# Seminar on Financial and Actuarial Mathematics

The seminar is organized by Prof. Biagini, Prof. Czado, Prof. Glau, 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 (Business Campus, Garching, Parkring 11)* *and the *Mathematical Institute of the LMU* (München, Theresienstraße 39).

Currently the Seminar takes place at the Mathematical Institute of the LMU (Mon 14:15 to 17:00 - Theresienstraße 39-B, Room B 349

The dates of the Graduate Seminar in Financial and Actuarial Mathematics (SoSe 2017) are:

May 8, 2017; June 19, 2017; July 24, 2017.

## Upcoming talks

## Previous talks

### 19.06.2017 16:30 Sascha Desmettre: Generalized Pareto processes and liquidity

Motivated by the modeling of liquidity risk in fund management in a dynamic setting, we propose and investigate a class of time series models with generalized Pareto marginals: the autoregressive generalized Pareto process (ARGP), a modified ARGP (MARGP) and a thresholded ARGP (TARGP). These models are able to capture key data features apparent in fund liquidity data and reflect the underlying phenomena via easily interpreted, low-dimensional model parameters. We establish stationarity and ergodicity, provide a link to the class of shot-noise processes, and determine the associated interarrival distributions for exceedances. Moreover, we provide estimators for all relevant model parameters and establish consistency and asymptotic normality for all estimators (except the threshold parameter, which as usual must be dealt with separately). Finally, we illustrate our approach using real-world fund redemption data, and we discuss the goodness-of-fit of the estimated models.

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### 08.05.2017 14:15 Aditi Dandapani: Martingales and strict local martingales

When discussing the nature of nonnegative solutions of SDEs,
making the distinction between a strict local martingale and a true
martingale can be very important. The papers of Delbaen, Shirakawa,
Mijatovic, Urusov, Lions, Musiela, Andersen, Piterbarg, Bernard, Cui, and
finally McLeish have studied the case of a one dimensional SDE, with or
without stochastic volatility. We present two concepts: how a solution of an
SDE which is a martingale can become a strict local martingale by the
addition of new information to the underlying filtration, and how various
components of a vector of SDEs can be strict local martingales for some
components of the system, and martingales for others. This is based on joint
work with Philip Protter, Professor at Columbia University.

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### 08.05.2017 15:00 Giorgia Riveccio (Parthenope University): Copula quantile regression for analysis of multiple time series

In financial researches and among risk management practitioners the analysis of multiple time-series is often conducted in a non-linear context. In addition, capturing the quantile conditional dependence structure could prove of interest in order to measure financial contagion risk. We propose a 3-stage estimation copula-based method applied to non-linear quantile dependence analysis of time-series vectors. This method aims to analyse the serial and cross-section dependence of time-series given specified quantiles, reducing the computational complexity. To the best of our knowledge, this is the first approach that combines the conditional quantile dependence analysis of multiple time-series with non-linear modelling by means of copula functions. Finally, we examine the conditional quantile behaviour of financial time-series with a non-linear copula quantile VAR model.
The talk is based on joint work with Giovanni De Luca.

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### 08.05.2017 16:05 Philipp Harms: Markovian representations of fractional Brownian motion and some applications in mathematical finance

A wide class of Gaussian processes, including fractional
Brownian motion, can be represented as linear functions of an
infinite-dimensional affine process. This opens the door to analyzing
such processes using tools from Markov processes and SPDEs. Moreover,
the affine structure makes computations tractable, and the
representation lends itself to numerical implementation. We will look
into some of this theory and its applications in mathematical finance.

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### 08.05.2017 16:50 Jeannette Woerner: Inference for the driving Lévy process of continuous-time moving average processes

Continuous-time moving average processes, defined as integrals of a deterministic kernel
function integrated with respect to a two sided Lévy process, provide a unifying framework
to different types of processes, including the popular examples of fractional Brownian
motion and fractional Lévy processes on the one side and Ornstein-Uhlenbeck processes
on the other side. The whole class of processes especially allows for a combination of a
given correlation structure with an infinitely divisible marginal distribution as it is desirable
for applications in finance, physics and hydrology.
So far inference for these processes is mainly concerned with estimating parameters
entering the kernel function which is responsible for the correlation structure. We now
consider the estimating problem for the driving Lévy process. We will provide two methods
working on different sets of conditions, one is based on a suitable integral transform, the
other on the Mellin transform.

