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Statistical Inference in Multifractal Random Walk Models for Financial Time Series

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Cristina Sattarhoff

The dynamics of financial returns varies with the return period, from high-frequency data to daily, quarterly or annual data. Multifractal Random Walk models can capture the statistical relation between returns and return periods, thus facilitating a more accurate representation of real price changes. This book provides a generalized method of moments estimation technique for the model parameters with enhanced performance in finite samples, and a novel testing procedure for multifractality. The resource-efficient computer-based manipulation of large datasets is a typical challenge in finance. In this connection, this book also proposes a new algorithm for the computation of heteroscedasticity and autocorrelation consistent (HAC) covariance matrix estimators that can cope with large datasets.

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2 The concept of multifractal volatility 23

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Chapter 2 The concept of multifractal volatility within the framework of financial markets theory 2.1 The Efficient Markets Hypothesis The Efficient Markets Hypothesis (EMH) introduced in 1965 by Fama provided a basis for the development of financial markets theory. According to the EMH: A market in which prices always “fully reflect” available information is called “efficient”. (Fama 1970, p. 383) The price of a security traded on an efficient financial market incorpor- ates at all times all the relevant information relating to it, with no piece of information left disregarded on the market. Hence it is impossible for an in- vestor to be better-informed than others and to make abnormal profits due to an information advantage. Price changes are caused solely by the arrival of news, which is by definition unanticipated. Investors can guess future prices and thereby make abnormal profits. Prices, however, cannot be systematically forecast. On average, investors earn fair payments for the risk level of their securities, with the expected value of any abnormal profits beyond this fair payment being zero (Cuthbertson 1997, Chapter 5). At first glance, the theory of efficient capital markets might discourage financial analysts and fund managers from pursuing their professions, since any attempt to predict future prices and to construct portfolios which outperform the market is a futile exercise. Currently, the prediction of price levels is an uninteresting topic within quantitative analysis of financial markets, which instead focuses on the modelling and prediction of price fluctuations (volatility) (Hanzon 2004, pp. 217-218). 24...

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