arXiv subjects

Ralf Korn

Publications and source records attributed to Ralf Korn.

14 recordsLinked to original sources

arXiv2023

Sig-Splines: universal approximation and convex calibration of time series generative models

We propose a novel generative model for multivariate discrete-time time series data. Drawing inspiration from the construction of neural spline flows, our algorithm incorporates linear transformations and the signature transform as a seamless substitution for traditional neural networks. This approach enables us to achieve not only the universality property inherent in neural networks but also introduces convexity in the model's parameters.

cs.LG↗

arXiv2021

Multi-Asset Spot and Option Market Simulation

We construct realistic spot and equity option market simulators for a single underlying on the basis of normalizing flows. We address the high-dimensionality of market observed call prices through an arbitrage-free autoencoder that approximates efficient low-dimensional representations of the prices while maintaining no static arbitrage in the reconstructed surface. Given a multi-asset universe, we leverage the conditional invertibility property of normalizing flows and introduce a scalable method to calibrate the joint distribution of a set of independent simulators while preserving the dynamics of each simulator. Empirical results highlight the goodness of the calibrated simulators and their fidelity.

q-fin.CP↗

arXiv2021

Estimating the Value-at-Risk by Temporal VAE

Estimation of the value-at-risk (VaR) of a large portfolio of assets is an important task for financial institutions. As the joint log-returns of asset prices can often be projected to a latent space of a much smaller dimension, the use of a variational autoencoder (VAE) for estimating the VaR is a natural suggestion. To ensure the bottleneck structure of autoencoders when learning sequential data, we use a temporal VAE (TempVAE) that avoids an auto-regressive structure for the observation variables. However, the low signal- to-noise ratio of financial data in combination with the auto-pruning property of a VAE typically makes the use of a VAE prone to posterior collapse. Therefore, we propose to use annealing of the regularization to mitigate this effect. As a result, the auto-pruning of the TempVAE works properly which also results in excellent estimation results for the VaR that beats classical GARCH-type and historical simulation approaches when applied to real data.

cs.LG↗

arXiv2021

A Generalised Linear Model Framework for $β$-Variational Autoencoders based on Exponential Dispersion Families

Although variational autoencoders (VAE) are successfully used to obtain meaningful low-dimensional representations for high-dimensional data, the characterization of critical points of the loss function for general observation models is not fully understood. We introduce a theoretical framework that is based on a connection between $β$-VAE and generalized linear models (GLM). The equality between the activation function of a $β$-VAE and the inverse of the link function of a GLM enables us to provide a systematic generalization of the loss analysis for $β$-VAE based on the assumption that the observation model distribution belongs to an exponential dispersion family (EDF). As a result, we can initialize $β$-VAE nets by maximum likelihood estimates (MLE) that enhance the training performance on both synthetic and real world data sets. As a further consequence, we analytically describe the auto-pruning property inherent in the $β$-VAE objective and reason for posterior collapse.

cs.LG↗

arXiv2020

Unifying the theory of storage and the risk premium by an unobservable intrinsic electricity price

In this paper we introduce a new concept for modelling electricity prices through the introduction of an unobservable intrinsic electricity price $p(τ)$. We use it to connect the classical theory of storage with the concept of a risk premium. We derive prices for all common contracts such as the intraday spot price, the day-ahead spot price, and futures prices. Finally, we propose an explicit model from the class of structural models and conduct an empirical analysis, where we find an overall negative risk premium.

q-fin.MF↗

arXiv2020

A lower bound for the ELBO of the Bernoulli Variational Autoencoder

We consider a variational autoencoder (VAE) for binary data. Our main innovations are an interpretable lower bound for its training objective, a modified initialization and architecture of such a VAE that leads to faster training, and a decision support for finding the appropriate dimension of the latent space via using a PCA. Numerical examples illustrate our theoretical result and the performance of the new architecture.

cs.LG↗

arXiv2019

Transforming public pensions: A mixed scheme with a credit granted by the state

Birth rates have dramatically decreased and, with continuous improvements in life expectancy, pension expenditure is on an irreversibly increasing path. This will raise serious concerns for the sustainability of the public pension systems usually financed on a pay-as-you-go (PAYG) basis where current contributions cover current pension expenditure. With this in mind, the aim of this paper is to propose a mixed pension system that consists of a combination of a classical PAYG scheme and an increase of the contribution rate invested in a funding scheme. The investment of the funding part is designed so that the PAYG pension system is financially sustainable at a particular level of probability and at the same time provide some gains to individuals. In this sense, we make the individuals be an active part to face the demographic risks inherent in the PAYG and re-establish its financial sustainability.

