Thorsten Schmidt

Thorsten Schmidt
University of Freiburg | Albert-Ludwigs-Universität Freiburg · Institute of Mathematics

Ph.D.

About

125
Publications
21,196
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799
Citations
Citations since 2017
60 Research Items
419 Citations
2017201820192020202120222023020406080100120
2017201820192020202120222023020406080100120
2017201820192020202120222023020406080100120
2017201820192020202120222023020406080100120
Additional affiliations
April 2015 - present
University of Freiburg
Position
  • Professor (Full)
October 2008 - present
Technische Universität Chemnitz
Position
  • Professor (Full)
January 2004 - September 2008
University of Leipzig
Position
  • Juniorprofessor

Publications

Publications (125)
Preprint
In most cases, insurance contracts are linked to the financial markets, such as through interest rates or equity-linked insurance products. To motivate an evaluation rule in these hybrid markets, Artzner et al. (2022) introduced the notion of insurance-finance arbitrage. In this paper we extend their setting by incorporating model uncertainty. To t...
Book
In the past decade, artificial intelligence (AI) has become a disruptive force around the world, offering enormous potential for innovation but also creating hazards and risks for individuals and the societies in which they live. This volume addresses the most pressing philosophical, ethical, legal, and societal challenges posed by AI. Contributors...
Preprint
In this work we consider one-dimensional generalized affine processes under the paradigm of Knightian uncertainty (so-called non-linear generalized affine models). This extends and generalizes previous results in Fadina et al. (2019) and L\"utkebohmert et al. (2022). In particular, we study the case when the payoff is allowed to depend on the path,...
Preprint
Estimating value-at-risk on time series data with possibly heteroscedastic dynamics is a highly challenging task. Typically, we face a small data problem in combination with a high degree of non-linearity, causing difficulties for both classical and machine-learning estimation algorithms. In this paper, we propose a novel value-at-risk estimator us...
Article
We study pricing and hedging under parameter uncertainty for a class of Markov processes which we call generalized affine processes and which includes the Black–Scholes model as well as the constant elasticity of variance (CEV) model as special cases. Based on a general dynamic programming principle, we are able to link the associated nonlinear exp...
Article
Longitudinal biomedical data are often characterized by a sparse time grid and individual‐specific development patterns. Specifically, in epidemiological cohort studies and clinical registries we are facing the question of what can be learned from the data in an early phase of the study, when only a baseline characterization and one follow‐up measu...
Article
In this paper, we study optimal mortgage decisions in a cross-currency setting. In particular, we address the question on how a household should optimally split its mortgage portfolio in a fixed rate mortgage in the domestic currency and an adjustable rate mortgage denominated in a foreign currency subject to some risk constraints. We propose an af...
Preprint
In the current reform of interest rate benchmarks, a central role is played by risk-free rates (RFRs), such as SOFR (secured overnight financing rate) in the US. A key feature of RFRs is the presence of jumps and spikes at periodic time intervals as a result of regulatory and liquidity constraints. This corresponds to stochastic discontinuities (i....
Article
While the estimation of risk is an important question in the daily business of banking and insurance, many existing plug-in estimation procedures suffer from an unnecessary bias. This often leads to the underestimation of risk and negatively impacts backtesting results, especially in small sample cases. In this article we show that the link between...
Preprint
While the estimation of risk is an important question in the daily business of banking and insurance, many existing plug-in estimation procedures suffer from an unnecessary bias. This often leads to the underestimation of risk and negatively impacts backtesting results, especially in small sample cases. In this article we show that the link between...
Book
Alle Systeme, Produkte und Dienstleistungen, die Künstliche Intelligenz nutzen oder von ihr gesteuert werden (im Folgenden: KI-Systeme), bergen gewisse Risi- ken. Da ihre Algorithmen von Menschen programmiert sind, sind wir Menschen auch verantwortlich, wenn sich ein solches Risiko verwirklicht. Die Staaten der internationalen Gemeinschaft müssen a...
Article
We study a Sparre Andersen model in which the business activity of the company is described by a compound renewal process with drift assuming that the capital reserves are invested in a risky asset. The price of the latter is assumed to evolve according to a geometric Lévy process. We prove that the asymptotic behavior of the ruin probability depen...
