Talks in Financial and Insurance Mathematics, Winter 1997/98

• Stephan Ommerborn (Feilmeier & Junker)
  Die Modellierung tarifloser Produkte - Der Aktuar der 4. Art?

• Wolfgang Runggaldier (University of Padua)
  On a factor model for the term structure
  
  Abstract: We consider a non-affine multifactor model for the term structure that does not allow direct observations of the state/factors from the yield curve. In this model, that can also be interpreted as an (infinite-dimensional) regression model, we consider explicitly a residual (''nuisance factor''). With the information of an economic agent given by the observations of a finite number of yields corresponding to (liquid) bonds with given maturities, the state variables/factors can then be filtered from these observations. Our objective is to compute, on the basis of this information, arbitrage-free prices of (illiquid) bonds and of bond derivatives by proceeding along two successive steps: (i) perform the calculations under the assumption that the parameters of the model are known; (ii) use the results in (i) to identify these parameters (calibrating the model to market data) and performing then the calculations with the identified parameters. We also show how, besides Monte-Carlo type simulations, (approximation) methods from stochastic filtering and optimal stochastic control can be useful to perform the calculations, whenever direct analytic computation is not feasible. (Based on work in progress related to a joint project with Tomas Björk and Andrea Gombani.)

• Serge Déteindre (SBC Warburg Dillon Read)
  Als Physiker im Derivatehandel
  
  (Seminar für Finanzmathematik, Angewandte Mathematik, Universität Zürich)

• Josef Kupper (ETH Zürich)
  Versicherungsmathematik und schweizerische Hochschulen
  
  

• Ivo Angehrn (EPF Lausanne, ETH Zürich)
  Diploma project: Some features of Lévy processes in finance
  
  Abstract: One argument against continuous option pricing models is that real data are sampled at discrete time points. There is no direct economic reason why the "underlying continuous-time process" should be continuous. Moreover, we want to suppose a stationary time structure of the returns, so that the only admissible continuous stock price model is geometric Brownian motion, but this is not flexible enough. Lévy processes provide a class of processes which is at the same time sufficiently large and still well tractable in calculations. Examples include time-changed Brownian motion via subordinators (positive Lévy processes) to model uncertainty in stock price processes and the use of gamma processes at the place of Brownian motion in interest rate models. Such models have been introduced for instance by Madan and are used in practice.

• Vladimir Anisimov (Bilkent University, Ankara, Turkey and Kiev University, Ukraine)
  Asymptotic analysis of extreme points of random functions and applications in statistics.
  
  Abstract: Asymptotic behaviour of extreme points (sets) of random functions (fields) as well as solutions of general stochastic equations is studied. Behaviour of trajectories of extreme points as set-valued processes in time is also considered. Applications to the analysis of maximum likelihood, moment's and least squares method estimators constructed by observations on trajectories of switching stochastic systems in transient and stationary conditions are considered. Analysis of estimators as set-valued processes depending on the length of the interval of observations is also carried out. Results given in applications are essentially based on the functional averaging principle and diffusion approximation type theorems for switching processes.

• Paul Embrechts (ETH Zürich)
  Actuarial versus financial pricing of insurance

• Laurens de Haan (Erasmus University, Rotterdam, The Netherlands)
  A bootstrap based method to achieve optimality in estimating the extreme value index

• Laurens de Haan (Erasmus University, Rotterdam, The Netherlands)
  Estimating the index of a stable distribution

• Laurens de Haan (Erasmus University, Rotterdam, The Netherlands)
  Sea and wind: multivariate extremes at work

• Freddy Delbaen (ETH Zürich)
  The passport option
  
  Abstract: A passport option gives the owner the right to take limited positions in an underlying security. At maturity the gains obtained by this strategy can be kept by the owner of the option, the losses will be paid by the writer of the option. The owner can use any strategy he wants as long as its position remains between fixed bounds (short or long). The problem is to determine the price, the optimal strategy as well as the hedge. The usual treatment of these problems is through PDE and viscosity solutions. The present presentation does not use these concepts and only uses stochastic calculus. It turns out the problem can be reduced to the calculation of an integral involving Brownian motion together with its local time at zero. Since this is well known, "explicit" solutions can be given.

• Hans Bühlmann (ETH Zürich)
  Risikoallokation und Finanzmärkte aus mathematischer Sicht
  
  Abstract: Gemäß Arrow kann in der Situation der Unsicherheit das Gleichgewicht einer Austauschökonomie durch ein zweistufiges Modell erreicht werden. Ich möchte vor allem die erste Stufe, den reinen Finanzmarkt diskutieren: Pareto Optima, Preisgleichgewichte und den Zusammenhang mit der Risikoaversion.

• Hélyette Geman (Paris)
  Transaction clock and asset price dynamics

• Claudio Albanese (University of Toronto, Canada)
  Equity options with stochastic volatility: efficient pricing algorithms and non-parametric dynamic hedging
  
  Abstract: Stochastic volatility models can explain the skew in implied volatility and allow for accurate calibration to option prices. We find a fourth-order finite difference equation that extends the Black-Scholes equation and accounts for the leading correction to prices in the limit of small volatility fluctuations. We have tested our approximate model on calibration problems in the literature and found that it performs as well as the full model. The talk also discusses some results on non-parametric delta and vega hedging.

