Talks in Financial and Insurance Mathematics

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

Autumn Semester 2016

Note: The highlighted event marks the next occurring event.

Date / Time Speaker Title Location
25 August 2016
No seminar
1 September 2016
Marcel Nutz 
Columbia University
A Mean Field Game of Optimal Stopping  HG G 43 
Abstract: We formulate a stochastic game of mean field type where the agents solve optimal stopping problems and interact through the proportion of players that have already stopped. Working with a continuum of agents, typical equilibria become functions of the common noise that all agents are exposed to, whereas idiosyncratic randomness can be eliminated by an Exact Law of Large Numbers. Under a structural monotonicity assumption, we can identify equilibria with solutions of a simple equation involving the distribution function of the idiosyncratic noise. Solvable examples allow us to gain insight into the uniqueness of equilibria and the dynamics in the population.
29 September 2016
Yan Dolinsky 
Hebrew University of Jerusalem
Super-Replication with Constant Transaction Costs  HG G 43 
Abstract: We study super-replication of contingent claims in markets with fixed transaction costs. First we prove that in reasonable continuous time financial market the super-replication price is prohibitively costly and leads to trivial buy-and-hold strategies. Our second result is deriving non trivial scaling limits of super-replication prices in the binomial models. joint work with Peter Bank
13 October 2016
Bruno Bouchard 
Ceremade, Université Paris-Dauphine
Stochastic invariance of closed sets with non-Lipschitz coefficients  HG G 43 
Abstract: We provide a new characterization of the stochastic invariance of a closed subset with respect to a diffusion. We extend the well-known inward pointing Stratonovich drift condition to the case where the diffusion matrix can fail to be differentiable: we only assume that the covariance matrix is. In particular, our result can be directly applied to construct affine and polynomial diffusions on any arbitrary closed set.
20 October 2016
Peter Tankov 
Université Paris-Diderot
Optimal importance sampling for Lévy processes  HG G 43 
Abstract: We develop importance sampling estimators for Monte Carlo pricing of European and path-dependent options in models driven by Lévy processes, extending earlier works focusing on the Black-Scholes and continuous stochastic volatility models. Using recent results from the theory of large deviations for processes with independent increments, we compute an explicit asymptotic approximation for the variance of the pay-off under an Esscher-style change of measure. Minimizing this asymptotic variance using convex duality, we then obtain an importance sampling estimator of the option price. Numerical tests in the variance gamma model show consistent variance reduction with a very small computational overhead. (Adrien Genin and Peter Tankov)
27 October 2016
Tim Boonen 
Universiteit van Amsterdam
Capital allocation for portfolios with non-linear risk aggregation  HG G 43 
Abstract: Existing risk capital allocation methods, such as the Euler rule, work under the explicit assumption that portfolios are formed as linear combinations of random loss/profit variables, with the firm being able to choose the portfolio weights. This assumption is unrealistic in an insurance context, where arbitrary scaling of risks is generally not possible. Here, we model risks as being partially generated by Lévy processes, capturing the non-linear aggregation of risk. The model leads to non-homogeneous fuzzy games, for which the Euler rule is not applicable. For such games, we seek capital allocations that are in the core, that is, do not provide incentives for splitting portfolios. We show that the Euler rule of an auxiliary linearized fuzzy game (non-uniquely) satisfies the core property and, thus, provides a plausible and easily implemented capital allocation. In contrast, the Aumann-Shapley allocation does not generally belong to the core. For the non-homogeneous fuzzy games studied, Tasche's (1999) criterion of suitability for performance measurement is adapted and it is shown that the proposed allocation method gives appropriate signals for improving the portfolio underwriting profit. This presentation is based on joint work with Andreas Tsanakas (Cass Business School) and Mario Wüthrich (ETH Zürich).
10 November 2016
Michaela Szölgyenyi 
Vienna University of Economics and Business
A numerical method for SDEs appearing in insurance and financial mathematics  HG G 43 
