We study the prices of European Emission Allowances (EUA), whereby we analyze their uncertainty and dependencies on related energy prices (natural gas, coal, and oil). We propose a probabilistic multivariate conditional time series model with a VECM-Copula-GARCH structure which exploits key characteristics of the data. Data are normalized with respect to inflation and carbon emissions to allow for proper cross-series evaluation. The forecasting performance is evaluated in an extensive rolling-window forecasting study, covering eight years out-of-sample. We discuss our findings for both levels- and log-transformed data, focusing on time-varying correlations, and in view of the Russian invasion of Ukraine.

We propose a new class of financial volatility models, which we call the REcurrent Conditional Heteroskedastic (RECH) models, to improve both the in-sample analysis and out-of-sample forecast performance of the traditional conditional heteroskedastic models. In particular, we incorporate auxiliary deterministic processes, governed by recurrent neural networks, into the conditional variance of the traditional conditional heteroskedastic models, e.g. the GARCH-type models, to flexibly capture the dynamics of the underlying volatility. The RECH models can detect interesting effects in financial volatility overlooked by the existing conditional heteroskedastic models such as the GARCH (Bollerslev, 1986), GJR (Glosten et al., 1993) and EGARCH (Nelson, 1991). The new models often have good out-of-sample forecasts while still explain well the stylized facts of financial volatility by retaining the well-established structures of the econometric GARCH-type models. These properties are illustrated through simulation studies and applications to four real stock index datasets. An user-friendly software package together with the examples reported in the paper are available at https://github.com/vbayeslab.

We study neural networks as nonparametric estimation tools for the hedging of options. To this end, we design a network, named HedgeNet, that directly outputs a hedging strategy. This network is trained to minimise the hedging error instead of the pricing error. Applied to end-of-day and tick prices of S&P 500 and Euro Stoxx 50 options, the network is able to reduce the mean squared hedging error of the Black-Scholes benchmark significantly. We illustrate, however, that a similar benefit arises by simple linear regressions that incorporate the leverage effect. Finally, we show how a faulty training/test data split, possibly along with an additional 'tagging' of data, leads to a significant overestimation of the outperformance of neural networks.

In high-dimensional data space, semi-supervised feature learning based on Euclidean distance shows instability under a broad set of conditions. Furthermore, the scarcity and high cost of labels prompt us to explore new semi-supervised learning methods with the fewest labels. In this paper, we develop a novel Minor Constraint Disturbances-based Deep Semi-supervised Feature Learning framework (MCD-DSFL) from the perspective of probability distribution for feature representation. There are two fundamental modules in the proposed framework: one is a Minor Constraint Disturbances-based restricted Boltzmann machine with Gaussian visible units (MCDGRBM) for modelling continuous data and the other is a Minor Constraint Disturbances-based restricted Boltzmann machine (MCDRBM) for modelling binary data. The Minor Constraint Disturbances (MCD) consist of less instance-level constraints which are produced by only two randomly selected labels from each class. The Kullback-Leibler (KL) divergences of the MCD are fused into the Contrastive Divergence (CD) learning for training the proposed MCDGRBM and MCDRBM models. Then, the probability distributions of hidden layer features are as similar as possible in the same class and they are as dissimilar as possible in the different classes simultaneously. Despite the weak influence of the MCD for our shallow models (MCDGRBM and MCDRBM), the proposed deep MCD-DSFL framework improves the representation capability significantly under its leverage effect. The semi-supervised strategy based on the KL divergence of the MCD significantly reduces the reliance on the labels and improves the stability of the semi-supervised feature learning in high-dimensional space simultaneously.

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.

The increasing importance of renewable energy, especially solar and wind power, has led to new forces in the formation of electricity prices. Hence, this paper introduces an econometric model for the hourly time series of electricity prices of the European Power Exchange (EPEX) which incorporates specific features like renewable energy. The model consists of several sophisticated and established approaches and can be regarded as a periodic VAR-TARCH with wind power, solar power, and load as influences on the time series. It is able to map the distinct and well-known features of electricity prices in Germany. An efficient iteratively reweighted lasso approach is used for the estimation. Moreover, it is shown that several existing models are outperformed by the procedure developed in this paper.