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### 23.01.2017 14:15 Cagin Ararat, Ph.D. (Bilkent University, Ankara): Multi-objective risk-averse stochastic optimization

Two-stage risk-averse stochastic optimization is concerned with the minimization of a risk measure of a random cost function over the feasible choices of a deterministic and a random decision variable. We study the multi-objective version of this problem in which case the cost function is vector-valued and its risk is quantified via a multivariate (set-valued) risk measure. We reformulate the resulting problem as a convex vector optimization problem with set-valued constraints and propose customized versions of Benson’s algorithm to solve it. In particular, by randomizing the deterministic decision variable, we develop convex duality-based decomposition methods to solve the scalar subproblems appearing in Benson’s algorithm. The algorithm is illustrated on examples including the multi-asset portfolio optimization problem with transaction costs.

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### 23.01.2017 15:00 Lorenz Schneider, Ph.D. (EM Lyon) : Seasonal Stochastic Volatility and Correlation together with the Samuelson Effect in Commodity Futures Markets

We introduce a multi-factor stochastic volatility model based on the CIR/Heston volatility process that incorporates seasonality and the Samuelson effect. First, we give conditions on the seasonal term under which the corresponding volatility factor is well-defined. These conditions appear to be rather mild. Second, we calculate the joint characteristic function of two futures prices for different maturities in the proposed model. This characteristic function is analytic. Finally, we provide numerical illustrations in terms of implied volatility and correlation produced by the proposed model with five different specifications of the seasonality pattern. The model is found to be able to produce volatility smiles at the same time as a volatility term-structure that exhibits the Samuelson effect with a seasonal component. Correlation, instantaneous or implied from calendar spread option prices via a Gaussian copula, is also found to be seasonal.

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### 23.01.2017 16:15 Christa Cuchiero, Ph.D. (University of Vienna): Cover's universal portfolio, stochastic portfolio theory and the numéraire portfolio

Cover's celebrated theorem states that the long run yield of a properly chosen "universal" constant rebalanced portfolio is as good as the long run yield of the best retrospectively chosen constant rebalanced portfolio. The "universality" pertains to the fact that this result is modelfree, i.e., not dependent on an underlying stochastic process. We extend Cover's theorem to the setting of stochastic portfolio theory as initiated by R. Fernholz: the rebalancing rule need not to be constant anymore but may depend on the present state of the stock market. This result is complimented by a comparison with the log-optimal numéraire portfolio when fixing a stochastic model of the stock market. Roughly speaking, under appropriate assumptions, the optimal long run yield coincides for the three approaches mentioned in the title. We present our results in discrete as well as in continuous time. The talk is based on joint work with Walter Schachermayer and Leonard Wong.

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### 12.12.2016 14:15 Patrick Cheridito, Ph.D. (ETH Zürich): Variable annuities with high water mark withdrawal benefit

We develop a continuous-time model for variable annuities that allow for periodic withdrawals proportional to the high water mark of the underlying account value as well as early surrender of the policy. We derive the Hamilton--Jacobi--Bellman equation characterizing the value of such a contract and the worst case policy holder behavior from an issuer's perspective. Based on these results, we construct a dynamic trading strategy which super-hedges the contract. To treat the problem numerically, we develop a semi-Lagrangian scheme based on a discretization of the underlying noise process.