q-fin.RM↗

arXiv2019

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. As an alternative, we introduce 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 long-range dependencies such as the presence of volatility clusters. 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.

q-fin.MF↗

arXiv2019

Machine Learning in Least-Squares Monte Carlo Proxy Modeling of Life Insurance Companies

Under the Solvency II regime, life insurance companies are asked to derive their solvency capital requirements from the full loss distributions over the coming year. Since the industry is currently far from being endowed with sufficient computational capacities to fully simulate these distributions, the insurers have to rely on suitable approximation techniques such as the least-squares Monte Carlo (LSMC) method. The key idea of LSMC is to run only a few wisely selected simulations and to process their output further to obtain a risk-dependent proxy function of the loss. In this paper, we present and analyze various adaptive machine learning approaches that can take over the proxy modeling task. The studied approaches range from ordinary and generalized least-squares regression variants over GLM and GAM methods to MARS and kernel regression routines. We justify the combinability of their regression ingredients in a theoretical discourse. Further, we illustrate the approaches in slightly disguised real-world experiments and perform comprehensive out-of-sample tests.

stat.ME↗

arXiv2019

Copula & Marginal Flows: Disentangling the Marginal from its Joint

Deep generative networks such as GANs and normalizing flows flourish in the context of high-dimensional tasks such as image generation. However, so far exact modeling or extrapolation of distributional properties such as the tail asymptotics generated by a generative network is not available. In this paper, we address this issue for the first time in the deep learning literature by making two novel contributions. First, we derive upper bounds for the tails that can be expressed by a generative network and demonstrate Lp-space related properties. There we show specifically that in various situations an optimal generative network does not exist. Second, we introduce and propose copula and marginal generative flows (CM flows) which allow for an exact modeling of the tail and any prior assumption on the CDF up to an approximation of the uniform distribution. Our numerical results support the use of CM flows.

cs.LG↗

arXiv2019

Optimal inflow control penalizing undersupply in transport systems with uncertain demands

We are concerned with optimal control strategies subject to uncertain demands. An Ornstein-Uhlenbeck process describes the uncertain demand. The transport within the supply system is modeled by the linear advection equation. We consider different approaches to control the produced amount at a given time to meet the stochastic demand in an optimal way. In particular, we introduce an undersupply penalty and analyze its effect on the optimal output in a numerical simulation study.

math.OC↗

arXiv2019

A structural Heath-Jarrow-Morton framework for consistent intraday, spot, and futures electricity prices

In this paper we introduce a flexible HJM-type framework that allows for consistent modelling of intraday, spot, futures, and option prices. This framework is based on stochastic processes with economic interpretations and consistent with the initial term structure given in the form of a price forward curve. Furthermore, the framework allows for existing day-ahead spot price models to be used in an HJM setting. We include several explicit examples of classical spot price models but also show how structural models and factor models can be formulated within the framework.

q-fin.MF↗

arXiv2018

Optimal control of electricity input given an uncertain demand

We consider the problem of determining an optimal strategy for electricity injection that faces an uncertain power demand stream. This demand stream is modeled via an Ornstein-Uhlenbeck process with an additional jump component, whereas the power flow is represented by the linear transport equation. We analytically determine the optimal amount of power supply for different levels of available information and compare the results to each other. For numerical purposes, we reformulate the original problem in terms of the cost function such that classical optimization solvers can be directly applied. The computational results are illustrated for different scenarios.

math.OC↗

arXiv2013

GARCH-extended models: theoretical properties and applications

This paper is concerned with some properties of the generalized GARCH models, obtained by extending GARCH models with exogenous variables, the so-called GARCH extended (GARCHX) models. For these, we establish sufficient conditions for some properties such as stationarity, existence of moments, ergodicity, geometric ergodicity, consistence and asymptotic normality of likelihood estimators of the model parameters. For some of these properties we show that the conditions that we propose are also necessary. We further provide examples and applications to illustrate and highlight the importance of our findings.

math.ST↗