Universität Rostock, 2005
Abstract: We concentrate on the stochastic description of processes in Physics and in the stock market. We analyze theoretical models, carry out numeric integration and compare results with empirical data. Therefore, we use high frequency data of the German stock market, delivered by the capital market database in Karlsruhe. We calculate the joint probability density distribution of velocity and position following the well known Ornstein-Uhlenbeck process. We depict the agreement of our solution with former results by Chandrasekhar. Furthermore, we underline the common characteristics of modelled processes in Physics and finance. We summarize the known stylized facts of the stock market data and describe selected properties with the help of our data. We illustrate the stock price dynamics and derive, why logarithmic stock price returns are always used. Despite this, we show quantitativly, that we can not reject the existence of the Markov property of the returns. For the description of the stock price dynamics we focus on the models with stochastic volatility. We introduce two well known models (Heston and Hull-White) and derive a solution of the returns for short time lags. We compare this solution with the empirical data and analyze the differences. That is why, we introduce new models, with the help of combination of known ones or transformation of models from Physics to finance. We emphasize the better agreement of some of them with the empirical stock market data.
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