Article
In this paper, we consider a market with a term structure of credit risky bonds in the single-name case. We aim at minimal assumptions extending existing results in this direction: first, the random field of forward rates is driven by a general semimartingale. Second, the Heath–Jarrow–Morton (HJM) approach is extended with an additional component c...
Preprint
We use the abstract method of (local) martingale problems in order to give criteria for convergence of stochastic processes. Extending previous notions, the formulation we use is neither restricted to Markov processes (or semimartingales), nor to continuous or cadlag paths. We illustrate our findings both, by finding generalizations of known result...
Article
This paper proposes a framework for the valuation and the management of complex life insurance contracts, whose design can be described by a portfolio of embedded options, which are activated according to one or more triggering events. These events are in general monitored discretely over the life of the policy, due to the contract terms. Similar d...
Preprint
Full-text available
We study pricing and hedging under parameter uncertainty for a class of Markov processes which we call {\it generalized affine processes} and which includes the Black-Scholes model as well as the constant elasticity of variance (CEV) model as special cases. Based on a general dynamic programming principle, we are able to link the associated nonline...
Preprint
In this work we consider a market with a term structure of credit risky bonds in the single-name case. We aim at minimal assumptions extending existing results in this direction. First, the random field of forward rates should be driven by general semimartingales. We show that in this case, it is necessary to extend the Heath-Jarrow-Morton approach...
Preprint
The second fundamental theorem of asset pricing characterizes completeness of a financial market by uniqueness of prices of financial claims. The associated super- and sub-hedging dualities give upper and lower bounds of the no-arbitrage interval. In this article we provide conditional versions of these results in discrete time. The main tool we us...
Article
Full-text available
The notion of statistical arbitrage introduced in Bondarenko (2003) is generalized to statistical ‐arbitrage corresponding to trading strategies which yield positive gains on average in a class of scenarios described by a ‐algebra . This notion contains classical arbitrage as a special case. Admitting general static payoffs as generalized strategie...
Preprint
Full-text available
We study a Sparre Andersen model in which the business activity of the company is described by a compound renewal process with drift assuming that the capital reserves are invested in a risky asset. The price of the latter is assumed to evolve according to a geometric L\'evy process. We prove that the asymptotic behavior of the ruin probability dep...
Preprint
Longitudinal biomedical data are often characterized by a sparse time grid and individual-specific development patterns. Specifically, in epidemiological cohort studies and clinical registries we are facing the question of what can be learned from the data in an early phase of the study, when only a baseline characterization and one follow-up measu...
Preprint
While the estimation of risk is an important question in the daily business of banks and insurances, it is surprising that efficient procedures for this task are not well studied. Indeed, many existing plug-in approaches for the estimation of risk suffer from an unnecessary bias which leads to the underestimation of risk and negatively impacts back...
Article
The goal of this article is to investigate infinite dimensional affine diffusion processes on the canonical state space. This includes a derivation of the corresponding system of Riccati differential equations and an existence proof for such processes, which has been missing in the literature so far. For the existence proof, we will regard affine p...
Preprint
Full-text available
This paper is an attempt to study fundamentally the valuation of insurance contracts. We start from the observation that insurance contracts are inherently linked to financial markets, be it via interest rates, or -- as in hybrid products, equity-linked life insurance and variable annuities -- directly to stocks or indices. By defining portfolio st...
Article
Full-text available
Calibration is a highly challenging task, in particular in multiple yield curve markets. This paper is a first attempt to study the chances and challenges of the application of machine learning techniques for this. We employ Gaussian process regression, a machine learning methodology having many similarities with extended Kálmán filtering, which ha...
Preprint
Calibration is a highly challenging task, in particular in multiple yield curve markets. This paper is a first attempt to study the chances and challenges of the application of machine learning techniques for this. We employ Gaussian process regression, a machine learning methodology having many similarities with extended Kalman filtering - a techn...
Article
Full-text available
We develop a general term structure framework taking stochastic discontinuities explicitly into account. Stochastic discontinuities are a key feature in interest rate markets, as for example the jumps of the term structures in correspondence to monetary policy meetings of the ECB show. We provide a general analysis of multiple curve markets under m...
