PRisMa 2006

Prof. Dr. Uwe Schmock
(FAM @ TU Wien)
Welcome and Presentation of the Christian Doppler Laboratory for Portfolio Risk Management (PRisMa Lab)

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Dr. Johann Strobl
(Chief Financial Officer and Chief Risk Officer, Member of the Board of Directors of Bank Austria Creditanstalt)
Forschungskooperation aus der Sicht der BA-CA

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Prof. Dr. Walter Schachermayer
(FAM @ TU Wien)
Introduction of Prof. Josef Teichmann, Laureate of the START Prize

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Prof. Dr. Josef Teichmann
(FAM @ TU Wien)
Flexibility of OU-Interest Rate Models

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Dr. Stefan Gerhold
(PRisMa Lab, FAM @ TU Wien)
An Implementation of the LIBOR Market Model for Pricing Exotic Constant Maturity Swaps

Abstract: The LIBOR market model is by now the standard device for pricing interest rate derivatives. It simulates a set of spanning forward interest rates. Each of these rates is log-normal under its appropriate forward measure, and satisfies SDEs under the forward measures associated to the other tenor dates. The volatilities and correlations of the forward rates are calibrated to market data. Upon discretizing the SDEs, interest rate products can be priced by Monte Carlo simulation. The payoffs of the products we price depend on various swap rates and forward rates.
The LIBOR market model has lots of variants. There are several parametric shapes for the volatility and correlation of the forward rates. Moreover, the correlations are sometimes calculated from swaption volatilities, and sometimes from historical time series. The volatility parameters are always obtained from market cap volatilities. Other important parameters are the number of factors and the step size in the discretization of the SDEs. In the talk we present some of the consequences of these choices via computational results for swaptions and exotic products.

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Dr. Irina Slinko
(FAM @ TU Wien)
Finite State Space Representation of Forward Interest Rates on a Foreign Exchange Market

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Dr. Friedrich Hubalek
(FAM @ TU Wien)
Simple Explicit Variance-Optimal Hedging for Path-Dependent and Multi-Asset Derivatives

Abstract: We consider variance-optimal hedging and Markowitz-type portfolio optimization for processes with stationary independent increments in discrete and continuous time. We recapitulate earlier work for single assets and extend the approach to the multivariate case with and without trading constraints. This allows to determine the optimal hedge for combine options as e.g. baskets and spreads. Moreover, the results are applied to surrogate hedging, where options on untradable assets are hedged by trading in correlated securities. Then a further extension to path-dependent options is developed. The formulas involve the moment or cumulant generating function of the underlying and a multivariate Laplace transform representation of the payoff. These expressions are known in closed form for typically considered Levy processes and exotic options. Therefore the approach is well adapted for numerical evaluation. This talk is based on joint work with Jan Kallsen.

For slides of the talk please contact Dr. Friedrich Hubalek via e-mail (friedrich.hubalek@fam.tuwien.ac.at).

Dr. Jan Palczewski
(School of Mathematics, University of Leeds, UK)
Portfolio Optimisation with Economic Factors and Transaction Costs (full Abstract)

Abstract: In the talk we approach the problem of portfolio optimisation. We deal with existence of optimal strategies and smoothness of solutions to appropriate Bellman equations, which is a starting point of development of good numerical algorithms calculating optimal porfolios.
We present two results concerning portfolio optimisation problems on the markets with transaction costs and economic factors. The costs are bounded away from zero (e.g. constant plus proportional costs), which requires the use of impulsive control methodology. First problem treats a continuous-time model with the functional which is a sum of the reward of the consumption and the integral-type term depending on the portfolio and the state of the market. Second problem deals with a discrete-time model, where we maximise average growth rate of the portfolio. A characteristic trait of both results is that the methods used in the proofs are of functional character. We do not exploit the differential approach (quasi-variational inequalities), which allows us to obtain results in greater generality.

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Dr. Gregory Temnov
(PRisMa Lab, FAM @ TU Wien)
Combined Methodology for Modelling and Measuring Operational Risk

Abstract: We approach the classical problem of modelling and quantifying of the operational risk, using methods and experiences of the area of insurance mathematics. Models of Extreme Value Theory, based on the use of generalized Pareto distribution (GPD) and such common counting processes for number of events, as homogeneous Poisson process and negative binomial distribution, are adjusted to the given data.
More flexible models, taking into account the special characteristics of operational loss data, are considered. In particular,
• for fitting the severity distributions combinations of GPD with another heavy tailed distributions, such as the Weibull distribution, are used,
• for modelling the frequency of events inhomogeneous Poisson models (i.e. the intensity changes in time) are adopted.
• bayesian estimators for the parameters of distributions are applied as a method for proper mixing internal and external data.
The application of the methods mentioned allows to increase the accuracy of OR modelling, in order to avoid overestimation of the probability of extremely high losses, and also gives the view of dynamics of loss models. For the calculation of aggregate loss distribution and VaR two alternative methods were used: Monte Carlo simulations and methodology based on Fourier-transformation (FFT algorithm). Final VaR results, obtained with the three methods, show good correspondence. We also make an approach to the problem of analysis of operational data with respect to its two-dimensional classification in a relation to insurance problem, i.e. which cells of ”lines of business — types of events” matrix correspond to the most risky contents and thus worth being insured.

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DI Christian Bayer
(FAM @ TU Wien)
Discretization of SDEs: Euler Methods and Beyond (full Abstract)

Abstract: Stochastic differential equations (SDEs) are used in financial mathematics as models for the time-evolution of stocks. Furthermore, the Feynman-Kac-formula implies that an important class of PDEs can be represented using the solution to SDEs. Naturally, explicit solutions to SDEs can usually not be given.
The topic of the talk is approximation methods for SDEs. We focus on weak approximation, i. e. approximation of the expected value of a function of the solution of the SDE. We start with discussing construction of the approximating processes using a straightforward generalisation of the classical Euler method for ODEs, continue with higher order methods and also discuss some generalisations to other situations.

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DI Barbara Forster
(FAM @ TU Wien)
Computation of Price Sensitivities (full Abstract)

Abstract: Typically referred to as The Greeks, sensitivities in financial markets are defined as partial derivatives of the price of a contingent claim with respect to the underlying model parameters. To simulate these sensitivities numerically, one usually applies finite difference approximations. Unfortunately, this procedure can become very inefficient, especially when the payoff function is complex or discontinuous.
To by-pass this difficulty, Fournie et al. proposed to apply Malliavin calculus to shift the differential operator from the expected payoff function to the underlying diffusion kernel, using integration by parts and introducing a weighting function which is independent of the payoff function.
The objective of this talk is to extend this approach to jump-diffusion models, by conditioning on the jump times and relying on methods of Malliavin calculus for diffusion processes.
Moreover, we give some explicit examples for the calculation of Greeks and compare simulation results of price sensitivities obtained by finite difference approximations and by Malliavin-Monte Carlo methods.

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