Abstract: When solving certain stochastic control problems in insurance- or financial mathematics, the optimal control policy sometimes turns out to be of threshold type, meaning that the control depends on the controlled process in a discontinuous way. The stochastic differential equations (SDEs) modeling the underlying process then typically have a discontinuous drift coefficient. This motivates the study of a more general class of such SDEs. We prove an existence and uniqueness result, based on a certain transformation of the state space by which the drift is “made continuous”. As a consequence the transform becomes useful for the construction of a numerical method. The resulting scheme is proven to converge with strong order 1/2. This is the first scheme for which strong convergence is proven for such a general class of SDEs with discontinuous drift. Joint work with G. Leobacher (JKU Linz)
17 November 2016
Pavel Shevchenko 
Macquarie University
Crunching Mortality and Annuity Portfolios with extended CreditRisk+  HG G 43 
Abstract: Using an extended version of the credit risk model CreditRisk+, we develop a flexible framework that provides a unified and stochastically sound approach to model mortality, allowing underlying stochastic risk factors on the one hand and risk aggregation in annuity portfolios on the other hand. Deaths are driven by common stochastic risk factors which may be interpreted as death causes like neoplasms, circulatory diseases or idiosyncratic components. These common factors introduce dependence between policyholders in the annuity portfolios or between death events in population. The approach provides an efficient and numerically stable algorithm for an exact calculation of the one-period loss distribution where various sources of risk are considered. As required by many regulators, we can then derive risk measures for the one-period loss distribution such as value-at-risk and expected shortfall. Using publicly available data, we provide estimation procedures for mortality model parameters including classical approaches, as well as Markov chain Monte Carlo methods. The model allows stress testing and offers insight into how certain health scenarios influence annuity payments of an insurer. Such scenarios may include outbreaks of epidemics, improvement in health treatment, or development of better medication. We conclude with a real world example using Australian death data and present long-term forecast for life expectancy and death probabilities due to different causes.The talk is based on recent papers: 1) J. Hirz, U. Schmock and P.V. Shevchenko and (2016). Crunching mortality and annuity portfolios with extended CreditRisk Plus. Preprint, 2) J. Hirz, U. Schmock and P.V. Shevchenko (2016). Modelling Annuity Portfolios and Longevity Risk with Extended CreditRisk Plus. Preprint, 3) P.V. Shevchenko, J. Hirz and U. Schmock (2015). Forecasting Leading Death Causes in Australia using Extended CreditRisk+. In T. Weber, M. J. McPhee, and R. S. Anderssen (Eds.), MODSIM2015, 21st International Congress on Modelling and Simulation. Modelling and Simulation Society of Australia and New Zealand, pp. 966-972. ISBN: 978-0-9872143-5-5.
24 November 2016
Johannes Muhle-Karbe 
University of Michigan
Equilibrium Liquidity Premia  HG G 43 
Abstract: In a continuous-time model with mean-variance investors and quadratic transaction costs, we show that the equilibrium expected return can be characterized as the solution of a system of coupled but linear forward-backward stochastic differential equations. Explicit formulas obtain in the small-cost limit, which allow to assess the comparative statics of equilibrium liquidity premia. (Joint work with Masaaki Fukasawa and Martin Herdegen)
8 December 2016
Marcus C Christiansen 
Heriot Watt University
Backward stochastic differential equations in life insurance mathematics  HG G 43 
Abstract: The concept of BSDEs primarily entered the life insurance literature as a tool to deal with financial risk, based on the fact that financial risk plays a major role in life insurance. Only recently BSDEs are also discussed as useful tools for distinctive insurance problems that are not just adopted from finance. The talk gives an overview of recent developments in the life insurance literature and discusses several BSDE applications in detail. In particular the talk covers the following problems: decomposition of risk, non-Markovian biometrical modelling, and circularly defined benefit payments. The latter question leads to a class of mean-field BSDEs that seem to be new in the literature.

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