Oberseminar Finanz- und Versicherungsmathematik LMU – TUM
F. Biagini, C. Czado, C. Klüppelberg, T. Meyer-Brandis, M. Scherer, R. Stelzer, R. Zagst
Wintersemester 2009 / 2010
Ort: Mathematisches Institut der Ludwig-Maximilians-Universität, Theresienstraße 39, 80333 München, Raum HS A 027
Zeit: Donnerstag, 17.15 - 18.15 Uhr (bei Doppelvorträgen 16.15 - 18.15 Uhr)
Alle Interessenten sind herzlich eingeladen!
29.10.2009
16.15 – 17.15
Nathalie Packham, Frankfurt School of Finance and Management
Latin hypercube sampling with dependence and applications in finance
Abstract: In Monte Carlo simulation, Latin hypercube sampling (LHS) (McKay et al., 1979) is a well-known variance reduction technique for vectors of Independent random variables. The method presented here, Latin hypercube sampling with dependence (LHSD), extends LHS to vectors of dependent random variables. The resulting estimator is shown to be consistent and asymptotically unbiased. For the bivariate case and under some conditions on the joint distribution, a Central Limit Theorem together with a closed formula for the limit variance is derived. It is shown that for a class of estimators satisfying some monotonicity condition, the LHSD limit variance is never greater than the corresponding Monte Carlo limit variance. In some valuation examples of financial payoffs, when compared to standard Monte Carlo simulation, a variance reduction of factors up to 200 is achieved. We illustrate that LHSD is suited for problems with rare events and for high-dimensional problems and that it may be combined with Quasi-Monte Carlo methods.
17.15 – 18.15
Johannes Muhle-Karbe, Universität Wien
A Characterization of the Martingale Property of Exponentially Affine Processes (joint work with Eberhard Mayerhofer and Alexander Smirnov)
Abstract: We consider local martingales of exponential form M = exp(X) or E(X) where X denotes one component of a multivariate affine process in the sense of Duffie, Filipovic and Schachermayer (2003). By completing the characterization of conservative affine processes, we give deterministic necessary and sufficient conditions in terms of the parameters of X for M to be a true martingale.
05.11.2009
17.15 – 18.15
Alexander Schnurr, TU Dortmund
A New Approach to the Analysis of Markov Semimartingales
Abstract: Every (nice) Feller process is a semimartingale. Using a probabilistic formula, we calculate the so called symbol of the process. What we obtain is a negative definite function (in the sense of Schoenberg) which depends on the starting point x. Let us mention that in the case of Levy processes the symbol coincides with the characteristic exponent. It is an interesting fact that - using the symbol - one directly obtains the semimartingale characteristics of the process X. Furthermore we show that the notion of the symbol can be generalized from (nice) Feller processes to Ito processes (in the sense of Cinlar, Jacod, Protter and Sharpe). This opens the door for a direct approach to some of the results of their (interesting but very technical) paper. We associate so called indices with the symbol which are - in a certain sense- generalizations of the well known Blumenthal-Getoor index and show that several (path-)properties of the process (e.g. p-variation, Hausdorff dimension) can be characterized in terms of these indices. As an example we consider the COGARCH process and the solution of a stochastic differential equation which is driven by a Levy process.
References:
[1] Cinlar, E., Jacod, J., Protter, P., Sharpe, M.: Semimartingales and Markov Processes. In: Z. Wahrscheinlichkeitstheorie verw. Gebiete 54 (1980), 161-219.
[2] Jacob, N., Schilling, R. L.: Levy-Type Processes and Pseudodifferential Operators, Barndorff-Nielsen, O. E., et al. (eds.) Levy processes: Theory and Applications, Birkhäuser, Boston, 139- 168, 2001.
[3] Schilling R. L., Schnurr, A.: The Symbol Associated to the Solution of an SDE, in preparation.
12.11.2009
17.15 – 18.15
Andrea Barth, CMA, Universität Oslo
Finite Element Method for Stochastic Partial Differential Equations and Applications
Abstract: In this talk we present convergence results for a Finite Element Method for parabolic and hyperbolic SPDE's driven by martingales. Convergence is shown in mean-square sense and almost surely. As an application for hyperbolic SPDE's we introduce an HJM-approach to forward prices in energy markets. We will conclude with simulations.
19.11.2009
17.15 – 18.15
Carole Bernard, University of Waterloo, Ontario (joint work with Phelim P. Boyle)
Explicit Representation of Cost Efficient Strategies
Abstract: This paper provides an explicit representation of cost efficient strategies in the context of investment choice among risky payoffs. It builds on and complements the earlier work of Dybvig (1988 a, b) and Cox and Leland (1982) which explored the relationship between efficiency and path-independence. We make the following assumptions:
(1) Agents preferences depend only on the probability distribution of terminal wealth.
(2) Agents prefer more to less.
(3) The market is perfect and frictionless.
(4) The market is arbitrage-free and all agents agree on a pricing operator.
Under these assumptions and given the desired probability distribution of final wealth, we provide an explicit expression for the optimal payoff. The key idea is that among the set of payoffs that have the same distribution we are able to explicitly characterize the one with the lowest cost. In particular, we are able to characterize the payoff of a financial derivative that strictly dominates any specified dynamic investment strategy with a path-dependent payoff in the sense of first order stochastic dominance. We illustrate these results by some new examples in the Black and Scholes framework. These examples show that some well known path independent payoffs such as put options are inefficient in our context. We also show that power options dominate geometric Asian options in the sense of first order stochastic dominance and characterize these power options explicitly. In the last section of the paper, we discuss why these results may not hold in practice.
