Departement Mathematik

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Herbstsemester 2022

Datum / Zeit Referent:in Titel Ort
22. September 2022
17:15-18:15 		Dr. Stephan Tobias Eckstein
ETH Zurich, Switzerland
Details

Talks in Financial and Insurance Mathematics

Titel Causal optimal transport with general information structures
Referent:in, Affiliation Dr. Stephan Tobias Eckstein, ETH Zurich, Switzerland
Datum, Zeit 22. September 2022, 17:15-18:15
Ort HG G 43
Abstract We present a framework of optimal transport aimed at probability measures arising from structural causal models, i.e., models that contain an information structure dictated by a directed graph. This builds on the recent string of literature (called causal or adapted optimal transport) studying this concept for time-series distributions, which has found many applications in financial contexts. We recover the adapted optimal transport problem for a particular temporal graph structure, and additionally show that other special cases like the standard optimal transport problem (fully connected graph) or problems related to Gromov-Wasserstein (graph without edges) are also included in the setting. The concept of coupling for a particular graph structure is characterized in various ways, giving rise to different interpretations and numerical possibilities. Further, we show that the resulting concept of Wasserstein distance can be used to control the difference between average treatment effects under different distributions, and is geometrically suitable to interpolate between different structural causal models.
Causal optimal transport with general information structuresread_more
HG G 43
29. September 2022
17:15-18:15 		Prof. Dr. Eckhard Platen
UTS Sydney
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Talks in Financial and Insurance Mathematics

Titel Modeling Long-Term Stock Market Dynamics via Entropy Maximization
Referent:in, Affiliation Prof. Dr. Eckhard Platen, UTS Sydney
Datum, Zeit 29. September 2022, 17:15-18:15
Ort HG G 43
Abstract By assuming the existence of the growth optimal portfolio (GP) and maximizing the entropy for a hierarchically structured stock market, the paper derives the GP dynamics as those of time-transformed squared Bessel processes of dimension four. The average of each of their squared volatility components is shown to converge toward a common level. The initial values of basis security accounts turn out to be gamma-distributed. The risk-adjusted return of the GP is not depending on a savings account and can be significantly higher than classical assumptions allow to explain.
Modeling Long-Term Stock Market Dynamics via Entropy Maximizationread_more
HG G 43
6. Oktober 2022
17:15-18:15 		Prof. Dr. Qiji Jim Zhu
Western Michigan University
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Talks in Financial and Insurance Mathematics

Titel Risk management in algorithmic trading
Referent:in, Affiliation Prof. Dr. Qiji Jim Zhu, Western Michigan University
Datum, Zeit 6. Oktober 2022, 17:15-18:15
Ort HG G 43
Abstract Algorithmic trading has become increasingly influential in financial industry. However, not all algorithmic trading is successful. We will analyze important ingredients of a typical algorithmic trading strategy: find a trading signal, backtest, money management, validation, and suspension. Risk management methods play important roles in algorithmic trading. In this introductory talk we will focus on the main ideas and limiting the technical details so that the talk is accessible to undergraduate students.
Risk management in algorithmic tradingread_more
HG G 43
13. Oktober 2022
17:15-18:15 		Prof. Dr. Marcel Nutz
Columbia University, New York, USA
Details

Talks in Financial and Insurance Mathematics

Titel Martingale Transports and Monge Maps
Referent:in, Affiliation Prof. Dr. Marcel Nutz, Columbia University, New York, USA
Datum, Zeit 13. Oktober 2022, 17:15-18:15
Ort HG G 43
Abstract It is well known that martingale transport plans between marginals $\mu\neq\nu$ are never given by Monge maps---with the understanding that the map is over the first marginal $\mu$, or forward in time. Here, we change the perspective, with surprising results. We show that any distributions $\mu,\nu$ in convex order with $\nu$ atomless admit a martingale coupling given by a Monge map over the \emph{second} marginal $\nu$. Namely, we construct a particular coupling called the barcode transport. Much more generally, we prove that such ``backward Monge'' martingale transports are dense in the set of all martingale couplings, paralleling the classical denseness result for Monge transports in the Kantorovich formulation of optimal transport. Various properties and applications are presented, including a refined version of Strassen's theorem and a mimicking theorem where the marginals of a given martingale are reproduced by a ``backward deterministic'' martingale, a remarkable type of process whose current state encodes its whole history. Joint work with Ruodu Wang (Waterloo) and Zhenyuan Zhang (Stanford).
Martingale Transports and Monge Mapsread_more
HG G 43
20. Oktober 2022
17:15-18:15 		Valentin Tissot-Daguette
Princeton University, USA
Details

