An introduction to Markov chains for interested high school students

Markov Chains are introduced by only assuming some knowledge of the notion of probability. The modelling of a situation in a context of biology gives the opportunity to students to approach various concepts of probability theory themselves. --

Recognizing mathematical talent : an approach using discriminant analysis

The aim of this paper is to extract mathematically talented students out of a group of arbitrary high school students. We do this by applying a stepwise discriminant analysis modified for ordinal data to the results of German high school students at the international mathematics competition ?Kangaroo of Mathematics?. It turns out that three of the thirty given problems are enough to discriminate between laureates, which we assume to be mathematically talented, and non-laureates. The three chosen problems are from different mathematical fields. --Discriminant analysis for ordinal data,discrete kernel estimation,testing mathematical talent,multiple choice competition

Pricing of options under different volatility models

In this paper we compare the price of an option with one year maturity of the German stock index DAX for several volatility models including long memory and leverage effects. We compute the price by applying a present value scheme as well as the Black-Scholes and Hull-White formulas which includes stochastic volatility. We find that long memory as well as asymmetry affect the Black-Scholes price significantly whereas the Hull-White price is hardly affected by long memory but still by including asymmetries. --Option Pricing,GARCH,Long Memory,Leverage Effect

S-estimators in the linear regression model with long-memory error terms

We investigate the behaviour of S - estimators in the linear regression model, when the error terms are long-memory Gaussian processes. It turns out that under mild regularity conditions S - estimators are still normally distributed with a similar variance - covariance structure as in the i.i.d - case. This assertion holds for the parameter estimates as well as for the scale estimates. Also the rate of convergence is for S - estimators the same as for the least squares estimator and for the BLUE

Tests of Bias in Log-Periodogram Regression

This paper proposes simple Hausman-type tests to check for bias in the log-periodogram regression of a time series believed to be long memory. The statistics are asymptotically standard normal on the null hypothesis that no bias is present, and the tests are consistent.Long memory, log periodogram regression, Hausman test.

Log-Periodogram estimation of the memory parameter of a long-memory process under trend

We show that small trends do not influence log-periodogram based estimators for the memory parameter in a stationary invertible long-memory process. In the case of slowly decaying trends which are easily confused with long-range dependence we show by Monte Carlo methods that the tapered periodogram is quite robust against these trends and thus provides a good alternative to standard log-periodogram methodology

S - estimators in the linear regression model with long - memory error terms

We investigate the behaviour of S - estimators in the linear regression model, when the error terms are long - memory Gaussian processes. It turns out that under mild regularity conditions S - estimators are still normally distributed with a similar variance - covariance structure as in the i.i.d. case. This assertion holds for the parameter estimates as well as for the scale estimates. Also the rate of convergence is for S - estimators the same as for the least squares estimator and for the BLUE

Spatial autoregressive fractionally integrated moving average model

In this paper, we introduce the concept of fractional integration for spatial autoregressive models. We show that the range of the dependence can be spatially extended or diminished by introducing a further fractional integration parameter to spatial autoregressive moving average models (SARMA). This new model is called the spatial autoregressive fractionally integrated moving average model, briefly sp-ARFIMA. We show the relation to time-series ARFIMA models and also to (higher-order) spatial autoregressive models. Moreover, an estimation procedure based on the maximum-likelihood principle is introduced and analysed in a series of simulation studies. Eventually, the use of the model is illustrated by an empirical example of atmospheric fine particles, so-called aerosol optical thickness, which is important in weather, climate and environmental science

Distinguishing between long-range dependence and deterministic trends

We provide a method for distinguishing long-range dependence from deterministic trends such as structural breaks. The method is based on the comparison of standard log-periodogram regression estimation of the memory parameter with its tapered counterpart. The difference of these estimators provides the desired test. Its asymptotic distribution depends on the true memory parameter under the null, and is therefore estimated by bootstrapping. The test is applied to inflation rates of three industrialized countries. --Long memory,trends,log-periodogram regression,inflation rates

The cost for the default of a loan : Linking theory and practice

When calculating the cost of entering into a credit transaction the predominant stochastic component is the expected loss. Often in the credit business the one-year probability of default of the liable counterpart is the only reliable parameter. We use this probability to calculating the exact expected loss of trades with multiple cash ows. Assuming a constant hazard rate for the default time of the liable counterpart we show that the methodology used in practice is a linear Taylor approximation of our exact calculus. In a second stage we can generalize the calculation to arbitrary hazard rates for which we prove statistical evidence and develop an estimate from historical data. --