26.11.2009
16.15 – 17.15
Eckhard Platen, University of Technology, Sydney
Asset Markets and Monetary Policy (joint research with Willy Semmler, The New School, New York)
Abstract: Monetary policy has pursued the concept of inflation targeting. This has been implemented in many countries. Here interest rates are supposed to respond to an inflation gap and output gap. Despite long term continuing growth of the world financial assets, recently, monetary policy, in particular in the U.S. during the financial crisis, was challenged by severe disruptions and a meltdown of the financial market. Subsequently, academics have been in search of a type of monetary policy that does allow to influence in an appropriate manner the investor's behavior and, thus, the dynamics of the economy and its financial market. We suggest here a dynamic portfolio approach. It allows one to study the interaction between investors` behavior and strategic monetary policy. The paper derives rules that explain how monetary authorities should set the short term interest rate in interaction with economic growth, asset prices, inflation rate, risk aversion, asset price volatility and consumption rates. Interesting is that the inflation rate needs to have a certain minimal level to allow the interest rate to be a viable control instrument.A particular target interest rate has been identified for the desirable regime. If the proposed monetary policy rule is applied properly, then the consumption rate will remain stable and the inflation rate can be kept close to a minimal possible level.
Literature:
Platen, E. & Heath, D.: A Benchmark Approach to Quantitative Finance,Springer Finance (2006).
Platen, E. & Semmler, W.: Asset markets and monetary policy, QFRC Research Paper 247 (2009).
17.15 – 18.15
Hans Manner, University of Maastricht
Dynamic stochastic copula models: Estimation, inference and applications (joint work with Christian M. Hafner)
Abstract: We propose a new dynamic copula model where the parameter characterizing dependence follows an autoregressive process. As this model class includes the Gaussian copula with stochastic correlation process, it can be viewed as a generalization of multivariate stochastic volatility models. Despite the complexity of the model, the decoupling of marginals and dependence parameters facilitates estimation. We propose estimation in two steps, where first the parameters of the marginal distributions are estimated, and then those of the copula. Parameters of the latent processes (volatilities and dependence) are estimated using efficient importance sampling (EIS). We discuss goodness-of-fit tests and ways to forecast the dependence parameter. For two bivariate stock index series, we show that the proposed model outperforms standard competing models.
03.12.2009
10.12.2009
17.12.2009
17.15 - 18.15
Martina Zähle, Universität Jena
Pathwise integrals and stochastic (partial) differential equations
Abstract: A survey on several methods for defining pathwise (stochastic) integrals is given, which lead to the same calculus. We demonstrate their application to SDE with fractal driving processes and give a new pathwise interpretation of classical stochastic versions for smooth coefficients.
By means of fractional calculus in Banach spaces and duality methods in function spaces the pathwise approach has been extended to certain parabolic SPDE in Euclidean and more general metric measure spaces (joint work with M. Hinz).
07.01.2010
14.01.2010
16.15 - 17.15
Peter Imkeller, Humboldt-Universität zu Berlin
Utility indifference hedging, BSDE of quadratic growth and measure solutions
Abstract: A financial market model is considered on which agents (e. g. insurers) are subject to an exogenous financial risk, which they trade by issuing a risk bond. They are able to invest in a market asset correlated with the exogenous risk. We investigate their utility maximization problem, and calculate bond prices using utility indifference. This hedging concept is interpreted by means of martingale optimality, and solved with BSDE with drivers of quadratic growth in the control variable. We investigate a new concept of solutions for BSDE of this type, which we call measure solutions and which corresponds to the concept of risk-neutral measures in arbitrage theory. We show that strong solutions of BSDE induce measure solutions, and present an algorithm by which measure solutions can be constructed without reference to strong ones. It yields solutions in new scenarios. For the case of unbounded terminal conditions existence and uniqueness questions become very difficult. We illustrate a wealth of different scenarios by giving examples and counterexamples. This is joint work with S. Ankirchner, A. Fromm, G. Heyne, Y. Hu, M. Müller, A. Popier, J. Zhang.
17.15 - 18.15
Stefan Tappe, ETH, Zürich
Option pricing in tempered stable stock models
Abstract: We apply tempered stable processes for modeling the evolution of stock prices. For this purpose, we perform a thorough investigation of tempered stable distributions, which is of independent interest. Afterwards we deal with option pricing in our financial model. Due to the incompleteness of the market, we shall discuss several choices of martingale measures.
21.01.2010
17.15 - 18.15
Frank Seifried, Universität Kaiserslautern
Optimal Investment for Worst-Case Crash Scenarios: A Martingale Approach
Abstract: We investigate the optimal portfolio problem under the threat of a financial market crash. We consider a multi-dimensional jump-diffusion market and devise a non-probabilistic crash model. Thus we assume that the investor seeks to maximize CRRA utility in the worst possible crash scenario. We recast the problem as a stochastic differential game between the investor and the market. With the help of the fundamental notion of indifference strategies, we are then able to completely solve the portfolio problem. It turns out that optimal investment strategies exhibit some interesting and empirically important properties.
28.01.2010
17.15 - 18.15
Carl Lindberg Charlmers, Universität Göteborg
Optimal Trading in Volatility
Abstract: We study the problem of how to trade in options, given that the investor has a view on the true future realized volatility. First, we derive the optimal liquidation strategy for a digital option. The problem is formulated as an optimal stopping problem, which we solve explicitly. Further, we analyze how to choose the strike price as one enters the position, and prove some monotonicity properties of the optimal liquidation strategy. We use the results for the digital option to find approximate optimal liquidation and switching strategies for a call spread and a call option. Finally, we investigate the optimal trading strategy if the investor is allowed to trade digital options and call options continuously.
04.02.2010
11.02.2010