Insurance Mathematics and Stochastic Finance

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Talks in Financial and Insurance Mathematics

This is the regular weekly research seminar on Insurance Mathematics and Stochastic Finance.

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Spring Semester 2022

Date / Time Speaker Title Location
3 March 2022
17:15-18:15 		Prof. Dr. Thorsten Schmidt
University of Freiburg
Event Details

Talks in Financial and Insurance Mathematics

Title Term Structure Modelling with Overnight Rates Beyond Stochastic Continuity
Speaker, Affiliation Prof. Dr. Thorsten Schmidt, University of Freiburg
Date, Time 3 March 2022, 17:15-18:15
Location HG G 43
Abstract 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.e., jumps occurring at predetermined dates) in the dynamics of RFRs. In this work, we propose a general modeling framework where RFRs and term rates can have stochastic discontinuities and characterize absence of arbitrage in an extended HJM setup. When the term rate is generated by the RFR itself, we show that it solves a BSDE, whose driver is determined by the HJM drift restrictions. In general, this BSDE may admit multiple solutions and we provide sufficient conditions ensuring uniqueness. We develop a tractable specification driven by affine semimartingales, also extending the classical short rate approach to the case of stochastic discontinuities. In this context, we show that a simple specification allows to capture stylized facts of the jump behavior of overnight rates. In a Gaussian setting, we provide explicit valuation formulas for bonds and caplets. Finally, we study hedging in the sense of local risk-minimization when the underlying term structures have stochastic discontinuities. This is joint work with Claudio Fontana and Zorana Grbac.
Term Structure Modelling with Overnight Rates Beyond Stochastic Continuityread_more
HG G 43
10 March 2022
17:15-18:15 		Prof. Dr. Umut Çetin
London School of Economics
Event Details

Talks in Financial and Insurance Mathematics

Title Power Laws in Market Microstructure
Speaker, Affiliation Prof. Dr. Umut Çetin, London School of Economics
Date, Time 10 March 2022, 17:15-18:15
Location HG G 43
Abstract We develop an equilibrium model for market impact of trades when investors with private signals execute via a trading desk. Fat tails in the distribution of the fundamental value lead to a power law for price impact, while the impact is logarithmic for lighter tails. Moreover, the tail distribution of the equilibrium trade volume obeys a power law, consistent with numerous empirical studies. The spread decreases with the degree of noise trading and increases with the number of insiders. In case of a monopolistic insider, the last slice traded against the limit order book is priced at the fundamental value of the asset reminiscent of continuous-time version of Kyle (1985). However, competition among insiders leads to aggressive trading, hence vanishing profit in the limit. The model also predicts that the order book flattens as the amount of noise trading increases converging to a model with proportional transaction costs with non-vanishing spread. Joint work with H. Waelbroeck.
Power Laws in Market Microstructureread_more
HG G 43
17 March 2022
17:15-18:15 		Prof. Dr. Martin Keller-Ressel
Technische Universität Dresden
Event Details

Talks in Financial and Insurance Mathematics

Title The Shape of the Term Structure in Affine Two-Factor Models
Speaker, Affiliation Prof. Dr. Martin Keller-Ressel, Technische Universität Dresden
Date, Time 17 March 2022, 17:15-18:15
Location HG G 43
Abstract We provide a full classification of all attainable shapes of the term structure of interest rates in the two-factor Vasicek model. In particular, we show that the shapes normal, inverse, humped, dipped and hump-dip are always attainable. In certain parameter regimes up to four additional shapes can be produced. Our results apply to both forward and yield curves and show that the correlation and the difference in mean-reversion speeds of the two factor processes play a key role in determining the scope of attainable shapes. In addition, we present recent results on the state-contingent analysis of the term structure shape, that is, on partitioning the model’s state space according to the shapes produced. We contrast two different approaches to the problem: One analytic and based on the theory of total positivity; the other geometric and based on the winding number of certain plane curves. We also discuss extensions to other affine models beyond the Vasicek model.
The Shape of the Term Structure in Affine Two-Factor Modelsread_more
HG G 43
24 March 2022
17:15-18:15 		Prof. Dr. Samuel Cohen
University of Oxford
Event Details