Talks in Financial and Insurance Mathematics

Titel Functional Expansions, Signature and Claim Decomposition
Referent:in, Affiliation Valentin Tissot-Daguette, Princeton University, USA
Datum, Zeit 20. Oktober 2022, 17:15-18:15
Ort HG G 43
Abstract Functionals are omnipresent in finance, whether it be the payoff of a claim, a hedging strategy, or path-dependent volatility. However, problems involving functionals are often infinite-dimensional and thus challenging from a conceptual and computational perspective. In the first part of the talk, we present a simulation algorithm based on the Karhunen-Loève expansion to efficiently price exotic options. In the second, we compare static expansions (Volterra/Wiener series as well as the novel intrinsic value expansion) and promote the functional Taylor expansion (FTE). The latter combines the Functional Itô Calculus with the signature to quantify the effect in a functional when a “shock" path is concatenated with the source path. The notions of analytic functionals and radius of convergence in the path space are then defined. We finally apply the FTE to decompose exotic claims. This is joint work with Bruno Dupire (Bloomberg LP).
Functional Expansions, Signature and Claim Decompositionread_more
HG G 43
27. Oktober 2022
17:15-18:15 		Prof. Dr. Julien Guyon
CERMICS, École des Ponts ParisTech
Details

Talks in Financial and Insurance Mathematics

Titel Volatility Is (Mostly) Path-Dependent
Referent:in, Affiliation Prof. Dr. Julien Guyon, CERMICS, École des Ponts ParisTech
Datum, Zeit 27. Oktober 2022, 17:15-18:15
Ort HG G 43
Abstract We learn from data that volatility is mostly path-dependent: at least 85-90% of the variance of the implied volatility of equity indexes is explained endogenously by past index returns, and around 60% for (noisy estimates of) future daily realized volatility. The path-dependency that we uncover is remarkably simple: a linear combination of a weighted sum of past daily returns and the square root of a weighted sum of past daily squared returns with different time-shifted power-law weights capturing both short and long memory. This simple model, which is homogeneous in volatility, is shown to consistently outperform existing models across equity indexes for both implied and realized volatility. It suggests a simple continuous-time path-dependent volatility (PDV) model that may be fed historical or risk-neutral parameters. The weights can be approximated by superpositions of exponential kernels to produce Markovian models. In particular, we propose a 4-factor Markovian PDV model which captures all the important stylized facts of volatility, produces very realistic price and volatility paths, and jointly fits SPX and VIX smiles remarkably well. We thus show, for the first time, that a continuous-time Markovian parametric stochastic volatility (actually, PDV) model can practically solve the joint SPX/VIX smile calibration problem.
Volatility Is (Mostly) Path-Dependentread_more
HG G 43
3. November 2022
17:15-18:15 		Prof. Dr. Łukasz Delong
SGH Warsaw School of Economics
Details

Talks in Financial and Insurance Mathematics

Titel The use of autoencoders for training neural networks with mixed categorical and numerical features
Referent:in, Affiliation Prof. Dr. Łukasz Delong, SGH Warsaw School of Economics
Datum, Zeit 3. November 2022, 17:15-18:15
Ort HG G 43
Abstract We focus on modelling categorical features and improving predictive power of neural networks with mixed categorical and numerical features in supervised learning problems. The goal is to challange the current dominant approach in actuarial data science with a new architecture of a neural network and a new training algorithm. The proposal is to use a joint embedding for all categorical features, instead of separate embeddings for each categorical features, and initalize the parameters of the neural network with parameters trained with a denoising autoencoder in a unsupervised learning problem, instead of random initialization. In other words, we propose a special initialization strategy for taining neural networks. We illustrate our ideas with experiments on a data set with insurance claim numbers. We demonstrate that we can achieve a predictive power on an independent test set higher than in the current approach. Moreover, we also show with experiments that our special initialization strategy performs better than other initialization strategies (with PCA, MCA, GLM) which can be alternatively proposed in our setting. Finally, we investigate the predictive power vs the bias of the predictions.
The use of autoencoders for training neural networks with mixed categorical and numerical featuresread_more
HG G 43
10. November 2022
17:15-18:15
Details

Talks in Financial and Insurance Mathematics

Titel No seminar
Referent:in, Affiliation
Datum, Zeit 10. November 2022, 17:15-18:15
Ort HG G 43
No seminar
HG G 43
17. November 2022
17:15-18:15 		Prof. Dr. Youri Kabanov
Université de Franche-Comté, Besançon, France
Details