• Damien Lamberton (Université de Marne la Vallée, France)
  Brownian optimal stopping problems and random walks
  
  Abstract: A natural numerical procedure to solve an optimal stopping problem along Brownian paths is to approximate Brownian motion by a random walk and solve the associated discrete optimal stopping problem via dynamic programming. The purpose of this talk is to present some results on the rate of convergence of this approximation. This study is motivated by American option pricing.

• Dr. Alois Gisler and Peter Mathis (Winterthur Insurance)
  Die WinCAT-Anleihe der Winterthur oder die Absicherung von Hagelschäden über der Kapitalmarkt

• Mete Soner (Carnegie Mellon University Pittsburg and University of Istanbul)
  Pricing derivative securities and quasi-variational inequalities
  
  Abstract: In this talk I will describe a pricing method for derivative securities in the presence of transaction costs. The result is a nonlinear parabolic equation whose solution is the price as a function of the current underlying asset and time to maturity. The mathematical tools used in this derivation are singular perturbations, stochastic optimal control and quasi-variational inequalities.

• Hans Föllmer (Humboldt-Universität, Berlin)
  Selected topics in mathematical finance (Minicourse, part IV)

• Marco Frittelli (University of Milan, Italy)
  Certainty equivalent and no arbitrage: a reconcilation via duality theory
  
  Abstract: We describe a general principle for the valuation problem in incomplete financial markets that reconciles the utility and martingale approaches. We provide a general criterion, for selecting one equivalent martingale measure, that requires minimizing an appropriate functional which depends on investors utility and we give sufficient conditions for the existence of the martingale measure that minimizes this functional. We then show that most existing financial criteria for pricing in incomplete markets are particular cases of our approach.

• Hans Föllmer (Humboldt-Universität, Berlin)
  Selected topics in mathematical finance (Minicourse, part III)

• Thorsten Rheinländer (TU Berlin)
  The family of q-optimal martingale measures and the minimal entropy martingale measure
  
  
• L. Krawzcyk (Universität Wien)
  Martingale inequalities in finance

• Susan Pitts (Cambridge University)
  Nonparametric estimation from busy and idle periods in queues
  
  Abstract: We consider the M/G/infinity queue and the M/G/1 queue. In each case we suppose that the input parameters, namely the Poisson arrival rate and the service time distribution, are unknown, but that data are available on the busy and idle periods for each queue. From these data, we construct estimators for quantities related to the unknown input parameters, and derive asymptotic properties of these estimators. This is joint work with Nick Bingham (Birkbeck College, University of London).

• Hans Föllmer (Humboldt-Universität, Berlin)
  Selected topics in mathematical finance (Minicourse, part II)

• Tomas Björk (University of Stockholm)
  Minimal realizations of forward rates
  
  Abstract: We consider interest rate models where the forward rates are allowed to be driven by a multidimensional Wiener process as well as by a marked point process. Assuming a deterministic volatility structure, and using ideas from systems and control theory, we investigate when the input-output map generated by such a model can be realized by a finite dimensional stochastic differential equation. We give necessary and sufficient conditions, in terms of the given volatility structure, for the existence of a finite dimensional realization and we provide a formula for the determination of the dimension of a minimal realization. The abstract state space for a minimal realization is shown to have an immediate economic interpretation in terms of a minimal set of benchmark forward rates, and we give explicit formulas for bond prices in terms of the benchmark rates as well as for the computation of derivative prices. (Joint work with Andrea Gombani, Padua)

• Hans and Peter Bühlmann (ETH Zürich)
  Model selection in credibility
  
  Abstract: Credibility estimation finds the least squares optimal fair premium as a linear Bayes estimate which results in a weighted avarage between collective and individual claims experience. The fair premium can often be modeled as remaining constant over time (Bühlmann-Straub model) but sometimes it is preferable to work with an underlying (linear) regression assumption (Hachemeister model). We propose to select the appropriate model on a data based criterion. In particular, we explicitely develop the criterion for the case "constant over time" against "simple linear regression". The criterion is obtained from straightforward least squares calculations and illustrated with simulation results. It is remarkable that this model selection approach in an empirical Bayes framework shows partially different phenomena than those obtained by classical statistical methods.

• Hans Föllmer (Humboldt-Universität, Berlin)
  Absicherung von Derivaten als wahrscheinlichkeitstheoretisches Problem: Was kommt nach der Black-Scholes-Formel?

• Freddy Delbaen (ETH Zürich)
  Coherent measures of risk

• Josef Kupper (Swiss Life)
  Geschichte der Versicherungsmathematik in der Schweiz

• Hans Föllmer (Humboldt-Universität, Berlin)
  Selected topics in mathematical finance (Minicourse, part I)

Page created and supported by Uwe Schmock until September 2003. Last update: September 13, 2003