Talks in Financial and Insurance Mathematics

Title Neural-SDE Options Market Models for Risk Modelling
Speaker, Affiliation Prof. Dr. Samuel Cohen, University of Oxford
Date, Time 24 March 2022, 17:15-18:15
Location HG G 43
Abstract Efficient and statistically accurate estimation of risk measures (for example, value at risk), is a critical part of the financial system. In this talk, we will consider how a model can be built and trained, which is flexible enough to capture observed market dynamics, is guaranteed not to produce static arbitrages (and dynamic arbitrages are well controlled), and can be simulated efficiently. This will exploit the use of neural networks as function approximators in an SDE, while giving an easily interpretable model. We will see that this approach leads to significant advantages when computing risks of options portfolios, both in terms of accuracy and computational cost. Based on joint work with Christoph Reisinger and Sheng Wang.
Neural-SDE Options Market Models for Risk Modellingread_more
HG G 43
31 March 2022
17:15-18:15
Event Details

Talks in Financial and Insurance Mathematics

Title Einführungsvorlesungen by Beatrice Acciaio and Dylan Possamaï
Speaker, Affiliation
Date, Time 31 March 2022, 17:15-18:15
Location HG F 30
Einführungsvorlesungen by Beatrice Acciaio and Dylan Possamaï
HG F 30
7 April 2022
17:15-18:15 		Prof. Dr. Paolo Guasoni
Dublin City University
Event Details

Talks in Financial and Insurance Mathematics

Title Incomplete-Market Equilibrium with Unhedgeable Fundamentals and Heterogenous Agents
Speaker, Affiliation Prof. Dr. Paolo Guasoni, Dublin City University
Date, Time 7 April 2022, 17:15-18:15
Location HG G 43
Abstract We solve a general equilibrium model of an incomplete market with heterogeneous preferences, identifying first-order and second-order effects. Several long-lived agents with different absolute risk-aversion and discount rates make consumption and investment decisions, borrowing from and lending to each other, and trading a stock that pays a dividend whose growth rate has random fluctuations over time. For small fluctuations, the first-order equilibrium implies no trading in stocks, the existence of a representative agent, predictability of returns, multi-factor asset pricing, and that agents use a few public signals for consumption, borrowing, and lending. At the second-order, agents dynamically trade stocks and no representative agent exist. Instead, both the interest rate and asset prices depend on the dispersion of agents' preferences and their shares of wealth. Dynamic trading arises from agents' intertemporal hedging motive, even in the absence of personal labor income.
Incomplete-Market Equilibrium with Unhedgeable Fundamentals and Heterogenous Agentsread_more
HG G 43
14 April 2022
17:15-18:15
Event Details

Talks in Financial and Insurance Mathematics

Title No seminar (Easter break)
Speaker, Affiliation
Date, Time 14 April 2022, 17:15-18:15
Location
No seminar (Easter break)
21 April 2022
17:15-18:15
Event Details

Talks in Financial and Insurance Mathematics

Title No seminar (Easter break)
Speaker, Affiliation
Date, Time 21 April 2022, 17:15-18:15
Location
No seminar (Easter break)
28 April 2022
17:15-18:15 		Prof. Dr. Eduardo Abi Jaber
Université Paris 1 Panthéon-Sorbonne
Event Details

Talks in Financial and Insurance Mathematics

Title Quadratic Gaussian Models: Analytic Expressions for Pricing and Portfolio Allocation
Speaker, Affiliation Prof. Dr. Eduardo Abi Jaber, Université Paris 1 Panthéon-Sorbonne
Date, Time 28 April 2022, 17:15-18:15
Location HG G 43
Abstract Stochastic models based on Gaussian processes, like fractional Brownian motion, are able to reproduce important stylized facts of financial markets such as rich autocorrelation structures, persistence and roughness of sample paths. This is made possible by virtue of the flexibility introduced in the choice of the covariance function of the Gaussian process. The price to pay is that, in general, such models are no longer Markovian nor semimartingales, which limits their practical use. We derive explicit analytic expressions for Fourier-Laplace transforms of quadratic functionals of Gaussian processes. Such analytic expression can be approximated by closed form matrix expressions stemming from Wishart distributions. We highlight the applicability of such result in the context of rough volatility modeling: (i) fast pricing and calibration in the (rough) fractional Stein-Stein model; (ii) explicit solutions for the Markowitz portfolio allocation problem in a multivariate rough Stein-Stein model. Based on joint works with Enzo Miller and Huyên Pham.
Quadratic Gaussian Models: Analytic Expressions for Pricing and Portfolio Allocationread_more
HG G 43
5 May 2022
17:15-18:15 		Dr. Chong Liu
ShanghaiTech University
Event Details

