# Talks in Financial and Insurance Mathematics

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

## Autumn Semester 2013

Note: The highlighted event marks the next occurring event and events marked with an asterisk (*) indicate that the time and/or location are different from the usual time and/or location.

Date / Time Speaker Title Location
12 September 2013
17:15-18:15
Prof. Dr. Yan Dolinsky
Hebrew University
HG G 43
Abstract: We prove limit theorems for the super-replication cost of European options in a Binomial model with friction. The examples covered are markets with proportional transaction costs and the illiquid markets. The dual representation for the super-replication cost in these models are obtained and used to prove the limit theorems. In particular, the existence of the liquidity premium for the continuous time limit of the model proposed by Cetein, Jarrow and Protter is proved. Hence, this paper extends the previous convergence result of Gokay and Soner to the general non-Markovian case. Moreover, the special case of small transaction costs yields, in the continuous limit, the $G$-expectation of Peng as earlier proved by Kusuoka.
* 19 September 2013
16:15-17:15
Prof. Dr. Andreas Tsanakas
Cass Business School, City University of London
HG G 19.2
Abstract: An internal risk model can be seen as consisting of (i) a random vector of risk factors and (ii) a business structure, understood as a real valued function mapping the risk factors to an output of interest, such as aggregate risk. In typical insurance applications, the risk factors can be high dimensional and the business structure non-linear. Such models are often black boxes, in the sense that the only information available to a model reviewer is a set of Monte-Carlo samples from risk factors and outputs. Sensitivity analyses are thus necessary, to provide a better understanding of the model. We propose a sensitivity measure that assesses the impact of each factor on the aggregate risk, as reflected by a distortion risk measure. It is then shown how the sensitivity measure can be calculated on a smooth approximation to the business structure, obtained by standard local linear regression methods. We demonstrate the proposed methodology through examples from life and non-life insurance and discuss its applicability to the measurement of parameter risk.
* 20 September 2013
15:15-16:15
Prof. Dr. Aleksandar Mijatovic
Imperial College London
HG G 26.5
Abstract: We consider the problem of identifying the optimal time to sell a defaultable asset in the sense of minimizing the “prophet’s drawdown” which is the ratio of the ultimate maximum (up to a random default time) and the value of the asset price at the moment of sale. We assume that default occurs at a constant rate, and that at the moment of default there is a random recovery value of $\rho(100)\%$. This problem is transformed to an optimal stopping problem, which we solve explicitly in the case that the asset price before default is modelled by a spectrally negative exponential Levy process. This is joint work with M. Pistorius.
3 October 2013
17:15-18:15
Dr. Curdin Ott
ETH Zurich
HG G 43
Abstract: We consider an option whose payoff corresponds to a “capped American lookback option with floating-strike” and solve the associated pricing problem (an optimal stopping problem) in a financial market whose price process is modeled by an exponential spectrally negative Lévy process. We will present some interesting features of the solution - in fact, it turns out that the continuation region has a feature that resembles a bottleneck and hence the name “Bottleneck option”. We will also come across some well-known optimal stopping problems such as the Russian optimal stopping problem and the American lookback optimal stopping problem.
10 October 2013
17:15-18:15
Prof. Dr. Frederic Abergel
Ecole Centrale Paris
HG G 43
Abstract: In this talk, I will present some recent results on the influence of taking liquidity costs and market impact into account when hedging a contingent claim. In a complete market, a fully non-linear partial differential equation is derived, and its well-posedness is characterized according to the value of a numerical parameter naturally interpreted as a relaxation coefficient for market impact. The more challenging case of stochastic volatility models is also investigated.
* 15 October 2013
15:15-16:00
Prof. Dr. Peter Markovich
University of Cambridge, KAUST, Universität Wien
HG G 19.1
Abstract: We present a Boltzmann-type model for the evolution of buyer and vendor densities in an economic market. In the limit of large transaction numbers the mean field parabolic free boundary problem as established by Lasry and Lions is obtained, in analogy to large-Knudsen number limits in gas-kinetics.
17 October 2013
17:15-18:15
Prof. Dr. Semyon Malamud
EPFL
HG G 43
Abstract: This paper develops an equilibrium model of decentralized trading that accommodates any coexisting exchanges including networks and more general, common market structures represented by hypergraphs. The model allows for any number of strategic traders and multiple divisible assets. We characterize equilibrium and welfare, and develop comparative statics with respect to preferences, assets, and market structures. Changes in market structure that increase price impact may increase utility of every agent. Equilibrium utility in a decentralized market can be strictly higher in the Pareto sense than in a centralized market with the same traders and assets. Agents with larger price impact may have higher equilibrium utility. Asset substitutability, or complementarity, is not determined by the primitive payoff covariance, but is endogenous and may differ across agents, depending on their participation in the exchanges.
24 October 2013
17:15-18:15
University of Oslo
HG G 43