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### 12.12.2016 15:00 Dr. Hajo Holzmann (Uni Marburg): Scoring functions for forecast evaluation and the role of the information set

Scoring functions are an essential tool to evaluate point forecasts, and scoring rules to evaluate probabilistic forecasts. We start by reviewing some recent results on the construction of scoring functions and scoring rules. Point forecasts are issued on the basis of certain information. If the forecasting mechanisms are correctly specified, a larger amount of available information should lead to better forecasts.
We show how the effect of increasing the information set on the forecast can be quantified by using strictly consistent scoring functions, and also discuss the role of the information set for evaluating probabilistic forecasts by using strictly proper scoring rules. Further, a method is proposed to test whether an increase in a sequence of information sets leads to distinct, improved $h$-step point forecasts. For the value at risk (VaR), we show that increasing the information set will result in VaR forecasts which lead to smaller expected shortfalls, unless an increase in the information set does not change the VaR forecast. The effect is illustrated in simulations and applications to stock returns for unconditional versus conditional risk management as well as univariate modeling of portfolio returns versus multivariate modeling of individual risk factors.
Reference: Holzmann, H., Eulert, M. (2014) The role of the information set for forecasting -- with applications to risk management. Annals of Applied Statistics 8, 595-621

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### 12.12.2016 16:15 Dr. Carsten Chong (TU München): Contagion in Financial Systems: A Bayesian Network Approach

We develop a structural default model for interconnected financial institutions in a probabilistic framework. For all possible network structures we characterize the joint default distribution of the system using Bayesian network methodologies. Particular emphasis is given to the treatment and consequences of cyclic financial linkages. We further demonstrate how Bayesian network theory can be applied to detect contagion channels within the financial network, to measure the systemic importance of selected entities on others, and to compute conditional or unconditional probabilities of default for single or multiple institutions.

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### 14.11.2016 14:15 Prof. Johannes Ruf (London School of Economics): Some remarks on functionally generated portfolios

In the first part of the talk I will review Bob Fernholz' theory of functionally generated portfolios. In the second part I will discuss questions related to the existence of short-term arbitrage opportunities.
This is joint work with Bob Fernholz and Ioannis Karatzas.

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### 14.11.2016 15:00 Prof. Teemu Pennanen (King's College London): Asset Valuation via Optimal Investment

We study optimal investment and the valuation of assets whose payouts cannot be replicated by trading other assets. Our market model allows for constraints and illiquidity effects that are encountered in practice. We review two hedging-based notions of asset value and relate them to the classical notions of risk neutral and net present values. Many classical results e.g. on attainability and duality are extended to the illiquid market setting. This is joint work with Ari-Pekka Perkkiö.

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### 14.11.2016 16:15 Lorenzo Toricelli : A Review on Voliatility Targeting and New Results

A target volatility (call) option is an option where the total underlying exposure is random and determined at maturity as the ratio of a predicted target volatility level with the volatility effectively realized by the underlying. This reduces the option price for a similar payoff if the volatility prediction was correct, thus allowing an affordable option position when vega hedging is too expensive or market implied volatilities are too high.
A target volatility strategy (TVS) is a bond-equity dynamic portfolio using the risky asset historical volatility as an allocation rule. High realized volatility decreases the equity exposure reducing the portfolio downside risk. Lower volatility increases the equity position as to benefit from the bearish market conditions. These adjustments over time should maintain the volatility of the investment constant around the investor’s desired target level.
In a market with stochastic volatility, we present a stochastic model for a TVS using a delayed differential system. We derive an approximate finite-dimensional Markovian approximation for the equations which we implement for the Heston model using a Euler scheme. This framework allows the valuation of guarantee costs of target volatility funds; if the constant volatility assumption is correct, such a value should be of Black-Scholes type. We investigate this claim within the presented model.

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### 08.11.2016 16:30 Prof. Dr. Christian Hipp: Stochstische Steuerung für Versicherungen: Alte und neue Ansätze und Probleme.

Stochastische Steuerung für Versicherungen ist mittlerweile ein lebendiges Teilgebiet der Mathematik, es hat sich nach anfänglichen Versuchen, Methoden aus dem Finance zu adaptieren, etwas eigenständiger weiterentwickelt. Dazu gehören Methoden zur Lösung von Integro-Differentialgleichungen, von Nebenbedingungen, von Modifizierungen der Argumente für Viskositätslösungen und nicht-stationäre Ansätze für Mischungsmodelle. Schwerpunkt liegt in der numerischen Umsetzung und der Interpretation der numerischen Resultate.

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