arXiv subjects

Gero Junike

Publications and source records attributed to Gero Junike.

14 recordsLinked to original sources

arXiv2026

How the interpolation of life tables affects the decomposition of life insurance surplus

The surplus of a life insurance policy depends on both systematic changes in mortality risk and financial changes. We propose to decompose the surplus by the axiomatically justified IASU decomposition, which is a continuous time version of the Shapley value. However, life tables are not updated continuously, but rather, only once per year. In this yearly update cycle of the life tables, we apply different interpolation methods to perform the IASU decomposition and analyze the effects of these methods on the surplus decomposition. Our results show that Lee-Carter and linear interpolation yield almost identical decompositions, whereas constant approximations results in substantially different decompositions. As a consequence, reporting standards and regulators should clarify how to interpolate mortality risks.

q-fin.ST↗

arXiv2025

Accuracy estimation of neural networks by extreme value theory

Neural networks are able to approximate any continuous function on a compact set. However, it is not obvious how to quantify the error of the neural network, i.e., the remaining bias between the function and the neural network. Here, we propose the application of extreme value theory to quantify large values of the error, which are typically relevant in applications. The distribution of the error beyond some threshold is approximately generalized Pareto distributed. We provide a new estimator of the shape parameter of the Pareto distribution suitable to describe the error of neural networks. Numerical experiments are provided.

stat.ML↗

arXiv2025

Batch normalization does not improve initialization

Batch normalization is one of the most important regularization techniques for neural networks, significantly improving training by centering the layers of the neural network. There have been several attempts to provide a theoretical justification for batch ormalization. Santurkar and Tsipras (2018) [How does batch normalization help optimization? Advances in neural information rocessing systems, 31] claim that batch normalization improves initialization. We provide a counterexample showing that this claim s not true, i.e., batch normalization does not improve initialization.

cs.LG↗

arXiv2025

Precise quantile function estimation from the characteristic function

We provide theoretical error bounds for the accurate numerical computation of the quantile function given the characteristic function of a continuous random variable. We show theoretically and empirically that the numerical error of the quantile function is typically several orders of magnitude larger than the numerical error of the cumulative distribution function for probabilities close to zero or one. We introduce the COS method for computing the quantile function. This method converges exponentially when the density is smooth and has semi-heavy tails and all parameters necessary to tune the COS method are given explicitly. Finally, we numerically test our theoretical results on the normal-inverse Gaussian and the tempered stable distributions.

math.ST↗

arXiv2024

Profit and loss decomposition in continuous time and approximations

Financial institutions and insurance companies that analyze the evolution and sources of profits and losses often look at risk factors only at discrete reporting dates, ignoring the detailed paths. Continuous-time decompositions avoid this weakness and also make decompositions consistent across different reporting grids. We construct a large class of continuous-time decompositions from a new extended version of Itô's formula and uniquely identify a preferred decomposition from the axioms of exactness, symmetry and normalization. This unique decomposition turns out to be a stochastic limit of recursive Shapley values, but it suffers from a curse of dimensionality as the number of risk factors increases. We develop an approximation that breaks this curse when the risk factors almost surely have no simultaneous jumps.

q-fin.MF↗

arXiv2024

Validation of machine learning based scenario generators

Machine learning (ML) methods are becoming increasingly important in the design economic scenario generators for internal models. Validation of data-driven models differs from classical theory-based models. We discuss two novel aspects of such a validation: first, checking dependencies between risk factors and second, detecting unwanted memorization effects. The first task becomes necessary since in ML-based methods dependencies are no longer derived from a financial-mathematical theory but are driven by data. The need for the latter task arises since it cannot be ruled out that ML-based models merely reproduce the empirical data rather than generating new scenarios. To address the first issue, we propose to use an existing test from the literature. For the second issue, we introduce and discuss a novel memorization ratio. Numerical experiments based on real market data are included and an autoencoder-based scenario generator is validated with these two methods.

q-fin.RM↗

arXiv2024

Enhancing Fourier pricing with machine learning

Fourier pricing methods such as the Carr-Madan formula or the COS method are classic tools for pricing European options for advanced models such as the Heston model. These methods require tuning parameters such as a damping factor, a truncation range, a number of terms, etc. Estimating these tuning parameters is difficult or computationally expensive. Recently, machine learning techniques have been proposed for fast pricing: they are able to learn the functional relationship between the parameters of the Heston model and the option price. However, machine learning techniques suffer from error control and require retraining for different error tolerances. In this research, we propose to learn the tuning parameters of the Fourier methods (instead of the prices) using machine learning techniques. As a result, we obtain very fast algorithms with full error control: Our approach works with any error tolerance without retraining, as demonstrated in numerical experiments using the Heston model.

