# Seminar on Financial and Actuarial Mathematics

The seminar is organized by Prof. Biagini, Prof. Czado, Prof. Klüppelberg, Prof. Meyer-Brandis, Prof. Scherer, Prof. Svindland and Prof. Zagst. The venue of the seminar changes on a regular basis between the TUM (Garching, Business Campus, Parkring 11)* *and the *Mathematical Institute of the LMU* (München, Theresienstraße 39).

Currently the Seminar takes place at the Technical University of Munich, Business Campus, Parkring 11, Garching, Room No. BC1 2.02.01 (on Mondays, 14:15 to 17:00)

The dates of the Graduate Seminar in Financial and Actuarial Mathematics (WiSe 2018/2019) are:

- November 5, 2018
- December 3, 2018
- January 28, 2019

## Upcoming talks

## Previous talks

### 05.11.2018 14:15 Thorsten Rheinländer : On the stochastic heat equation with mutiplicative noise

We study a parsimonious but non-trivial model of the latent limit order book where orders get placed with a fixed displacement from a center price process, i.e. some process in-between best bid and best ask, and get executed whenever this center price reaches their level. This mechanism corresponds to the fundamental solution of the stochastic heat equation with multiplicative noise for the relative order volume distribution, for which we provide a solution via a local time functional.
Moreover, we classify various types of trades, and introduce the trading excursion process which is a Poisson point process. This allows to derive the Laplace transforms of the times to various trading events under the corresponding intensity measure.

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### 05.11.2018 15:00 Gonçalo dos Reis: Large Deviations for McKean Vlasov Equations and Importance Sampling

We discuss two Freidlin-Wentzell large deviation principles for McKean-Vlasov equations (MV-SDEs) in certain path space topologies. The equations have a drift of polynomial growth and an existence/uniqueness result is provided. We apply the Monte-Carlo methods for evaluating expectations of functionals of solutions to MV-SDE with drifts of super-linear growth. We assume that the MV-SDE is approximated in the standard manner by means of an interacting particle system and propose two importance sampling (IS) techniques to reduce the variance of the resulting Monte Carlo estimator. In the "complete measure change"
approach, the IS measure change is applied simultaneously in the coefficients and in the expectation to be evaluated. In the "decoupling"
approach we first estimate the law of the solution in a first set of simulations without measure change and then perform a second set of simulations under the importance sampling measure using the approximate solution law computed in the first step.

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### 02.07.2018 15:00 Birgit Rudolf, Wien: Time consistency of the mean-risk problem

Multivariate risk measures appear naturally in markets with transaction costs or when measuring the systemic risk of a network of banks.
Recent research suggests that time consistency of these multivariate risk measures can be interpreted as a set-valued Bellman principle. And a more general structure emerges that might also be important for other applications and is interesting in itself.
In this talk I will show that this set-valued Bellman principle holds also for the dynamic mean-risk portfolio optimization problem. In most of the literature, the Markowitz problem is scalarized and it is well known that this scalarized problem does not satisfy the (scalar) Bellman principle. However, when we do not scalarize the problem, but leave it in its original form as a vector optimization problem, the upper images, whose boundary is the efficient frontier, recurse backwards in time under very mild assumptions. I will present conditions under which this recursion can be exploited directly to compute a solution in the spirit of dynamic programming and will state some open problems and challenges for the general case. (Joint work with Gabriela Kováčová)

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### 02.07.2018 16:00 Dr. Nils Detering (UCSB): An Integrated Model of Fire Sales and Default Contagion in Financial Systems

In (Detering et. al, 2016) and (Detering et. al, 2018) we developed a random graph model for 'default contagion' in financial networks and using 'law of large numbers' effects we were able to compute the size of the final default cluster induced by an arbitrarily given initial shock in a large system. Further, we were able to derive sufficient and necessary criteria for resilience of a system to small shocks. In that sense our model provides better insights than the popular Eisenberg-Noe model which is concerned with existence and uniqueness of a clearing vector (and hence the final state of the system) but gives no indication of favorable network structures and sufficient capital requirements to ensure resilience. The Eisenberg-Noe model, however, has proven to be flexible enough to be extended by contagion channels other than default contagion, the most important being 'fire sales' which describes contagion effects due to falling asset prices as institutions sell off their assets.
In this article, we first propose a model for fire sales that uses an Eisenberg-Noe like description for finite networks but allows to describe the final state of the system (size of the default cluster and final price impact) asymptotically. In particular, we are able to provide sufficient capital requirements that ensure resilience of the system. Furthermore, we integrate the channel of default contagion into our model applying results from (Detering et. al, 2018) and extending them to the non-continuous case induced by the fire sales. Finally, for this integrated setting, we provide criteria that determine whether a certain financial system is resilient or prone to small initial shocks and furthermore give sufficient capital requirements for financial systems to be resilient.
This is joint work with Thilo Meyer-Brandis, Konstantinos Panagiotou and Daniel Ritter

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### 11.06.2018 14:15 Erick Trevino Aguilar, Universidad de Guanajuato: Integral functionals of cadlag processes and partial superhedging of American options

In this talk we present advances in convex analysis and obtain a novel interchange rule for convex functionals defined over cadlag processes. This interchange rule allows to develop convex duality for a rich class of convex problems in general stochastic settings and requires a careful analysis of set valued mappings and its cadlag selections. As an application, we develop the dual problem of American option's partial hedging.