Article
In this paper, we develop a novel methodology for estimation of risk capital allocation. The methodology is rooted in the theory of risk measures. We work within a general, but tractable class of law-invariant coherent risk measures, with a particular focus on expected shortfall. We introduce the concept of fair capital allocations and provide expl...
Article
Calibration is a highly challenging task, in particular in multiple yield curve markets. This paper is a first attempt to study the chances and challenges of the application of machine learning techniques for this. We employ Gaussian process regression, a machine learning methodology having many similarities with extended Kálmán filtering, which ha...
Article
In this work, we develop polynomial processes where the coefficients of the process may depend on time. A full characterization of this model class is given by means of their semimartingale characteristics. While in the time-homogeneous case, moments can be calculated easily through matrix exponentials, we show that in the inhomogeneous case this i...
Article
This paper proposes a market consistent valuation framework for variable annuities (VAs) with guaranteed minimum accumulation benefit, death benefit and surrender benefit features. The setup is based on a hybrid model for the financial market and uses time-inhomogeneous Lévy processes as risk drivers. Further, we allow for dependence between financ...
Preprint
The goal of this article is to investigate infinite dimensional affine diffusion processes on the canonical state space. This includes a derivation of the corresponding system of Riccati differential equations and an existence proof for such processes, which has been missing in the literature so far. For the existence proof, we will regard affine p...
Preprint
Full-text available
Generalized statistical arbitrage concepts are introduced corresponding to trading strategies which yield positive gains on average in a class of scenarios rather than almost surely. The relevant scenarios or market states are specified via an information system given by a $\sigma$-algebra and so this notion contains classical arbitrage as a specia...
Article
Full-text available
This paper discusses ambiguity in the context of single-name credit risk. We focus on uncertainty in the default intensity but also discuss uncertainty in the recovery in a fractional recovery of the market value. This approach is a first step towards integrating uncertainty in credit-risky term structure models and can profit from its simplicity....
Preprint
This paper proposes a market consistent valuation framework for variable annuities with guaranteed minimum accumulation benefit, death benefit and surrender benefit features. The setup is based on a hybrid model for the financial market and uses time-inhomogeneous Lévy processes as risk drivers. Further, we allow for dependence between financial an...
Article
Full-text available
We develop a one-dimensional notion of affine processes under parameter uncertainty, which we call nonlinear affine processes. This is done as follows: given a set Θ of parameters for the process, we construct a corresponding nonlinear expectation on the path space of continuous processes. By a general dynamic programming principle, we link this no...
Preprint
Full-text available
In this paper we develop a novel methodology for estimation of risk capital allocation. The methodology is rooted in the theory of risk measures. We work within a general, but tractable class of law-invariant coherent risk measures, with a particular focus on expected shortfall. We introduce the concept of fair capital allocations and provide expli...
Preprint
The goal of the paper is twofold. On the one hand, we develop the first term structure framework which takes stochastic discontinuities explicitly into account. Stochastic discontinuities are a key feature in interest rate markets, as for example the jumps of the term structures in correspondence to monetary policy meetings of the ECB show. On the...
Preprint
Time homogeneous polynomial processes are Markov processes whose moments can be calculated easily through matrix exponentials. In this work, we develop a notion of time inhomogeneous polynomial processes where the coeffiecients of the process may depend on time. A full characterization of this model class is given by means of their semimartingale c...
Preprint
Full-text available
We develop a one-dimensional notion of affine processes under parameter uncertainty, which we call non-linear affine processes. This is done as follows: given a set of parameters for the process, we construct a corresponding non-linear expectation on the path space of continuous processes. By a general dynamic programming principle we link this non...
Article
In this paper we study time-inhomogeneous affine processes beyond the common assumption of stochastic continuity. In this setting times of jumps can be both inaccessible and predictable. To this end we develop a general theory of finite dimensional affine semimartingales under very weak assumptions. We show that the corresponding semimartingale cha...
