. There is almost no application of these powerful tools to problems having small dimensional output spaces. Regression problems involving the inductive learning of a map, $y=f(x,z)$, $z$ denoting noise, $f:\mathbb{R}^n\times \mathbb{R}^k \rightarrow \mathbb{R}^m$, with $m$ small (e.g., $m=1$ or just a few) is one good case in point. The standard approach to solve regression problems is to probabilistically model the output $y$ as the sum of a mean function $m(x)$ and a noise term $z$; it is also usual to take the noise to be a Gaussian. These are done for convenience sake so that the likelihood of observed data is expressible in closed form. In the real world, on the other hand, stochasticity of the output is usually caused by missing or noisy input variables. Such a real world situation is best represented using an implicit model in which an extra noise vector, $z$ is included with $x$ as input. CGAN is naturally suited to design such implicit models. This paper makes the first step in this direction. Through several artificial and real world datasets, we demonstrate CGAN to be an effective approach for solving regression problems. We compare against Gaussian Processes and show that CGAN has excellent output likelihood properties and possesses the ability to model complex noise forms in a better way.

### Deep Generative Quantile-Copula Models for Probabilistic Forecasting

We introduce a new category of multivariate conditional generative models and demonstrate its performance and versatility in probabilistic time series forecasting and simulation. Specifically, the output of quantile regression networks is expanded from a set of fixed quantiles to the whole Quantile Function by a univariate mapping from a latent uniform distribution to the target distribution. Then the multivariate case is solved by learning such quantile functions for each dimension's marginal distribution, followed by estimating a conditional Copula to associate these latent uniform random variables. The quantile functions and copula, together defining the joint predictive distribution, can be parameterized by a single implicit generative Deep Neural Network.

### Multivariate time-series modeling with generative neural networks

Generative moment matching networks (GMMNs) are introduced as dependence models for the joint innovation distribution of multivariate time series (MTS). Following the popular copula-GARCH approach for modeling dependent MTS data, a framework allowing us to take an alternative GMMN-GARCH approach is presented. First, ARMA-GARCH models are utilized to capture the serial dependence within each univariate marginal time series. Second, if the number of marginal time series is large, principal component analysis (PCA) is used as a dimension-reduction step. Last, the remaining cross-sectional dependence is modeled via a GMMN, our main contribution. GMMNs are highly flexible and easy to simulate from, which is a major advantage over the copula-GARCH approach. Applications involving yield curve modeling and the analysis of foreign exchange rate returns are presented to demonstrate the utility of our approach, especially in terms of producing better empirical predictive distributions and making better probabilistic forecasts. All results are reproducible with the demo GMMN_MTS_paper of the R package gnn.