Graduate Seminar Financial- and Actuarial Mathematics LMU and TUM (WS 2013/14)
Organized By
Prof. F. Biagini, Prof. C. Czado, Prof. K. Glau, Prof. C. Klüppelberg, Prof. T. Meyer-Brandis, Prof. M. Scherer, Prof. G. Svindland, Prof. R. Zagst
Time and Venue
Mon 14:15 to 17:00 - Business Campus, Parkring 11, Garching-Hochbrück
Date, Time and Venue | Speaker | Title |
---|---|---|
31.10.13 | ||
14:00 - 15:00 | Lorenz Schneider | |
04.11.13 |
| |
14:15 - 15:00 | Michael Eichler | |
15:00 - 15:45 | Antoine Hakassou | |
Break | ||
16:15 - 17:00 | Ilia Negri | Optimal prediction-based estimating function for COGARCH(1,1) models |
02.12.13 | ||
14:15 - 15:00 | Ernst Eberlein | Lévy driven two price valuation with applications to long-dated contracts |
15:00 - 15:45 | Axel Bücher | |
Break | ||
16:15 - 17:00 | Michael Kupper | tba |
13.01.14 | ||
14:15 - 15:00 | Carole Bernard | Assessing Model Risk With Partial Information on Dependence In High Dimensions |
15:00 - 15:45 | Johanna Ziegel | |
Break | ||
16:15 - 17:00 | Markus Reiss | Semiparametrically efficient estimation of the quadratic covariation matrix under noise |
03.02.14 | ||
14:15 - 15:00 | Mathias Beiglböck | Model-Independent Finance, Optimal Transport and Skorokhod Embedding |
Model-Independent Finance, Optimal Transport and Skorokhod Embedding
by Mathias Beiglböck
Model-independent pricing has grown into an independent field in Mathematical Finance during the last 15 years. A driving inspiration in this area has been the fruitful connection to the Skorokhod embedding problem. We discuss a more recent approach to model-independent pricing, based on a link to Monge-Kantorovich optimal transport. This transport-viewpoint also sheds new light on Skorokhod's classical problem.
Assessing Model Risk With Partial Information on Dependence In High Dimensions
by Carole Bernard
A central problem in quantitative risk management concerns the evaluation of the risk of a portfolio, ie the sum S of n individual risks Xi. Solving this problem is mainly a numerical task once the joint distribution of (X1,X2, . . . ,Xd) is completely specified. Unfortunately, while the marginal distributions of the risks Xi are often known, their interaction (dependence) is usually either unknown or only partially known, implying that any computed risk measure of S is subject to model error. Previous academic research has provided us with maximum and minimum possible values for risk measures when only the marginal distributions are assumed to be known (unconstrained bounds). This approach leads to wide bounds as all information on the dependence is ignored. In this paper, we are able to also consider the availability of dependence information in the computation of the bounds. We provide analytic bounds that are easy to compute but not always sharp. We also provide algorithms that allow to obtain sharp bounds approximately. Interestingly, the approximate sharp bounds match closely the ones that are obtained analytically. Numerical illustrations show that our approach leads to bounds that are significantly tighter than the (unconstrained) ones available in the literature.
This is joint work with Steven Vanduffel.
Nonparametric inference on Lévy measures and copulas
by Axel Bücher
In this talk nonparametric methods to assess the multivariate Lévy measure are introduced. Starting from high frequency observations of a Lévy process X, we construct estimators for its tail integrals and the Pareto Lévy copula and prove weak convergence of these estimators in certain function spaces.
Lévy driven two price valuation with applications to long-dated contracts
by Ernst Eberlein
In the classical valuation theory the law of one price prevails and market participants trade freely in both directions at the same price. This approach is appropriate for highly liquid markets. In the absence of perfect liquidity the law of one price should be replaced by a two price valuation theory where market participants continue to trade freely with the market but the terms of trade now depend on the direction of the trade. We develop here a static as well as a continuous time theory for two price economies. The two prices are termed bid and ask or lower and upper price but they should not be confused with the literature relating bid-ask spreads to transaction costs or other frictions involved in modeling financial markets. The bid price arises as the infimum of test valuations whereas the ask price is the supremum of such valuations. The two prices are related to nonlinear expectations. Probability as well as measure distortions are used to make this approach operational. We consider specific models where the uncertainty is given by purely discontinuous Lévy processes. The approach is illustrated to price stochastic perpetuities, i.e. contracts with no apparent maturity, and to value compound Poisson processes of insurance loss liabilities. This is joint work with Dilip Madan, Martijn Pistorius, Wim Schoutens and Marc Yor.