Article
Full-text available
The estimation of risk measures recently gained a lot of attention, partly because of the backtesting issues of expected shortfall related to elicitability. In this work we shed a new and fundamental light on optimal estimation procedures of risk measures in terms of bias. We show that once the parameters of a model need to be estimated, one has to...
Article
We introduce the concept of no-arbitrage in a credit risk market under ambiguity considering an intensity-based framework. We assume the default intensity is not exactly known but lies between an upper and lower bound. By means of the Girsanov theorem, we start from the reference measure where the intensity is equal to $1$ and construct the set of...
Chapter
Shot-Noise processes constitute a useful tool in various areas, in particular in finance. They allow to model abrupt changes in a more flexible way than processes with jumps and hence are an ideal tool for modelling stock prices, credit portfolio risk, systemic risk, or electricity markets. Here we consider a general formulation of shot-noise proce...
Book
This book, dedicated to Winfried Stute on the occasion of his 70th birthday, presents a unique collection of contributions by leading experts in statistics, stochastic processes, mathematical finance and insurance. The individual chapters cover a wide variety of topics ranging from nonparametric estimation, regression modelling and asymptotic bound...
Article
Shot-Noise processes constitute a useful tool in various areas, in particular in finance. They allow to model abrupt changes in a more flexible way than processes with jumps and hence are an ideal tool for modelling stock prices, credit portfolio risk, systemic risk, or electricity markets. Here we consider a general formulation of shot-noise proce...
Chapter
Full-text available
The intensity of a default time is obtained by assuming that the default indicator process has an absolutely continuous compensator. Here we drop the assumption of absolute continuity with respect to the Lebesgue measure and only assume that the compensator is absolutely continuous with respect to a general \(\sigma \)-finite measure. This allows f...
Chapter
We investigate default-free bond markets and relax assumptions on the numéraire, which is typically chosen to be the bank account. Considering numéraires different from the bank account allows us to study bond markets where the bank account process is not a valid numéraire or does not exist at all. We argue that this feature is not the exception, b...
Article
The estimation of risk measured in terms of a risk measure is typically done in two steps: in the first step, the distribution is estimated by statistical methods, either parametric or non-parametric. In the second step, the estimated distribution is considered as true distribution and the targeted risk-measure is computed. In the parametric case t...
Article
We consider the problem of modelling the term structure of bonds subject to default risk, under minimal assumptions on the default time. In particular, we do not assume the existence of a default intensity and we therefore allow for the possibility of default at predictable times. It turns out that this requires the introduction of an additional te...
Article
The intensity of a default time is obtained by assuming that the default indicator process has an absolutely continuous compensator. Here we drop the assumption of absolute continuity with respect to the Lebesgue measure and only assume that the compensator is absolutely continuous with respect to a general $\sigma$-finite measure. This allows for...
Article
Der Industriestandard bei der Messung des Downside Risks von Finanzpositionen, Value at Risk (VaR), weist schwerwiegende Defizite auf. Der Artikel diskutiert Eigenschaften von Risikomaßen und Alternativen zu VaR in systematischer Weise. Gute Eigenschaften werden bei Utiliy-based Shortfall Risk (UBSR) nachgewiesen. Abstract The industry standard...
Article
The two main approaches in credit risk are the structural approach pioneered in Merton (1974) and the reduced-form framework proposed in Jarrow & Turnbull (1995) and in Artzner & Delbaen (1995). The goal of this article is to provide a unified view on both approaches. This is achieved by studying reduced-form approaches under weak assumptions. In p...
Article
Full-text available
Shot-noise processes generalize compound Poisson processes in the following way: a jump (the shot) is followed by a decline (noise). This constitutes a useful model for insurance claims in many circumstances; claims due to natural disasters or self-exciting processes exhibit similar features. We give a general account of shot-noise processes with t...
Article
Full-text available
We investigate default-free bond markets where the standard relationship between a possibly existing bank account process and the term structure of bond prices is broken, i.e. the bank account process is not a valid num\'eraire. We argue that this feature is not the exception but rather the rule in bond markets when starting with, e.g., terminal bo...