Talks in Financial and Insurance Mathematics

Title Distribution Regression via Higher Rank Signatures
Speaker, Affiliation Dr. Chong Liu, ShanghaiTech University
Date, Time 5 May 2022, 17:15-18:15
Location HG G 43
Abstract Using the universality of signature one can approximate weakly continuous functions on laws of stochastic processes, e.g., the pricing functionals for path-dependent payoffs, by performing a linear regression on expected signatures. This classical approach fails when we deal with Optimal Stopping Problems (American option pricing in finance) as the value functions of OSP are discontinuous for the weak topology. In this talk we propose a novel approach by using the so-called higher rank signature and adapted topology, which allows us to use linear regression to solve Optimal Stopping Problems. Thanks to the intrinsic dynamics of higher rank signatures, this infinite dimensional linear regression scheme is eventually reduced to the task of solving low dimensional PDEs. This approach reveals an interesting combination of (Hopf) algebra, adapted topology and kernel learning. Based on joint works with Patric Bonnier, Harald Oberhauser, Maud Lemercier and Cris Salvi.
Distribution Regression via Higher Rank Signaturesread_more
HG G 43
12 May 2022
17:15-18:15 		Prof. Dr. Michael Kupper
University of Konstanz
Event Details

Talks in Financial and Insurance Mathematics

Title Markovian Semigroups under Model Uncertainty
Speaker, Affiliation Prof. Dr. Michael Kupper, University of Konstanz
Date, Time 12 May 2022, 17:15-18:15
Location HG G 43
Abstract When considering stochastic processes for the modelling of real world phenomena, a major issue is so-called model uncertainty. In this talk, we present two ways to incorporate model uncertainty into a Markovian dynamics. One approach considers parameter uncertainty in the generator of a Markov process while the other considers perturbations of a reference model within a Wasserstein proximity. We show that, in typical situations, these two a priori different approaches lead to the same convex transition semigroup. In the second part, we focus on semigroups of convex monotone operators on spaces of continuous functions and their behaviour with respect to Gamma-convergence. In particular, we discuss the role of Lipschitz sets and provide a comparison result. The talk is based on joint works with Daniel Bartl, Jonas Blessing, Robert Denk, Stephan Eckstein, Sven Fuhrmann and Max Nendel.
Markovian Semigroups under Model Uncertaintyread_more
HG G 43
12 May 2022
18:15-19:15 		Prof. Dr. Aleš Černý
City, University of London
Event Details

Talks in Financial and Insurance Mathematics

Title Simplified Stochastic Calculus via Semimartingale Representations
Speaker, Affiliation Prof. Dr. Aleš Černý, City, University of London
Date, Time 12 May 2022, 18:15-19:15
Location HG G 43
Abstract We introduce a simple way of recording and manipulating general stochastic processes without explicit reference to a probability measure. The resulting calculus makes it easy to capture a variety of predictable transformations of semimartingales such as changes of variables, stochastic integrals, and their compositions. The new calculus is very effective when it comes to computing drifts and expected values that possibly involve a change of measure. Such drift calculations yield, for example, partial integro-differential equations, Hamilton–Jacobi–Bellman equations, Feynman–Kac formulae, or exponential moments needed in numerous applications. We provide several illustrations of the new technique, among them (1) a novel result on the Margrabe option to exchange one defaultable asset for another; and (2) a novel technique for computing the distribution of a signed stochastic exponential with application in mean-variance analysis. The talk is based on a series of joint works with Johannes Ruf; see ssrn.com/abstract=3500384, ssrn.com/abstract=3633638, ssrn.com/abstract=3633622, and arxiv.org/abs/1909.03020.
Simplified Stochastic Calculus via Semimartingale Representationsread_more
HG G 43
19 May 2022
17:15-18:15 		Prof. Dr. Sergey Nadtochiy
Illinois Institute of Technology
Event Details

Talks in Financial and Insurance Mathematics

Title Probabilistic Solutions to Stefan Equations
Speaker, Affiliation Prof. Dr. Sergey Nadtochiy, Illinois Institute of Technology
Date, Time 19 May 2022, 17:15-18:15
Location HG G 43
Abstract This talk is concerned with the probabilistic methods for solving Stefan free-boundary PDEs (a.k.a. laplacian growth models). The latter equations appear in many models of fundamental physical and biological processes, such as: phase transition (i.e., melting/freezing), phase segregation (e.g., aging of alloys), crystal growth, neurons interaction, etc.. Despite their importance, to date, there is no general existence and uniqueness theory for such equations due to the potential singularity of their solutions, which makes it difficult to apply the classical analytical methods. Recently, novel probabilistic methods, based on the analysis of associated mean-field particle systems and McKean-Vlasov equations, were successfully used to tackle these mathematical challenges yielding new well-posedness results for certain types of Stefan equations. I will present an overview of the recent results and will focus on the well-posedness of the Stefan equation with surface tension. This talk is based on joint works with F. Delarue, M. Shkolnikov, and X. Zhang.
Probabilistic Solutions to Stefan Equationsread_more
HG G 43
26 May 2022
17:15-18:15
Event Details