q-fin.MF↗

arXiv2024

Failure of Fourier pricing techniques to approximate the Greeks

The Greeks Delta and Gamma of plain vanilla options play a fundamental role in finance, e.g., in hedging or risk management. These Greeks are approximated in many models such as the widely used Variance Gamma model by Fourier techniques such as the Carr-Madan formula, the COS method or the Lewis formula. However, for some realistic market parameters, we show empirically that these three Fourier methods completely fail to approximate the Greeks. As an application we show that the Delta-Gamma VaR is severely underestimated in realistic market environments. As a solution, we propose to use finite differences instead to obtain the Greeks.

q-fin.CP↗

arXiv2024

On the number of terms in the COS method for European option pricing

The Fourier-cosine expansion (COS) method is used to price European options numerically in a very efficient way. To apply the COS method, one has to specify two parameters: a truncation range for the density of the log-returns and a number of terms N to approximate the truncated density by a cosine series. How to choose the truncation range is already known. Here, we are able to find an explicit and useful bound for N as well for pricing and for the sensitivities, i.e., the Greeks Delta and Gamma, provided the density of the log-returns is smooth. We further show that the COS method has an exponential order of convergence when the density is smooth and decays exponentially. However, when the density is smooth and has heavy tails, as in the Finite Moment Log Stable model, the COS method does not have exponential order of convergence. Numerical experiments confirm the theoretical results.

q-fin.CP↗

arXiv2023

Profit and loss attribution: An empirical study

The profit and loss (p&l) attrition for each business year into different risk or risk factors (e.g., interest rates, credit spreads, foreign exchange rate etc.) is a regulatory requirement, e.g., under Solvency 2. Three different decomposition principles are prevalent: one-at-a-time (OAT), sequential updating (SU) and average sequential updating (ASU) decompositions. In this research, using financial market data from 2003 to 2022, we demonstrate that the OAT decomposition can generate significant unexplained p&l and that the SU decompositions depends significantly on the order or labeling of the risk factors. On the basis of an investment in a foreign stock, we further explain that the SU decomposition is not able to identify all relevant risk factors. This potentially effects the hedging strategy of the portfolio manager. In conclusion, we suggest to use the ASU decomposition in practice.

q-fin.PM↗

arXiv2023

Scenario generation for market risk models using generative neural networks

In this research, we show how to expand existing approaches of using generative adversarial networks (GANs) as economic scenario generators (ESG) to a whole internal market risk model - with enough risk factors to model the full band-width of investments for an insurance company and for a one year time horizon as required in Solvency 2. We demonstrate that the results of a GAN-based internal model are similar to regulatory approved internal models in Europe. Therefore, GAN-based models can be seen as a data-driven alternative way of market risk modeling.

cs.LG↗

arXiv2023

Cost-efficient Payoffs under Model Ambiguity

Dybvig (1988a,b) solves in a complete market setting the problem of finding a payoff that is cheapest possible in reaching a given target distribution ("cost-efficient payoff"). In the presence of ambiguity, the distribution of a payoff is, however, no longer known with certainty. We study the problem of finding the cheapest possible payoff whose worst-case distribution stochastically dominates a given target distribution ("robust cost-efficient payoff") and determine solutions under certain conditions. We study the link between "robust cost-efficiency" and the maxmin expected utility setting of Gilboa and Schmeidler, as well as more generally with robust preferences in a possibly non-expected utility setting. Specifically, we show that solutions to maxmin robust expected utility are necessarily robust cost-efficient. We illustrate our study with examples involving uncertainty both on the drift and on the volatility of the risky asset.

q-fin.PM↗

arXiv2023

From characteristic functions to multivariate distribution functions and European option prices by the damped COS method

We provide a unified framework to obtain numerically certain quantities, such as the distribution function, absolute moments and prices of financial options, from the characteristic function of some (unknown) probability density function using the Fourier-cosine expansion (COS) method. The classical COS method is numerically very efficient in one-dimension, but it cannot deal very well with certain integrands in general dimensions. Therefore, we introduce the damped COS method, which can handle a large class of integrands very efficiently. We prove the convergence of the (damped) COS method and study its order of convergence. The method converges exponentially if the characteristic function decays exponentially. To apply the (damped) COS method, one has to specify two parameters: a truncation range for the multivariate density and the number of terms to approximate the truncated density by a cosine series. We provide an explicit formula for the truncation range and an implicit formula for the number of terms. Numerical experiments up to five dimensions confirm the theoretical results.

q-fin.CP↗

arXiv2022

Precise option pricing by the COS method--How to choose the truncation range

The Fourier cosine expansion (COS) method is used for pricing European options numerically very fast. To apply the COS method, a truncation range for the density of the log-returns need to be provided. Using Markov's inequality, we derive a new formula to obtain the truncation range and prove that the range is large enough to ensure convergence of the COS method within a predefined error tolerance. We also show by several examples that the classical approach to determine the truncation range by cumulants may lead to serious mispricing. Usually, the computational time of the COS method is of similar magnitude in both cases.

q-fin.CP↗