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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.

We consider three applications of impulse control in financial mathematics, a cash management problem, optimal control of an exchange rate, and portfolio optimisation under transaction costs. We sketch the different ways of solving these problems with the help of quasi-variational inequalities. Further, some viscosity solution results are presented.

Abstract:We consider the determination of optimal portfolios under the threat of a crash. Our main assumption is that upper bounds for both the crash size and the number of crashes occurring before the time horizon are given. We make no probabilistic assumption on the crash size or the crash time distribution. The optimal strategies in the presence of a crash possibility are characterized by a balance problem between insurance against the crash and good performance in the crash-free situation. Explicit solutions for the log-utility case are given. Our main finding is that constant portfolios are no longer optimal ones.

Illumina reads of the SSU-rDNA-V9 region obtained from the circumglobal Tara Oceans expedition allow the investigation of protistan plankton diversity patterns on a global scale. We analyzed 6,137,350 V9-amplicons from ocean surface waters and the deep chlorophyll maximum, which were taxonomically assigned to the phylum Ciliophora. For open ocean samples global planktonic ciliate diversity is relatively low (ca. 1,300 observed and predicted ciliate OTUs). We found that 17% of all detected ciliate OTUs occurred in all oceanic regions under study. On average, local ciliate OTU richness represented 27% of the global ciliate OTU richness, indicating that a large proportion of ciliates is widely distributed. Yet, more than half of these OTUs shared <90% sequence similarity with reference sequences of described ciliates. While alpha-diversity measures (richness and exp(Shannon H)) are hardly affected by contemporary environmental conditions, species (OTU) turnover and community similarity (β-diversity) across taxonomic groups showed strong correlation to environmental parameters. Logistic regression models predicted significant correlations between the occurrence of specific ciliate genera and individual nutrients, the oceanic carbonate system and temperature. Planktonic ciliates displayed distinct vertical distributions relative to chlorophyll a. In contrast, the Tara Oceans dataset did not reveal any evidence that latitude is structuring ciliate communities.

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The Solvency II directive asks insurance companies to derive their solvency capital requirement from the full loss distribution over the coming year. While this is in general computationally infeasible in the life insurance business, an application of the Least-Squares Monte Carlo (LSMC) method offers a possibility to overcome this computational challenge. We outline in detail the challenges a life insurer faces, the theoretical basis of the LSMC method and the necessary steps on the way to a reliable proxy modeling in the life insurance business. Further, we illustrate the advantages of the LSMC approach via presenting (slightly disguised) real-world applications.