Abstract
This talk is organized in two parts. In the first part we present the calibration of the deterministic volatility Libor market model to at-the-money cap and swaption volatilities. In the second part we discuss a variant of the Libor market model with stochastic volatility. As for the deterministic volatility Libor market model we require the parameterization of the model to be as time homogenous as possible. Here, this is achieved by using time homogenous mean reversion levels and speeds for the stochastic volatilities of the respective forward rates. Correct (perfect) pricing of the (at-the-money) caplets corresponds then to non-stationary initial values of the forward rate volatilities. However, demanding a time homogenous model restricts possible caplet smile surfaces.

slides

Abstract
The concept of symbolic dynamics, entropy and complexity measures has been widely utilized for the analysis of measured time series. However, little attention as been devoted to investigate the effects of choosing different partitions to obtain the coarse-grained symbolic sequences. Because the theoretical concepts of generating partitions mostly fail in the case of empirical data, one commonly introduces a homogeneous partition which ensures roughly equidistributed symbols. We will show that such a choice may lead to spurious results for the estimated entropy and will not fully reveal the randomness of the sequence.

preprint

Abstract
We consider time series of financial data as the Dow Jones Index with respect to the existence of local order. The basic idea is that in spite of the high stochasticity in average there might be special local situations where there local order exist and the predictability is considerably higher than in average. In order to check this assumption we discretise the time series and investigate the frequency of the continuation of definite words of length n first. We prove the existence of relatively long-range correlations under special conditions. The higher order Shannon entropies and the conditional entropies (dynamical entropies) are calculated, characteristic fluctuations are found. Instead of the dynamic entropies which yield mean values of the uncertainty/predictability we finally investigate the local values of the uncertainty/predictability and the distribution of these quantities.We consider time series of financial data as the Dow Jones Index with respect to the existence of local order. The basic idea is that in spite of the high stochasticity in average there might be special local situations where there local order exist and the predictability is considerably higher than in average. In order to check this assumption we discretise the time series and investigate the frequency of the continuation of definite words of length n first. We prove the existence of relatively long-range correlations under special conditions. The higher order Shannon entropies and the conditional entropies (dynamical entropies) are calculated, characteristic fluctuations are found. Instead of the dynamic entropies which yield mean values of the uncertainty/predictability we finally investigate the local values of the uncertainty/predictability and the distribution of these quantities.

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Abstract
The structure of time series and letter sequences is investigated using the concepts of entropy and complexity. First conditional entropy and transinformation are introduced and several generalizations are discussed. Further several measures of complexity are introduced and discussed. The capability of these concepts to describe the structure of time series and letter sequences generated by nonlinear maps, data series from meteorology, astrophysics, cardiology, cognitive psychology and finance is investigated. The relation between the complexity and the predictability of informational strings is discussed. The relation between local order and the predictability of time series is investigated.

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Abstract
In the given paper, we consider the scaling features of long letter sequences like human writings, discretized images and discretized financial data. Using several approaches we show that the symbolic strings and time series being analyzed have a complex multiscale structure and demonstrate different scalings for large and small fluctuations. We discuss complex phenomena in the scaling behavior of partition functions in the case of high frequency DAX-future data.

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Abstract
On experimental data the historical volatility is usually calculated by averaging the local variance (or its generalizations) over a finite time window. Already in the case of a constant volatility in the Gaussian model the resulting historical volatility is non-Gaussian distributed. We will calculate historical volatility distributions in the Gaussian and GARCH(1,1) model for different time window sizes and compare them with those obtained from the S&P500 data.

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Abstract
Linear and nonlinear correlations between price changes of the German stock index DAX and the implied volatility index VDAX on a daily basis, January 92 - September 98, are studied. A rescaling of the underlying stock index changes by the implied volatility reduces the non-Gaussian nature of the probability distribution of these price changes. The nonlinear time correlations are reduced as well.

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Abstract
We present a framework that allows for a systematic assessment of risk given a specific model and belief on the market. Within this framework the time evolution of risk is modeled in a twofold way. On the one hand, risk is modeled by the time discrete and nonlinear garch(1,1) process, which allows for a (time-)local understanding of its level, together with a short term forecast. On the other hand, via a diffusion approximation, the time evolution of the probability density of risk is modeled by a Fokker-Planck equation. Then, as a final step, using Bayes theorem, beliefs are conditioned on the stationary probability density function as obtained from the Fokker-Planck equation. We believe this to be a highly rigorous framework to integrate subjective judgments of future market behavior and underlying models. In order to demonstrate the approach, we apply it to risk assessment of empirical interest rate scenario methodologies, i.e. the application of Principal Component Analysis to the the dynamics of bonds.

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Abstract
Factor based interest rate models are widely used for risk managing purposes, for option pricing and for identifying and capturing yield curve anomalies. The movements of a term structure of interest rates are commonly assumed to be driven by a small number of orthogonal factors such as SHIFT, TWIST and BUTTERFLY (BOW). These factors are usually obtained by a Principal Component Analysis (PCA) of historical bond prices (interest rates). Although PCA diagonalizes the covariance matrix of either the interest rates or the interest rate changes, it does not use both covariance matrices simultaneously. Furthermore higher linear and nonlinear correlations are neglected. These correlations as well as the mean reverting properties of the interest rates become crucial, if one is interested in a longer time horizon (infrequent hedging or trading). We will show that Independent Component Analysis (ICA) is a more appropriate tool than PCA, since ICA uses the covariance matrix of the interest rates as well as the covariance matrix of the interest rate changes simultaneously. Additionally higher linear and nonlinear correlations may be easily incorporated. The resulting factors are uncorrelated for various time delays, approximately independent but nonorthogonal. This is in contrast to the factors obtained from the PCA, which are orthogonal and uncorrelated for identical times only. Although factors from the ICA are nonorthogonal, it is sufficient to consider only a few factors in order to explain most of the variation in the original data. Finally we will present examples that ICA based hedges outperforms PCA based hedges specifically if the portfolio is sensitive to structural changes of the yield curve.

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Abstract
The structure of high-frequency time series of financial data taking the DAX future as an example is investigated with respect to the existence of local order on a time horizon of a few minutes. We will show that there might be special local situations where local order exists and where the predictability is considerably higher than average. We discretize the time series and investigate the continuation frequency of definite words of length n first. Besides higher order Shannon entropies and conditional entropies (dynamic entropies) which yield mean values of the uncertainty/predictability, we study the local values of the uncertainty/predictability and the distribution of these quantities. The local order significance is treated by means of surrogate sequences with identical short memory as the original data.

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Abstract
This work is devoted to applications of the Ehrenfest urn model to innovation and search processes. In the first part we discuss systems of two urns serving as models of innovation processes. The elementary act of innovation is considered as a transition from old (technologies, way of production, behavior, decisions) to new. The survival probability of the new under the influence of stochastic effects is discussed. In the second part we study systems of s>>1 urns serving as models for optimal solution searching in optimization problems. The problem is to find the minimum on a large set of real numbers U_i using a total of N seekers (N~2-100) simultaneously. The potential U_i is defined on the integer set i=1,…,s, where s is extremely large. In particular, we consider the frustrated periodic strings and the merit problem. The known equations for thermodynamic search processes and for simple models of biological evolution are unified by defining a two-parameter family of equations which embeds both cases. The search parameters are controlled by means of seeker ensemble dispersion.

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