Talks in Financial and Insurance Mathematics

Titel Recent results in the ruin theory with investments
Referent:in, Affiliation Prof. Dr. Youri Kabanov, Université de Franche-Comté, Besançon, France
Datum, Zeit 17. November 2022, 17:15-18:15
Ort HG G 43
Abstract In the classical collective risk theory it is usually assumed that the capital reserve of a company is placed in the bank account paying zero interest. In the recent two decades the theory was extended to cover a more realistic situation where the reserve is invested, fully or partially, in a risky asset (e.g., in a portfolio evolving as a market index). This natural generalization generates a huge variety of new ruin problems. Roughly speaking, each “classical” ruin problem, e.g., a version of the Cramer-Lundberg model (for the non-life insurance, for the annuity payments etc., with a specific assumption) can be combined a model of price of the risky security (geometric Brownian motion, geometric Lévy process, various models with stochastic volatilities, etc.). In the talk we present new asymptotic results for the ruin probabilities, in particular, for the Sparre Andersen type models with risky investments having the geometric Lévy dynamics and for Cramér-Lundberg type models with investments in a risky asset with a regime switching price.
Recent results in the ruin theory with investmentsread_more
HG G 43
24. November 2022
17:15-18:15 		Prof. Dr. John Armstrong
King's College London
Details

Talks in Financial and Insurance Mathematics

Titel A rough path approach to replicating derivatives and the importance of gamma hedging
Referent:in, Affiliation Prof. Dr. John Armstrong, King's College London
Datum, Zeit 24. November 2022, 17:15-18:15
Ort HG G 43
Abstract We will examine the theory of replicating derivatives and in particular the Fundamental Theorem of Derivative Trading using the theory of rough paths. No model of stock prices will ever be completely accurate, so it is important that any theory of replication is robust to small perturbations of the stock price process. The fundamental theorem of derivative trading one such robustness result which applies to perturbations of diffusion models. In this talk we will consider perturbations of the stock price that go beyond diffusion models, and indeed, beyond probabilistic models using rough path theory which has excellent continuity properties. We will see that the gamma hedging strategy which is widely used by practitioners emerges naturally from this theory, and indeed is in some sense essential to make the replication strategy work. This and the corresponding robustness results provide a mathematical explanation for the popularity of gamma hedging.
A rough path approach to replicating derivatives and the importance of gamma hedgingread_more
HG G 43
1. Dezember 2022
17:15-18:15 		Dr. Gabriele Visentin
ETH Zurich, Switzerland
Details

Talks in Financial and Insurance Mathematics

Titel Universal approximation of credit portfolio losses using Restricted Boltzmann Machines
Referent:in, Affiliation Dr. Gabriele Visentin, ETH Zurich, Switzerland
Datum, Zeit 1. Dezember 2022, 17:15-18:15
Ort HG G 43
Abstract In this talk we investigate the use of Restricted Boltzmann Machines (RBMs) in credit risk management. RBMs are stochastic neural networks capable of universal approximation of loss distributions. We test this model on an empirical dataset of default probabilities of 30 investment-grade US companies and show that it outperforms commonly used parametric factor copula models across several credit risk management tasks. In particular, the model leads to better out-of-sample fits for the empirical loss distribution and more accurate risk measure estimations. We introduce an importance sampling procedure which allows risk measures to be estimated at high confidence levels in a computationally efficient way. Furthermore, we show that the statistical factors extracted by the model admit an interpretation in terms of the underlying portfolio sector structure and provide practitioners with quantitative tools for the management of concentration risk. Finally, we showcase the usefulness of the model for stress testing by estimating stressed risk measures (e.g. stressed VaR) for our empirical portfolio under various macroeconomic stress test scenarios, such as those specified by the FRB's Dodd-Frank Act stress test.
Universal approximation of credit portfolio losses using Restricted Boltzmann Machinesread_more
HG G 43
8. Dezember 2022
17:15-18:15
Details

Talks in Financial and Insurance Mathematics

Titel No seminar
Referent:in, Affiliation
Datum, Zeit 8. Dezember 2022, 17:15-18:15
Ort HG G 43
No seminar
HG G 43
15. Dezember 2022
17:15-18:15 		Dr. Mehdi Christian Talbi
ETH Zurich, Switzerland
Details

Talks in Financial and Insurance Mathematics

Titel Mean field optimal stopping
Referent:in, Affiliation Dr. Mehdi Christian Talbi, ETH Zurich, Switzerland
Datum, Zeit 15. Dezember 2022, 17:15-18:15
Ort HG G 43
Abstract We are interested in the study of the mean field optimal stopping problem, that is the optimal stopping of a McKean-Vlasov diffusion, when the criterion to optimize is a function of the distribution of the stopped process. This problem models the situation where a central planner controls a continuous infinity of interacting agents by assigning a stopping time to each of them, in order to maximize some criterion which depends on the distribution of the system. We study this problem via a dynamic programming approach, which allows to characterize its value function by a partial differential equation on the space of probability measures, that we call obstacle problem (or equation) on Wasserstein space by analogy with the classical obstacle problem, which arises in particular in standard optimal stopping. We introduce a notion a viscosity solution for this equation and show that, under appropriate assumptions, the value function of the mean field optimal stopping problem is the unique viscosity solution of the obstacle problem on Wasserstein space. We also study the corresponding finite population problem and prove its convergence to the mean field problem.
Mean field optimal stoppingread_more
HG G 43

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