Fractal Dimensions For Fractal Structures And Their Applications To Financial Markets
- ISBN/EAN
- 9788854866188
- Editore
- Aracne
- Formato
- Brossura
- Anno
- 2013
- Pagine
- 268
Disponibile
16,00 €
This book provides a new approach to fractal dimension theory from the point of view of Asymmetric Topology through fractal structures. Fractal dimension is the main invariant of each fractal set which gives information about the irregularities that it presents when being examined with enough level of detail. In this work, the authors develop new models to calculate the fractal dimension for any subspace with respect to a fractal structure which generalize the classical fractal dimension definitions, namely, both the Hausdorff dimension and the box-counting dimension. They also include some specific results for self-similar sets. In addition, the new definitions of fractal dimension can be calculated in empirical applications unlike it happens with the Hausdorff dimension, which may result hard or even impossible to calculate. The second part of the book contains an interesting application of fractal dimension for fractal structures to financial markets. The authors introduce some tools and prove some results which connect the fractal dimension with the Hurst exponent which has been classically used to look for long-range dependence in financial time series. These theoretical results allow to provide new algorithms especially accurate to estimate the self-similarity exponent of a wide range of processes which includes fractional Brownian motions and Lévy stable motions as particular cases. Moreover, some specific analysis have also been carried out for real stocks and international indexes.
Maggiori Informazioni
| Autore | Fernandez Marti'nez Manuel; Sanchez Granero Miguel Angel; Trinidad Segovia Juan Evangelista |
|---|---|
| Editore | Aracne |
| Anno | 2013 |
| Tipologia | Libro |
| Num. Collana | 0 |
| Lingua | Inglese |
| Indice | I Fractal Dimensions for Fractal Structures 5 Introduction of Part I 7 Main goals of Part I 13 1 Preliminaries of Part I 17 1.1 Quasi-pseudometrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2 Fractal structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.3 Iterated function systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.4 The box-counting dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.5 The Hausdorff dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2 Fractal dimension for fractal structures 29 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2 Fractal dimension I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3 Fractal dimension II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3 Fractal dimension for fractal structures: applications to the domain of words 55 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 1 5 15 17 23 27 29 30 31 34 35 41 43 44 51 69 713.2 Applications of fractal dimension in non-Euclidean contexts: the domain of words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2.1 Quasi-metrics, fractal structures and the domain of words . . . . . . . 57 3.2.2 The fractal dimension of a language generated by a regular expression 58 3.2.3 The fractal dimension as a tool to study the efficiency of an encoding system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2.4 An empirical application to search trees . . . . . . . . . . . . . . . . . 63 4 Fractal dimension for fractal structures: a Hausdorff approach 69 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2 A new Hausdorff’s model to calculate the fractal dimension . . . . . . . . . . 71 4.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2.2 Key concept and results . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.3 The fractal dimension of a curve . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.3.1 An application of the fractal dimension to study the complexity of space-filling curves ............................. 95 4.3.2 The fractal dimension as a tool to study the complexity of a curve which fills a whole self-similar set . . . . . . . . . . . . . . . . . . . . . . . . 99 4.3.3 The fractal dimension of the devil’s staircase .............. 100 5 Fractal dimension for fractal structures: a Hausdorff approach revisited 103 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.2 Motivation of the new fractal dimension models . . . . . . . . . . . . . . . . . 104 5.3 Key concepts and first results . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.4 Analytical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.5 Generalizing the Hausdorff dimension . . . . . . . . . . . . . . . . . . . . . . 125 5.6 Fractal dimension for self-similar sets . . . . . . . . . . . . . . . . . . . . . . . 132 6 Contents 72 72 73 76 78 83 83 86 86 90 106 110 114 115 119 121 121 128 139 142 1493.2 Applications of fractal dimension in non-Euclidean contexts: the domain of words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.2.1 Quasi-metrics, fractal structures and the domain of words . . . . . . . 57 3.2.2 The fractal dimension of a language generated by a regular expression 58 3.2.3 The fractal dimension as a tool to study the efficiency of an encoding system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2.4 An empirical application to search trees . . . . . . . . . . . . . . . . . 63 4 Fractal dimension for fractal structures: a Hausdorff approach 69 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2 A new Hausdorff’s model to calculate the fractal dimension . . . . . . . . . . 71 4.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2.2 Key concept and results . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.3 The fractal dimension of a curve . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.3.1 An application of the fractal dimension to study the complexity of space-filling curves ............................. 95 4.3.2 The fractal dimension as a tool to study the complexity of a curve which fills a whole self-similar set . . . . . . . . . . . . . . . . . . . . . . . . 99 4.3.3 The fractal dimension of the devil’s staircase .............. 100 5 Fractal dimension for fractal structures: a Hausdorff approach revisited 103 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.2 Motivation of the new fractal dimension models . . . . . . . . . . . . . . . . . 104 5.3 Key concepts and first results . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.4 Analytical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.5 Generalizing the Hausdorff dimension . . . . . . . . . . . . . . . . . . . . . . 125 5.6 Fractal dimension for self-similar sets . . . . . . . . . . . . . . . . . . . . . . . 132 6 Conclusions of Part I 135 7 Bibliography of Part I 139 II Applications to Financial Markets 147 Introduction of Part II 149 Main goals of Part II 153 8 Preliminaries of Part II 155 8.1 Self-similar processes ................................ 156 8.1.1 Random functions and their increments. Self-affinity properties .... 156 8.1.2 Fractional Brownian motions . . . . . . . . . . . . . . . . . . . . . . . 158 8.1.3 Stable processes and fractional L´evy stable motions . . . . . . . . . . 161 9 Hurst exponent as a tool to analyze the efficiency of capital markets 165 9.1 The Efficient Market Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . 166 9.2 Three efficiency hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 9.3 Memory processes ................................. 169 9.4 Algorithms to estimate the self-similarity exponent of a random process . . . 173 9.4.1 (Hurst) Rescaled range analysis ...................... 174 9.4.2 Detrended fluctuation analysis . . . . . . . . . . . . . . . . . . . . . . 174 9.4.3 Absolute value method . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 9.4.4 Geometric method-based procedures . . . . . . . . . . . . . . . . . . . 176 10 Geometric method-based procedures to calculate the Hurst exponent 179 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Contents 7 153 157 165 167 173 177 179 179 181 184 187 189 190 192 196 197 197 198 199 201 2034 10.2 Justifying GM procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 10.2.1 GM1 algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 10.2.2 GM2 algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 10.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 11 GM procedures to measure the self-similarity exponent of LSMs 187 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 11.2 Testing the accuracy of the estimators . . . . . . . . . . . . . . . . . . . . . . 191 11.3 Empirical application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 11.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 12 FD methods to calculate the Hurst exponent of financial time series 205 12.1 Hurst exponent estimation using fractal dimension . . . . . . . . . . . . . . . 206 12.1.1 A first approach to the fractal dimension: FD1 method . . . . . . . . 207 12.1.2 Two new methods based on statistical moments: FD2 & FD3 . . . . . 210 12.2 Testing the accuracy of FD methods . . . . . . . . . . . . . . . . . . . . . . . 217 12.3 Empirical application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 12.3.1 Exploring long-memory in stock market indexes . . . . . . . . . . . . . 221 12.3.2 The self-similarity exponent for individual stocks . . . . . . . . . . . . 223 12.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 13 Conclusions of Part II 229 14 Bibliography of Part II 233 8 Contents 253 257 |
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| Stato editoriale | In Commercio |
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