外汇市场：收益率分布、多重分形、反常多重分形与Epps效应
The foreign exchange market: return distributions, multifractality, anomalous multifractality and Epps effect
摘要：
我们提出了一个系统的研究，从外汇市场的高频回报的各种统计特征。本研究基于欧元-英镑-美元和英镑-瑞士法郎-日元这两个三角形的六种汇率。结果表明，汇率收益率波动的所有考虑对很好地描述的非广泛统计的q-高斯。对于不同的汇率，非扩展性q参数值存在一些小的定量变化，这可能与给定汇率在世界货币贸易中的重要性有关。然而，时间相关性组织了一系列收益，使得它们为所有具有不同对称程度的奇异谱f（alpha）的汇率发展了多重分形特征。最对称的频谱是针对英镑/美元的。我们还形成了三角形的剩余收益的时间序列，并发现其波动的分布发展不成比例地较重的尾巴相比，小的波动，排除了描述的q-高斯。这些剩余收益的多重分形特征揭示了负奇异指数甚至负奇异谱等异常性质。这种反常的多重分形措施迄今已被认为是在文献中的扩散限制聚集和湍流。我们发现，市场在短时间尺度上的无效率导致发生的Epps效应在更长的时间尺度上。虽然货币市场的流动性比股票市场高得多，交易频率也高得多，但相关性的建立需要几个小时--这与在股票市场中观察到的时间相差不大。这可能表明，交易的非同步性不是观察到的效果的唯一来源。
英文摘要：
We present a systematic study of various statistical characteristics of high-frequency returns from the foreign exchange market. This study is based on six exchange rates forming two triangles: EUR-GBP-USD and GBP-CHF-JPY. It is shown that the exchange rate return fluctuations for all the pairs considered are well described by the nonextensive statistics in terms of q-Gaussians. There exist some small quantitative variations in the nonextensivity q-parameter values for different exchange rates and this can be related to the importance of a given exchange rate in the world's currency trade. Temporal correlations organize the series of returns such that they develop the multifractal characteristics for all the exchange rates with a varying degree of symmetry of the singularity spectrum f(alpha) however. The most symmetric spectrum is identified for the GBP/USD. We also form time series of triangular residual returns and find that the distributions of their fluctuations develop disproportionately heavier tails as compared to small fluctuations which excludes description in terms of q-Gaussians. The multifractal characteristics for these residual returns reveal such anomalous properties like negative singularity exponents and even negative singularity spectra. Such anomalous multifractal measures have so far been considered in the literature in connection with the diffusion limited aggregation and with turbulence. We find that market inefficiency on short time scales leads to the occurrence of the Epps effect on much longer time scales. Although the currency market is much more liquid than the stock markets and it has much larger transaction frequency, the building-up of correlations takes up to several hours - time that does not differ much from what is observed in the stock markets. This may suggest that non-synchronicity of transactions is not the unique source of the observed effect.
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