Article
The goal of this article is to study in detail the pricing and calibration in mar-ket models for credit portfolios. Starting from the framework of market models driven by time-inhomogeneous Lévy processes in a top-down approach proposed in Eberlein, Grbac, and Schmidt (2012) we consider a slightly simplified setup which eases calibration. This lead...
Article
The goal of this paper is to specify dynamic term structure models with discrete tenor structure for credit portfolios in a top-down setting driven by time-inhomogeneous Lévy processes. We provide a new framework, conditions for absence of arbitrage, explicit examples, an affine setup which includes contagion, and pricing formulas for single tranch...
Article
Full-text available
This paper considers general term structure models like the ones appearing in portfolio credit risk modelling or life insurance. We give a general model starting from families of forward rates driven by infinitely many Brownian motions and an integer-valued random measure, generalizing existing approaches in the literature. Then we derive drift con...
Article
Full-text available
The recent financial crisis, responsible for massive accumulations of credit events, emphasizes the urgent need for adequate portfolio default models. Due to the high dimensionality of real credit portfolios, balancing flexibility and numerical tractability is of uttermost importance. To acknowledge this, a multivariate default model with interesti...
Article
Background: The aim of this study was to determine the prognostic utility of serial measurement of the cardiovascular biomarker midregion proadrenomedullin (MR-proADM) in patients admitted with lower respiratory tract infection. In a previous trial in dyspneic patients (BACH trial) we could show that serial measurement of MR-proADM proves useful f...
Article
The goal of this paper is to specify market models for credit portfolios in a top-down setting driven by time-inhomogeneous Levy processes. We provide a new framework, conditions for absence of arbitrage, explicit examples, an affine setup which includes contagion and pricing formulas for STCDOs and options on STCDOs. A calibration to iTraxx data w...
Article
Full-text available
In this paper, we propose a new, information-based approach for modelling the dynamic evolution of a portfolio of credit risky securities. In our setup, market prices of traded credit derivatives are given by the solution of a nonlinear filtering problem. The innovations approach to nonlinear filtering is used to solve this problem and to derive th...
Article
Full-text available
This paper studies Galerkin approximations applied to the Zakai equation of stochastic fil-tering. The basic idea of this approach is to project the infinite-dimensional Zakai equation onto some finite-dimensional subspace; this leads to a finite-dimensional system of stochastic differential equations that can be solved numerically. The contributio...
Article
Statistik ist die Wissenschaft, die Regeln und Verfahren für die Erhebung, Beschreibung, Analyse und Interpretation von numerischen Daten entwickelt. Der Schwerpunkt dieses Buches liegt auf der Entwicklung und Darstellung von statistischen Analyseverfahren. Dazu werden stochastische Modelle vorgestellt, die von unbekannten Parametern abhängen. Um...
Article
Dieses Kapitel beschäftigt sich mit der Optimalität von Schätzern. Hierfür wird der klassische Zugang der Effizienz, welche am mittlerem quadratischen Abstand von dem zu schätzenden Parameter gemessen wird, betrachtet. Es stellt sich heraus, das zusätzlich zu einem Abstandskriterium eine zweite Bedingung, die Unverzerrtheit, gefordert werden muss,...
Article
In diesem Kapitel studieren wir die Optimalität von Tests. Zu Beginn werden die zentralen Resultate von Neyman und Pearson vorgestellt, welche eine Klasse von optimalen Tests basierend auf Likelihood-Quotienten behandeln. Diese Optimalität gilt zunächst nur unter ganz einfachen Hypothesen θ = θ 0 gegen θ = θ 1. Allerdings lassen sich diese Ergebnis...
Article
Ziel von linearen Modellen ist es, Abhängigkeiten zwischen einer Zielvariablen und beobachteten Einflussgröβen zu studieren. Die Zielvariable Y wird auch als abhängige oder endogene Variable bezeichnet, im Englischen wird der Begriff Response verwendet. Die bekannten Einflussgröβen x 1,...,x k werden als Kovariablen, unabhängige oder exogene Variab...
Article
Hierbei bezeichnet x den Vektor der Messergebnisse oder Beobachtungen und X die zugehörige Zufallsvariable. Der Parameter θ ist unbekannt und typischerweise möchte man θ selbst schätzen. Es kommt allerdings vor, dass man nicht direkt den Parameter θ schätzen möchte, Dies wird mit den folgenden beiden Beispielen illustriert. Qualitätssicherung aus B...
