Coupled continuous time random walks in finance

被引:162
|
作者
Meerschaert, Mark M. [1 ]
Scalas, Enrico
机构
[1] Univ Otago, Dept Math & Stat, Dunedin 9001, New Zealand
[2] Univ Piemonte Orientale, Dipartimento Sci & Tecnol Avanzate, Alessandria, Italy
关键词
anomalous diffusion; continuous time random walks; heavy tails; fractional calculus;
D O I
10.1016/j.physa.2006.04.034
中图分类号
O4 [物理学];
学科分类号
0702 ;
摘要
Continuous time random walks (CTRWs) are used in physics to model anomalous diffusion, by incorporating a random waiting time between particle jumps. In finance, the particle jumps are log-returns and the waiting times measure delay between transactions. These two random variables (log-return and waiting time) are typically not independent. For these coupled CTRW models, we can now compute the limiting stochastic process (just like Brownian motion is the limit of a simple random walk), even in the case of heavy-tailed (power-law) price jumps and/or waiting times. The probability density functions for this limit process solve fractional partial differential equations. In some cases, these equations can be explicitly solved to yield descriptions of long-term price changes, based on a high-resolution model of individual trades that includes the statistical dependence between waiting times and the subsequent log-returns. In the heavy-tailed case, this involves operator stable space-time random vectors that generalize the familiar stable models. In this paper, we will review the fundamental theory and present two applications with tick-by-tick stock and futures data. (c) 2006 Elsevier B.V. All rights reserved.
引用
收藏
页码:114 / 118
页数:5
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