Causal inference from multiple time series
by Michael Eichler
In time series analysis, inference about cause-effect relationships among multiple time series is commonly based on the concept of Granger causality, which exploits temporal structure to achieve causal ordering of dependent variables. One major and well known problem in the application of Granger causality for the identification of causal relationships is the possible presence of latent variables that affect the measured components and thus lead to so-called spurious causalities.
We present a new graphical approach for describing and analysing Granger-causal relationships in multivariate time series that are possibly affected by latent variables. It is based on mixed graphs in which directed edges represent direct influences among the variables while dashed edges - directed or undirected - indicate associations that are induced by latent variables. We show how such representations can be used for inductive causal learning from time series and discuss the underlying assumptions and their implications for causal learning. Finally we will discuss tetrad constraints in the time series context and how the can be exploited for causal inference.
Recent results and perspectives on IDT processes
by Antoine Hakassou
This talk deals with IDT processes, i.e. processes which are infinitely divisible with respect to time. A survey of some well-known properties of these processes is done. Based on notions of Lévy sheet, Sato sheet, Gaussian sheet, numerous examples of IDT processes and their associated Lévy processes are given. Some transformations of IDT processes are discussed and some open questions are exposed.
Optimal prediction-based estimating function for COGARCH(1,1) models
by Ilia Negri
The COGARCH (Continuous Generalized Auto-Regressive Conditional Heteroschedastic) model was introduced by Kluppelberg et al. (2004) as a continuous-time version of the GARCH model. Log-returns and volatility are modelled as solutions of stochastic differential equations driven by a Lévy processes. To estimate the parameters of the model, Haug et al. (2007) proposed moment estimators. Prediction-based estimating functions (PBEFs) were introduced by Sorensen (2000 and 2011) as a generalization of martingale estimating functions. They are based on linear predictors, have some of the most attractive properties of the martingale estimating functions, moreover an optimal prediction-based estimating function can be found. In this work (jointly with Enrico Bibbona, University of Torino) PBEFs are derived to draw statistical inference about the COGARCH(1,1) model from discretely observed data. To find the optimal PBEF, the explicit expression of some higher order moments of the COGARCH(1,1) model are needed. We propose iterative formulae to calculate the n-th order moments of the volatility process and of the log-returns under suitable hypothesis. Some simulation studies are presented to investigate the empirical quality of the proposed estimators compared with the one obtained with the method of moment.
Semiparametrically efficient estimation of the quadratic covariation matrix under noise
by Markus Reiß
An efficient estimator is constructed for the quadratic covariation or integrated covolatility matrix of a multivariate continuous martingale based on noisy and non-synchronous observations under high-frequency asymptotics. Our approach relies on an asymptotically equivalent continuous-time observation model where a local generalised method of moments in the spectral domain turns out to be optimal. Asymptotic semiparametric efficiency is established in the Cramér-Rao sense. The efficient covariance structure shows surprising geometric features. Main findings are that non-synchronicity of observation times has no impact on the asymptotics and that major efficiency gains are possible under correlation.
Based on joint work with Markus Bibinger, Nikolaus Hautsch and Peter Malec.
Entropy, Relative Entropy and Variance Swaps
by Lorenz Schneider
We study the problem of finding probability densities that match given European call option prices. To allow prior information about such a density to be taken into account, we generalize a previously presented algorithm to find the maximum entropy density of an asset price to the relative entropy case. This is applied to study the impact the choice of prior density has in two market scenarios. In the first scenario, call option prices are prescribed at only a small number of strikes, and we see that the choice of prior, or indeed its omission, yields notably different densities. The second scenario is given by CBOE option price data for S&P 500 index options at a large number of strikes. Prior information is now considered to be given by calibrated Heston, Schöbel-Zhu or Variance Gamma models. We find that the resulting digital option prices are essentially the same as those given by the (non-relative) Buchen-Kelly density itself. In other words, in a sufficiently liquid market the influence of the prior density seems to vanish almost completely. Finally, we study variance swaps and derive a simple formula relating the fair variance swap rate ( entropy. Then we show, again, that the prior loses its influence on the fair variance swap rate as the number of strikes increases.
Coherence and elicitability
by Johanna Ziegel
The risk of a financial position is usually summarized by a risk measure. As this risk measure has to be estimated from historical data, it is important to be able to verify and compare competing estimation procedures. In statistical decision theory, risk measures for which such verification and comparison is possible, are called elicitable. It is known that quantile based risk measures such as value-at-risk are elicitable. However, the coherent risk measure expected shortfall is not elicitable. Hence, it is unclear how to perform forecast verification or comparison. We address the question whether coherent and elicitable risk measures exist (other than minus the expected value). We show that one positive answer are expectiles, and that they play a special role amongst all elicitable law-invariant coherent risk measures.