Article
Dieses Kapitel stellt zunächst Konfidenzintervalle im ein- und mehrdimensionalen Fall vor und behandelt danach Hypothesentests nach dem Ansatz von Neyman und Pearson. Abschließend wird die Dualität zwischen den beiden Begriffen erläutert.
Chapter
Die Formulierung von statistischen Modellen bildet die Grundlage der Statistik. Hierbei werden Modelle ausgewählt, welche der Realität zum einen möglichst gut entsprechen sollen, zum anderen die für die statistische Analyse notwendige Handhabbarkeit besitzen. Das statistische Modell beschreibt stets das Ergebnis eines Zufallsexperiments, etwa die W...
Article
This paper provides a unifying approach for valuing contingent claims on a portfolio of credits, such as collateralized debt obligations (CDOs). We introduce the defaultable (T, x)-bonds, which pay one if the aggregated loss process in the underlying pool of the CDO has not exceeded x at maturity T, and zero else. Necessary and sufficient condition...
Article
Full-text available
This paper provides a general framework for doubly stochastic term structure models for portfolio of credits, such as collateralized debt obli-gations (CDOs). We introduce the defaultable (T, x)-bonds, which pay one if the aggregated loss process in the underlying pool of the CDO has not exceeded x at maturity T , and zero else. Necessary and suffi...
Chapter
Copulas are a general tool for assessing the dependence structure of random variables. Important properties as well as several examples are discussed, including Archimedean copulas and the Marshall–Olkin copula. As measures of dependence, we consider linear correlation, rank correlation, the coefficients of tail dependence, and association.
Chapter
Correlation is a well-established concept to capture the linear relationship between two or more variables. This article covers basic properties, fallacies of correlation, and correlation risk in financial applications.
Article
In this work we study a form of shot-noise processes which is driven by Levy subordinators. The main focus is on applications to portfolios which are subject to credit risk. We show how to augment an arbitrary model for credit risk (e.g. an ane model) with shot-noise processes. This introduceds clustering of defaults into the original model, which...
Article
This chapter studies structural and reduced-form credit risk models under incomplete information. Applying stochastic filtering techniques we tackle the aspect of incomplete information in different settings: starting with a brief introduction to stochastic filtering we thereafter cover the pricing of corporate securities (debt and equity) in struc...
Article
Full-text available
This paper considers the modelling of collateralized debt obligations (CDOs). We propose a top-down model via forward rates generalizing D. Filipović, L. Overbeck and T. Schmidt [Math. Finance 21, No. 1, 53–71 (2011; Zbl 1229.91306)] to the case where the forward rates are driven by a finite dimensional Lévy process. The contribution of this work i...
Article
Full-text available
We present the SPA framework, a novel approach to the modeling of the dynamics of portfolio default losses. In this framework, models are specified by a two-layer process. The first layer models the dynamics of portfolio loss distributions in the absence of information about default times. This background process can be explicitly calibrated to the...
Article
This paper considers the pricing and hedging of collateralized debt obligations (CDOs). CDOs are complex derivatives on a pool of credits which we choose to analyse in the top down model proposed in Filipovic et al. (2009). We reflect on the implied forward rates and bring them in connection with the top-down framework in Lipton and Shelton (2009)...
Article
This paper considers the pricing of corporate securities of a given firm, in particular equity, when investors do not have full information on the firm's asset value. We show that under noisy asset information, the pricing of corporate securities leads to a nonlinear filtering problem. This problem is solved by a Markov chain approximation, leading...
Chapter
Full-text available
This paper proposes a top-down model for pricing Collateralized Debt Obligation CDOs). Our proposal is both treatable and realistic, in the sense we are able to obtain closed-form solutions to single tranche CDOs and capturing extreme credit events. We use as key ingredients the so-called (T, x)-bonds, as proposed in Filipovic, Overbeck, and Schmid...

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Project
describe by means of Bayesian techniques platonic financial markets, i.e. markets where trades do not have full information on prices.