Talks in Financial and Insurance Mathematics

Title No seminar (Ascension Day)
Speaker, Affiliation
Date, Time 26 May 2022, 17:15-18:15
Location
No seminar (Ascension Day)
2 June 2022
17:15-18:15 		Dr. Cristopher Salvi
Imperial College London
Event Details

Talks in Financial and Insurance Mathematics

Title From Neural SDEs to Neural SPDEs
Speaker, Affiliation Dr. Cristopher Salvi, Imperial College London
Date, Time 2 June 2022, 17:15-18:15
Location HG G 43
Abstract Many standard deep learning architectures may be interpreted as approximations to differential equations. The resulting continuous models are dubbed neural differential equations. ResNets is the approximation of ODEs while RNNs and variants (LSTM-GRU) are approximations of controlled differential equations (CDEs). In the first half of the talk, I will show how tools from rough path theory allows to understand properties of Neural CDEs to process time-varying data. Despite being an elegant solution to model temporal dynamics, Neural CDEs are not designed to process streams varying both in space and in time, such as videos. Stochastic PDEs (SPDEs) are the mathematical tool of choice to model many physical, biological, and economic systems subject to the influence of randomness. In the second half of the talk, I will present the Neural SPDE model capable of learning solution operators of PDEs with (possibly stochastic) forcing from partially observed data. Experiments on various semilinear SPDEs, including the stochastic Navier-Stokes equations, demonstrate how the Neural SPDE model is capable of learning complex spatiotemporal dynamics in a resolution-invariant way, with better accuracy and lighter training data requirements compared to alternative models, and up to 3 orders of magnitude faster than traditional solvers.
From Neural SDEs to Neural SPDEsread_more
HG G 43
2 June 2022
18:15-19:15 		Prof. Dr. Agostino Capponi
Columbia University
Event Details

Talks in Financial and Insurance Mathematics

Title Goal Based Investment Management
Speaker, Affiliation Prof. Dr. Agostino Capponi, Columbia University
Date, Time 2 June 2022, 18:15-19:15
Location HG G 43
Abstract We develop a continuous time framework for goals-based investing, where the objective is to maximize the priority-weighted fundedness of the clients' goals. We show that the value function of the control problem can be recovered as the unique viscosity solution to a nonlinear Hamilton-Jacobi-Bellman equation. Our analysis highlights the fundamental tradeoff between immediate goal consumption versus savings toward future goal liabilities. We show that it is optimal to fund an expiring goal up to the level where the marginal benefit of consuming wealth is exceeded by the marginal cost of reducing the fundedness of future goals. This tradeoff depends crucially on the interplay between goal amounts, goal deadlines, client's initial wealth and guaranteed income stream. Our comparative statics analysis reveals that the risk aversion towards a goal increases if the goal's deadline is approaching or if the priority of a goal relative to future goals increases. (joint work with Yuchong Zhang).
Goal Based Investment Managementread_more
HG G 43
23 June 2022
17:15-18:15 		Prof. Dr. Ying Chen
National University of Singapore
Event Details

Talks in Financial and Insurance Mathematics

Title Deep Switching State Space Model (DS3M) for Nonlinear Time Series Forecasting with Regime Switching
Speaker, Affiliation Prof. Dr. Ying Chen, National University of Singapore
Date, Time 23 June 2022, 17:15-18:15
Location HG G 43
Abstract We propose a deep switching state space model (DS3M) for efficient inference and forecasting of nonlinear time series with irregularly switching among various regimes. The switching among regimes is captured by both discrete and continuous latent variables with recurrent neural networks. The model is estimated with variational inference using a reparameterization trick. We test the approach on a variety of simulated and real datasets. In all cases, achieves competitive performance compared to several state-of-the-art methods (e.g. GRU, SRNN, DSARF, SNLDS), with superior forecasting accuracy, convincing interpretability of the discrete latent variables, and powerful representation of the continuous latent variables for different kinds of time series. Specifically, the MAPE values increase by 0.09% to 15.71% against the second-best performing alternative models. This is a joint work with Xiuqin Xu. https://arxiv.org/abs/2106.02329
Deep Switching State Space Model (DS3M) for Nonlinear Time Series Forecasting with Regime Switchingread_more
HG G 43

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