Abstract: We consider portfolio theory and portfolio separation properties for a class of possibly non-symmetric heavy-tailed distributions under restrictions of no short sale or no riskless opportunity. The talk will review the classic result and show new generalizations, some of which possibly interesting for insurance and some of which likely interesting only to the curious, including a one-fund separation result for an asymmetric 1-stable distribution. The talk will discuss preliminary results on when separation actually implies existence of a CAPM, and when the distributions are incompatible with a market equilibrium; this part is based on work in progress and will pose some open questions for discussion. The approach will be simple from a technical point of view, as even the continuous-time case can be covered without Itô stochastic calculus - provided we can fit the distributions to continuous time, with or without infinite divisibility.
31 October 2013
17:15-18:15
Prof. Dr. Jean-Charles Rochet
Universität Zürich
HG G 43
Abstract: the objective of the paper is to develop a structural (yet tractable) model of corporate default in continuous time, in the presence of fi nancial frictions. We build on DeMarzo and Sannikov (2006) and Biais, Mariotti, Plantin, and Rochet (2007). The difference with these papers is that the cash flow of the firm is absolutely continuous and persistent. The shareholders select a dividend policy and a closure policy that maximizes the value of the firm. Our results encompass the real option model of Dixit Pyndick and the moral hazard model of DeMarzo and Sannikov (2006) and Biais, Mariotti, Plantin, and Rochet (2007).
* 5 November 2013
15:15-17:00
Prof. Dr. Ruodu Wang
University of Waterloo, CA
HG G 19.1
Abstract: Inappropriate modelling and misused quantitative methods in financial risk management could lead to severe model risk. One of the most challenging model risk lies in modelling the dependence between individual risks. The risk arises from a statistical challenge - we usually do not practically know how the individual risks are dependent. To give a proper mathematical framework to study the model risk in dependence, we introduce the Admissible Risk Class as the set of all possible risk aggregation when the marginal distributions of individual risks are fixed the dependence structure is uncertain. The concept provides flexibility for the analysis of dependence uncertainty. We will also discuss convex ordering bounds over an admissible risk class, which can be used to identify extreme scenarios for risk aggregation and calculate bounds on VaR, convex risk measures and other quantities of interest.
21 November 2013
No talk due to special event
28 November 2013
17:15-18:15
Prof. Dr. Mathieu Rosenbaum
Université Pierre et Marie Curie and École Polytechnique
HG G 43
Abstract: Because of their tractability and their natural interpretations in term of market quantities, Hawkes processes are nowadays widely used in high frequency finance. However, in practice, the statistical estimation results seem to show that very often, only nearly unstable Hawkes processes are able to fit the data properly. By nearly unstable, we mean that the L1 norm of their kernel is close to unity. We study in this work such processes for which the stability condition is almost violated. Our main result states that after suitable rescaling, they asymptotically behave like integrated Cox Ingersoll Ross models. Thus, modeling financial order flows as nearly unstable Hawkes processes may be a good way to reproduce both their high and low frequency stylized facts. We then extend this result to the Hawkes based price model introduced by Bacry et al. We show that under a similar criticality condition, this process converges to a Heston model. Again, we recover well known stylized facts of prices, both at the microstructure level and at the macroscopic scale.
5 December 2013
17:15-18:15
Prof. Dr. Stefan Ankirchner
Universität Bonn
HG G 43
Abstract: We consider the Skorokhod embedding problem (SEP) for a general time-homogeneous diffusion X: given a distribution ρ, is there a stopping time τ such that the stopped process X has the distribution ρ? We present a solution method that makes use of martingale representations and draws on law uniqueness of weak solutions of SDEs. Then we ask if there exist solutions of the SEP which are respectively finite almost surely, integrable or bounded, and when does our proposed construction have these properties. We provide conditions that guarantee existence of finite time solutions. Moreover, we fully characterize the distributions that can be embedded with integrable stopping times, and we derive necessary, respectively sufficient, conditions under which there exists a bounded embedding. Finally we apply the results to winner-take-all contests where agents aim at stopping a process at a highest possible value. The talk is based on joint work with David Hobson and Philipp Strack.
12 December 2013
17:15-18:15
Prof. Dr. Thomas Mikosch
University of Copenhagen
HG G 43
Abstract: This is joint work with Richard A. Davis (Columbia University) and Yuwei Zhao (University of Copenhagen). Davis, Mikosch (Bernoulli 2009; J. Econometrics 2012) proposed the extremogram and its sample analog as a simple means to measure and estimate serial extremal dependence in a time series. The extremogram is a lag-wise tail dependence coefficient and can be understood as an autocorrelation function of the underlying time series which takes into account only the extremal events in the series. In view of the latter interpretation, it is also possible to consider the corresponding spectral density and to estimate it by using some kind of a periodogram. We study the properties of this periodogram and its integrated versions and illustrate how to detect periodic cyles in the time series of extremal returns. The latter approach is advocated in Mikosch, Zhao (Bernoulli 2013, to appear).

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