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### 07.05.2018 14:15 Prof. Michael Ludkowski: Marrying Stochastic Control and Machine Learning: from Bermudan Options to Natural Gas Storage and Microgrids

Simulation-based strategies bring the machine learning toolbox to numerical resolution of stochastic control models. I will begin by reviewing the history of this idea, starting with the seminal work by Longstaff-Schwartz and through the popularized Regression Monte Carlo framework. I will then describe the Dynamic Emulation Algorithm (DEA) that we developed, which unifies the different existing approaches in a single modular template and emphasizes the two central aspects of regression architecture and experimental design. Among novel DEA implementations, I will discuss Gaussian process regression, as well as numerous simulation designs (space-filling, sequential, adaptive, batched). The overall DEA template is illustrated with multiple examples drawing from Bermudan option pricing, natural gas storage valuation, and optimal control of back-up generator in a power microgrid. This is partly joint work with Aditya Maheshwari (UCSB).

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### 07.05.2018 15:00 Dr. Tobias Kley, Berlin: Quantile-Based Spectral Analysis of Time Series

Classical methods for the spectral analysis of time series account for covariance-related serial dependencies. This talk will begin with a brief introduction to these traditional procedures. Then, an alternative method is presented, where, instead of covariances, differences of copulas of pairs of observations and the independence copula are used to quantify serial dependencies. The Fourier transformation of these copulas is considered and used to define quantile-based spectral quantities. They allow to separate marginal and serial aspects of a time series and intrinsically provide more information about the conditional distribution than the classical location-scale model. Thus, quantile-based spectral analysis is more informative than the traditional spectral analysis based on covariances. For an observed time series the new spectral quantities are then estimated. The asymptotic properties, including the order of the bias and process convergence, of the estimator (a function of two quantile levels) are established. The results are applicable without restrictive distributional assumptions such as the existence of finite moments and only a weak form of mixing, such as alpha-mixing, is required.

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### 07.05.2018 16:00 Dr. Gregor Kastner, Wien: Bayesian Time-Varying Covariance Estimation in Many Dimensions using Sparse Factor Stochastic Volatility Models

We address the curse of dimensionality in dynamic covariance estimation by modeling the underlying co-volatility dynamics of a time series vector through latent time-varying stochastic factors. The use of a global-local shrinkage prior for the elements of the factor loadings matrix pulls loadings on superfluous factors towards zero. To demonstrate the merits of the proposed framework, the model is applied to simulated data as well as to daily log-returns of 300 S&P 500 members. Our approach yields precise correlation estimates, strong implied minimum variance portfolio performance and superior forecasting accuracy in terms of log predictive scores when compared to typical benchmarks. Furthermore, we discuss the applicability of the method to capture conditional heteroskedasticity in large vector autoregressions.

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### 04.12.2017 14:15 Alexander Szimayer: Rating Under Asymmetric Information

We analyze how a firm’s reputation and track record affect its rating and cost of debt. We model a setting in which outsiders such as a rating agency and the firm’s creditors continuously update their assessment of the firm’s true state described by its cash flow. They observe the latter only imperfectly due to asymmetric information. Other things equal, the rating agency optimally rates a firm with the same observed cash flow higher, if the historical minimum is sufficiently low. Thus, the rating is not only driven by the most recent information, but history matters. The rating agency refines its unbiased cash flow estimate by ruling out the most overestimated types, leading to an overestimation at default. In response, the firm delays default and lower asset values are available to creditors upon default.

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### 04.12.2017 15:00 Harry Joe: Estimation of tail dependence coefficients and extreme joint tail probabilities

Let C be a d-dimensional copula. With a random sample from this copula, several methods are introduced for estimation of the upper and lower tail dependence coefficients, as well as extreme joint tail probabilities such as the probability that all variables exceed their 0.99 quantiles and all variables are below their 0.01 quantiles.
The main theory is based on (i) a tail expansion of the distribution D() of maximum or minimum of the random vector on the copula scale and (ii) a tail expansion of an integral of D(). Item (ii) comes from investigating a tail-weighted dependence measure that is related to an estimate of the extremal index for multivariate extreme value data. The estimation methods for extreme joint tail probabilities consist of (a) likelihood-based threshold methods (for observations of appropriate maxima/minima that lie beyond a threshold, or (b) weighted regression
methods. Examples will be used for illustration of the main ideas.

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